Deep learning driven inverse design of multi-material structures with tailored anisotropic mechanical responses

Deep learning driven inverse design of multi-material structures with tailored anisotropic mechanical responses | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Deep learning driven inverse design of multi-material structures with tailored anisotropic mechanical responses Jigang Huang, Zhengda Chen, XiangJun Zha, Haocheng Yang, Lin Chen, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8760116/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted You are reading this latest preprint version Abstract Multi-material structures are complex architectures formed by the selective distribution of diverse materials at the microstructural level, enabling the anisotropic designs with customized mechanical responses. Although these structures offer significant advantages via their immense design freedom, the vast design space and the non-linear coupling of multi-phase mechanical properties pose formidable challenges for inverse design of such structures. This study develops a deep learning-driven framework for the rapid design of voxelized multi-material structures. By integrating a high-fidelity forward surrogate model with an inverse neural network, our method can generate spatial material distributions that satisfy targeted stress-strain curves within seconds. Validated through multi-material 3D printing and quasi-static compression tests, the framework achieves an accuracy of over 95%. Notably, we introduce a joint loss function coupled with a Hard Constraint Check (HCC) strategy, allowing the model to selectively bias designs toward specific soft-to-hard material ratios without compromising mechanical performance. Furthermore, a weighted multi-objective optimization scheme is implemented to incorporate priorities when managing anisotropic responses. Experimental results demonstrate the immense potential of deep learning in the spatial distribution design of multi-material structures, paving the way for advanced material applications in fields such as medicine, robotics, aerospace, civil engineering, and vehicle engineering. Physical sciences/Engineering/Mechanical engineering Physical sciences/Materials science/Structural materials/Composites multi-material 3D printing inverse design anisotropic responses deep learning Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 1. Introduction Anisotropic structures, characterized by direction-dependent mechanical properties, serve as the foundation for the development of advanced functional devices [1-6] . Structures exhibiting significant anisotropic features hold broad application prospects in fields such as acoustics [7-9] , biomedicine [10-17] , and aerospace [18-22] . However, designing components that exhibit precise, pre-defined stress-strain behaviors in different directions presents a significant complexity. This challenge is compounded not only by the non-linear constitutive relationships of the materials involved but also by the non-intuitive mapping between microstructural geometry and macroscopic performance. Multi-material architectures offer a powerful pathway to achieve such complex anisotropy. People can create digital materials [23-26] or hierarchical structures [27-29] by strategically distributing materials with distinct moduli. These voxel-based designs provide immense freedom to tune local stiffness and global deformation mechanisms. However, this flexibility comes at a cost: as the resolution of the material distribution increases, the combinatorial design space explodes exponentially [30, 31] . Current design methodologies predominantly rely on forward trial-and-error approaches or physics-based topology optimization (TO). Empirical designs, dependent on human intuition, are time-consuming and often fail to exploit the full potential of the design space. While topology optimization has proven effective for maximizing stiffness or compliance, it often incurs prohibitive computational costs [32] . Furthermore, as the demand for "customized" mechanical responses grows—requiring the matching of entire stress-strain curves rather than single scalar values like modulus—traditional methods struggle to meet the requirements for both accuracy and design efficiency. In recent years, Machine Learning (ML) has emerged as a transformative paradigm for solving inverse problems in mechanics and materials science [33-39] . Zheng et al. [40] introduced a generative ML pipeline capable of creating mechanical metamaterials that replicate nearly all possible uniaxial compressive stress-strain curves, achieving almost 90% fidelity between target and experimental results. Similarly, Zong et al. [41] proposed a machine-learning-powered approach for multi-target inverse design based on graded triply periodic minimal surface (TPMS) architectures, achieving rapid and accurate results within seconds for complex applications such as customized soles. Moving beyond static metamaterials, Matthews et al. [42] developed a gradient-based optimization method capable of designing functional robots from scratch in under 30 seconds, demonstrating that complex physical interdependencies can be co-optimized far more efficiently than through traditional evolutionary algorithms. Furthermore, the capability to design for precise spatial responses has been explored by Ge et al. [27] , who pioneered the inverse design of strain fields in 2D multi-material structures using recurrent neural networks (RNNs) and evolutionary algorithms. Although these works have leveraged machine learning to significantly enhance the accuracy and efficiency of inverse design, they still lack sufficient degrees of freedom regarding spatial distribution. In this work, we present a deep learning-driven inverse design framework capable of instantly generating multi-material microstructures based on user-defined anisotropic stress-strain curves. Unlike iterative optimization methods, our data-driven model learns the intrinsic structure-property relationships, reducing the design cycle from hours to mere seconds while achieving a curve-fitting accuracy of over 95%. We address the "non-uniqueness" or "overlap" problem inherent in multi-material design, where different material distributions might yield similar mechanical responses. We propose customized joint loss function that integrates a Mean Squared Error (MSE) term for performance matching with a material ratio constraint term. This approach, combined with "hard constraint check," enables the explicit control of the volume fraction of soft and hard materials without compromising the target mechanical performance. By adjusting the loss weights across different loading directions, our model enables the flexible customization of anisotropic behaviors, paving the way for the efficient design of high-performance metamaterials. 2. Experimental section 2.1. Material Selection and Additive Manufacturing We selected commercial polylactic acid (PLA Basic) and thermoplastic polyurethane (TPU for AMS) produced by Bambu Lab (Shenzhen, China) as the constituent materials. PLA Basic serves as the rigid phase, providing structural support and high stiffness, with a density of 1.24g/cm 3 and a Young's modulus of 2750MPa. Conversely, TPU for AMS serves as the soft phase to induce large deformations and energy absorption. The significant contrast in mechanical properties between these two materials enables a vast design space for tuning the global mechanical behavior of the composite structures. The physical specimens were fabricated using the Bambu Lab X1 Carbon Fused Deposition Modeling (FDM) 3D printer. This printer features a build volume of 256 mm × 256 mm × 256 mm and is equipped with an Automatic Material System (AMS), which allows for the seamless printing of up to four different materials within a single layer. To ensure optimal interlayer adhesion and interface bonding between the PLA and TPU phases, the nozzle temperature was set to 220℃ for PLA and 230℃ for TPU, with a heated bed temperature of 50℃. The layer thickness was set to 0.2mm with 100% infill density to replicate the solid voxel structure used in the simulations. More details about the materials used in this work are provided in Table S1. 2.2. Finite Element Simulation and Dataset Generation To train the deep learning model, a large-scale dataset linking material distributions to their mechanical responses was generated via Finite Element Analysis (FEA). We utilized a custom Python script to randomly generate 8,000 unique material distribution combinations. Each sample was modeled as a cubic structure with overall dimensions of 36 × 36 × 36 mm 3 . The structure was discretized into a 3 × 3 × 3 voxel array, consisting of 27 individual unit cells, where each unit cell (voxel) had dimensions of 12 × 12 × 12 mm 3 . The numerical simulations were performed using the commercial FEA software Abaqus (Dassault Systems SIMULIA Established Products 2022, Johnston, United States). In the simulation environment, each voxel was assigned the material properties of either PLA or TPU based on the generated binary matrix (0 for hard material, 1 for soft material). A uniaxial compression test was simulated for each case. The boundary conditions were defined as follows: the bottom surface of the cubic structure was fully fixed (Ux = Uy = Uz = 0), while a uniform vertical displacement was applied to the top surface to achieve a global compressive strain of 20% (equivalent to a displacement of 7.2 mm). An implicit solver was employed to ensure convergence. The resulting reaction forces and displacements were extracted to compute the stress-strain curves, which served as the ground truth labels for the neural network training. A more detailed description about this part is provided in Note S1. 2.3. Deep Learning Framework Implementation The deep learning framework, including the forward prediction network and the inverse design model, was implemented using the Tensorflow library. All model training and computational tasks were conducted on a desktop workstation. The hardware configuration consisted of an Intel Core i5-10500 CPU, 16 GB of RAM, and an NVIDIA GeForce RTX 3060 Ti graphics processing unit (GPU) to accelerate the tensor operations. The dataset of 8,000 samples was randomly split into a training set (70%), a validation set (30%). The input to the network was the 1 × 27 binary matrix representing the material distribution, and the output was the corresponding stress-strain sequence. The model was optimized using the Adam optimizer. To prevent overfitting, we monitored the validation loss and employed early stopping techniques. 2.4 Experiment methods Compression tests were performed at room temperature with a universal testing machine (Instron 5967) and the testing speed was set to be 10 mm/min. We adopted a quasi-static testing method the strain rate was set to be 0.01/s. 3. Results and discussion 3.1 The deep learning-driven design framework. The proposed framework integrates feature extraction, deep learning-driven inverse prediction, and multi-material additive manufacturing into a closed-loop system, enabling the precise translation of complex stress-strain targets into physical realities (Fig. 1 ). The design process initiates with the definition of target mechanical behaviors. The mechanical performance of the anisotropic structure is defined by distinct quasi-static compression stress-strain curves in orthogonal directions (X, Y, and Z-axis. Figure 1 a). To process these complex non-linear responses, we employ a feature extraction mechanism that condenses the continuous stress-strain data into discrete characteristic points (Fig. 1 b). This feature extraction is beneficial for ensuring the efficiency of subsequent neural network processing while retaining key mechanical characteristics, such as modulus and yield strength. These extracted features are then fed into our pre-trained deep learning driven inverse design model (Fig. 1 c). This model identifies the intrinsic mapping between macroscopic mechanical responses and microscopic material configurations, with its output being a digital material representation in the form of a 3 \(\:\times\:\:\) 3 \(\:\times\:\) 3 spatial vector matrix. In this binary matrix, "0" designates the hard phase (PLA), which provides structural rigidity and load-bearing capacity, while "1" designates the soft phase (TPU), which facilitates large deformation and energy absorption (Fig. 1 d). This discrete, voxel-based representation allows for the generation of complex, heterogeneous architectures that would be counter-intuitive to design using traditional trial-and-error methods. Fig. S2 provide more details about the base unit used in this work. The final optimized design is fabricated using a multi-material 3D printer. Upon testing, the resulting multi-material structure will demonstrate the ability to accurately replicate prescribed stress-strain responses under quasi-static compressive loading. The proposed deep learning framework is bifurcated into two primary modules: the forward prediction model and the inverse design model. The forward model serves as a high-fidelity surrogate framework, effectively bypassing the computationally prohibitive Finite Element Analysis (FEA) simulations. It takes the voxel-based material distribution matrix as input, executes sequential processing across three hidden layers, and yields discrete stress characteristic values which are further reconstructed into complete macroscopic stress-strain curves. To train this model, a comprehensive dataset comprising 8,000 unique samples was generated via automated FEA scripts (Fig. S3). This dataset covers a diverse spectrum of material volume fractions to ensure the model captures the full complexity of the design space (Fig. S4). As illustrated in Fig. 2 c, the loss values for both training and validation sets converge to a remarkably low level (10 − 2 ) as the number of training epochs increases. Notably, the close alignment between training and validation loss curves indicates the absence of overfitting, confirming the model’s robust generalization capabilities across unseen material configurations. To rigorously validate the predictive accuracy, the model was tested against 2,000 novel datasets. As shown in Fig. 2 d, the predicted stress values exhibit high consistency with the FEA ground truth, with data points tightly clustered around the y = x identity line. This correlation is quantitatively confirmed by a coefficient of determination (R 2 ) of 0.9874, underscoring the model's precision and reliability as a surrogate for physical simulations. Furthermore, we systematically investigated the influence of various hyperparameters on model efficacy (Fig. S3 and Table S2), including the training set size, batch size, learning rate and hidden layer dimensions. By fine-tuning these parameters, we optimized the forward model to achieve a balance between computational efficiency and predictive accuracy, ensuring stable performance across diverse structural scenarios. Inverse Model Training is trained by utilizing stress characteristic values as inputs to generate corresponding material distribution matrices. During the training phase, the pre-trained (frozen) forward model is employed as a rapid evaluator to generate predicted stress responses from the inverse model’s output. The error between these predicted responses and the ground truth labels is then backpropagated to update the inverse network’s weights. Similar to the forward module, the inverse model exhibits excellent convergence performance, with the convergence error reaching 10 − 2 (Fig. 2 e), and achieves a high R 2 value of 0.9707 on 2,000 new test cases (Fig. 2 f). The influences of various hyperparameters on inverse model efficacy are demonstrated in Fig. S4 and Table S3. In the deployment of the complete framework, the system takes user-defined stress characteristic points as the target inputs. The inverse model generates potential design candidates, which are subsequently fed into the forward model for the estimation of their respective mechanical behaviors. The optimal design is identified by evaluating the discrepancy between these estimated curves and the predefined target. Since the final selection is based on a direct comparison of the holistic curve features, the proposed framework ensures the uniqueness of the derived solution. This strategy effectively circumvents the "one-to-many" mapping challenge inherent in inverse problems, where multiple distinct microstructures may yield nearly identical macroscopic mechanical responses [ 40 ] . 3.2 Inverse Design with Material Ratio Constraints The mechanical performance of multi-material structures bears no simple linear relationship to the proportion of constituent materials; instead, prominent "overlap zones" emerge within the design space (Fig. 3 a). Taking the proposed multi-material architecture as an example, a higher volume fraction of soft material generally leads to reduced structural stiffness, yet certain configurations with high soft-material ratios maintain remarkably high mechanical performance due to the formation of localized support zones through strategic spatial distribution. Conversely, designs with lower soft-material ratios exhibit macroscopic flexibility because their deformation zones—comprised of soft phases—absorb the majority of the strain during compression (Fig. 3 b). Consequently, inverse design tasks targeting stress-strain curves within these overlap zones face a non-uniqueness challenge, where the solution may fall into either a high or low soft-material proportion region. To explicitly regulate the material usage in the final design, we propose a joint loss function that simultaneously considers curve fitting accuracy and material volume fractions. This function incorporates a material ratio constraint term into the original objective, with a weighting coefficient n used to modulate the priority of the ratio constraint (Fig. 3 c): $$\:L={nloss}_{constraint}+MSE$$ 1 By training the model with this joint loss, the framework can generate spatial distributions that satisfy a predefined material ratio range while maintaining high fidelity to the target stress-strain curve. In scenarios where multiple distinct microstructures could potentially yield the same target response, the model is biased toward designs that comply with the specified material constraints (Fig. 3 d). We further investigated the influence of the weighting coefficient n on the training efficacy. Figure 3 e illustrates the evolution of the coefficient of determination (R 2 ) and the ratio accuracy—defined as the proportion of samples satisfying the predefined material limits—as n increases. As n rises, the ratio accuracy improves significantly from less than 30% to 100%, demonstrating that the weighting coefficient effectively drives the model toward compliant designs. However, we observed a trade-off: the R 2 values decrease from over 90% to below 60% at excessive n values, indicating that the model becomes overly focused on material proportions at the expense of performance matching. To resolve this conflict, we introduced a Hard Constraint Check (HCC) strategy. The core of this strategy involves a post-processing step before the final design output: for results that do not satisfy the material ratio requirements, the m voxels with the lowest probability of belonging to their assigned phase are identified as "replacement points." These points are then toggled (e.g., from soft to hard or vice-versa) until the material ratio strictly meets the constraint. This mechanism ensures absolute adherence to manufacturing requirements while preventing the performance degradation associated with overly high n values. Figure 3 f demonstrates an application of this approach within the 0–50% and 50–100% overlap zones, where the inverse model successfully identified two distinct, highly accurate optimal designs (NRMSE(A) = 0.03150, NRMSE(B) = 0.04655) for the same target curve across different material ratio regimes. 3.3 Anisotropic Inverse Design with Weighted Multi-Objective Optimization To comprehensively address the anisotropy of multi-material structures and enable the inverse design of customized mechanical responses over all spatial directions, we significantly expanded the training dataset for the proposed model. Beyond the original Z-direction compression simulations, we conducted supplementary simulations for compression along the X and Y axes, resulting in a compiled dataset of three sets of stress characteristic arrays (totaling 63 feature points). The anisotropy of the structure is thus explicitly characterized by the stress-strain features across these three orthogonal directions. We retrained the deep learning model using this augmented dataset comprising the new multi-directional stress features and material distributions. As illustrated in Fig. 4 a, the network architecture was modified to handle directional dependencies: while the material distribution generated by the inverse model is directly fed into the forward model to predict Z-direction stress features, predictions for the X and Y directions are obtained by passing the material distribution through a Coordinate Transformation module before entering the shared forward model. Furthermore, to guide the optimization process effectively, we introduced a weighted joint loss function: $$\:Loss={\alpha\:loss}_{X}+{\beta\:loss}_{Y}+{\gamma\:loss}_{Z}$$ 2 where weights α , β , and γ assign priority to the stress features of specific directions. Figure 4 b presents the performance metrics (R 2 and MSE) of the model trained with uniform weights ( α = β = γ ). Compared to the previous inverse design tasks targeting a single direction, the design efficacy of the multi-directional model declined, with R 2 values across all three directions falling below 0.9. Figure 4 e displays the target versus actual stress-strain curves for a representative sample under this uniform weighting scheme. A consistent but mediocre fitting performance is observed across all three directions (X, Y, and Z). This performance degradation is likely attributed to the increased complexity of the optimization landscape, where the simultaneous fitting of excessive feature points (63 points) without prioritization hinders the model's convergence. By significantly increasing the weighting coefficient for the Z-direction ( α : β : γ = 1:1:1000), the trained model exhibited a marked prioritization for satisfying Z-direction compression features. As shown in Fig. 4 c, the R 2 and MSE for the Z-direction improved significantly compared to the X and Y directions during prediction. We observed a trade-off mechanism: while the biased weights successfully guided the model to meet the high-priority Z-target, this came at the cost of increased MSE in the X and Y directions. Figure 4 f visualizes a typical sample predicted by this Z-weighted model, clearly demonstrating a high degree of fidelity between the predicted and target curves in the Z-direction, while deviations remain in the secondary directions. To further refine this control, we adjusted the weights to a hierarchical distribution ( α : β : γ = 1:1000:1000000). The resulting model demonstrated a fitting priority order of Z > Y > X, satisfying stress features in the Z-direction first, followed by Y, and finally X. This characteristic is quantitatively reflected in the MSE distribution shown in Fig. 4 d. Figure 4 g presents a representative sample generated by this gradient-weight model, where the fitting accuracy clearly degrades sequentially from Z to Y to X, aligning perfectly with the imposed weight hierarchy. This strategy of utilizing deep learning to inversely design anisotropic multi-material structures, coupled with a weighted loss function to prioritize mechanical performance in specific directions, represents a novel approach not previously reported in existing literature. Our methodology provides a new paradigm for designing metamaterials with complex, directionally tailored mechanical properties. Quadrotor Unmanned Aerial Vehicles (UAVs) typically feature four exposed propellers, rendering them highly vulnerable to structural damage upon impact with rigid surfaces such as walls. Furthermore, free-fall incidents from significant altitudes may result in catastrophic fuselage damage (Fig. 5 a). To address these challenges, we categorized the collision dynamics of a UAV into two distinct scenarios: horizontal impact and vertical fall. Horizontal impact refers to collisions within the X-Y plane, where the UAV strikes a vertical obstacle (e.g., a wall) with a specific initial velocity, resulting in the rapid dissipation of horizontal kinetic energy. Conversely, a vertical fall represents a collision along the Z-axis, where the drone undergoes free-fall from a specific height, impacting the ground and dissipating potential energy converted into kinetic energy. Both scenarios pose significant risks of irreversible structural failure. To mitigate these risks, we developed a specialized propeller protective cage for quadrotors. This assembly consists of an external supporting frame integrated with the proposed anisotropic multi-material structures (Fig. 5 b). The kinetic energy generated during impact is dissipated through the deformation of multi-material lattices. To quantify the energy absorption requirements, we conducted a dynamic analysis of a commercial UAV under both collision scenarios (Table. S3). The resulting relationships between impact time and the evolution of the drone's kinetic energy are presented in Figs. 5 c-e. To ensure the UAV's kinetic energy is entirely absorbed through the deformation of the protective cage, we denote the initial kinetic energy of the drone as E 0 and the final kinetic energy as E 1 : $$\:{E}_{0}=\frac{1}{2}m{v}^{2}\:;\:\:{E}_{1}=0$$ 3 Where m is the mass of the UAV and v is the velocity of the UAV. The entire energy absorption process is determined by calculating the integral of the stress-strain curve of the protective cage: $$\:{W}_{total}=V{\int\:}_{0}^{{\epsilon\:}_{max}}\delta\:{d}_{\epsilon\:}=\:{E}_{0}-{E}_{1}$$ 4 Where V is the volume of the energy absorption module, ε is the strain, and ε is the stress. The energy absorbed by each individual energy absorption unit within the protective cage is: $$\:{W}_{1}=\frac{{W}_{total}}{n}$$ 5 Where n is the number of units that participate in energy absorption during the impact process and W 1 was used as the basis to design the target stress-strain curve. To maximize energy absorption efficiency, we tailored the target stress-strain characteristics of the metamaterial in each spatial direction to match the specific energy dissipation profiles required by the different collision modes. Given that vertical drop incidents occur with higher frequency than horizontal collisions in practical operation, we prioritized energy absorption in the vertical direction. Consequently, the weighting coefficients for the inverse design model were assigned as α : β : γ = 1:1:1000 to bias the optimization toward the Z-axis performance while maintaining sufficient protection in lateral directions. Utilizing the trained weighted inverse model, we generated an optimized multi-material microstructure. As illustrated in Figs. 5 f-h, the resulting structure exhibits excellent agreement with the target response in the Z-direction compression test, demonstrating high fidelity in the primary load-bearing direction. Simultaneously, the structure satisfies the energy absorption criteria for the X and Y directions, balancing anisotropy with functional requirements. Figures 5 i-k illustrate the two collision processes of the UAV equipped with the protective shell along the three orthogonal directions. The protective assembly effectively safeguards the core components of the UAV, including the propellers. 4. Conclusion In this study, we established an efficient deep learning-driven framework for the rapid inverse design of anisotropic multi-material structures. By integrating a high-fidelity forward surrogate model with an inverse generation network, we effectively mapped the complex relationship between mechanical responses and voxelized material distributions. A key innovation of this work lies in the introduction of a weighted multi-objective optimization strategy and a material ratio constraint mechanism. These strategies endow the model with specific design preferences, allowing it to prioritize directional mechanical performance weights or specific material composition ratios within the vast solution space, thereby achieving highly customized structural generation according to practical requirements. Our framework significantly enhances design efficiency, reducing the cycle from hours to seconds while maintaining a prediction accuracy exceeding 90%. We demonstrated the practical application value of the proposed method through the actual design of a protective cage for a quadrotor UAV. This data-driven paradigm overcomes the computational limitations of traditional approaches and provides a flexible, efficient tool for the design of advanced metamaterials in fields such as soft robotics, aerospace, and personalized protective equipment. Declarations Funding This work was supported by the National Natural Science Foundation of China (grant No. 52305398) and Key Research and Development Project of Sichuan Province (No. 2024YFFK0043). Author contributions Zhengda Chen: Investigation, Methodology, Software, Validation. Haocheng Yang: Visualization, Investigation. Xiangjun Zha: Data curation, Methodology, Writing-Original draft preparation. Lin Chen: Methodology. Xianglei Li: Data curation, Software. Jigang Huang: Conceptualization, Supervision, Resources, Writing- Reviewing and Editing. Data availability statement The raw data required to reproduce these findings are available upon request. Disclosure statement No potential conflict of interest was reported by the author(s). References Cheng J, Yu K L, Xu J Y, et al. Two-dimensional anisotropic semiconductors: from structure and properties to device applications [J]. Nanoscale, 2025, 17(25): 15086-109. Gao Y, Zhao X Q, Han X Y, et al. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8760116","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":593488543,"identity":"ec9fd635-baaf-4682-b2d4-0c24cb22d39c","order_by":0,"name":"Jigang Huang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/ElEQVRIiWNgGAWjYBACxmYInQAmP0A4BsRrYZxBjBYYAGth5iFGC3M7+8XPBb8Y8gyOnz382uaPXWIDe/M2CYaaO3gcxlMsPbOPodjgTF6adW5bcmIDz7EyCYZjz/BpSZDm7WFI3HAgx8w4t+FAYoNEjpkEY8NhfFqSf4O1nH9jZmzxB6hF/g0hLezHpHl+ALXcyDF+zMAGsoWHoC1s1rwNEokzb7wxY+xtSzZu40krtkg4hluLYf/xx7d5/tgk9p3PMf7w44+dbD/74Y03PtTg0dLAY8DA2CYBYrNBSBCRgFMDA4M8A/sDBoY/YDbzBzwKR8EoGAWjYAQDAO7GVixRUgB4AAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-8756-2262","institution":"Sichuan University","correspondingAuthor":true,"prefix":"","firstName":"Jigang","middleName":"","lastName":"Huang","suffix":""},{"id":593488544,"identity":"83ea7ff3-174f-4b97-ae3b-f53f3ff9cf2c","order_by":1,"name":"Zhengda Chen","email":"","orcid":"","institution":"Sichuan University","correspondingAuthor":false,"prefix":"","firstName":"Zhengda","middleName":"","lastName":"Chen","suffix":""},{"id":593488545,"identity":"c6b96e0f-d11d-43fd-9109-d1c442ad8c91","order_by":2,"name":"XiangJun Zha","email":"","orcid":"","institution":"The Third People's Hospital of Chengdu","correspondingAuthor":false,"prefix":"","firstName":"XiangJun","middleName":"","lastName":"Zha","suffix":""},{"id":593488546,"identity":"6f479ab2-9401-4b4e-bf4e-fb0b3665f08a","order_by":3,"name":"Haocheng Yang","email":"","orcid":"","institution":"Sichuan University","correspondingAuthor":false,"prefix":"","firstName":"Haocheng","middleName":"","lastName":"Yang","suffix":""},{"id":593488547,"identity":"beb79b35-1001-47c6-a4ee-d536e4c8d3fe","order_by":4,"name":"Lin Chen","email":"","orcid":"","institution":"Sichuan University","correspondingAuthor":false,"prefix":"","firstName":"Lin","middleName":"","lastName":"Chen","suffix":""},{"id":593488548,"identity":"70ce0702-dd9a-49d4-9e2e-68e3b6a7dda0","order_by":5,"name":"Xianglei Li","email":"","orcid":"","institution":"Sichuan University","correspondingAuthor":false,"prefix":"","firstName":"Xianglei","middleName":"","lastName":"Li","suffix":""},{"id":593488549,"identity":"8f8b000a-06e0-458c-8fe0-574ab95e018c","order_by":6,"name":"Qingyi Qi","email":"","orcid":"","institution":"Sichuan University","correspondingAuthor":false,"prefix":"","firstName":"Qingyi","middleName":"","lastName":"Qi","suffix":""}],"badges":[],"createdAt":"2026-02-02 03:45:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8760116/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8760116/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103081073,"identity":"ba4c46c2-9944-4f89-8bb0-7945f843013d","added_by":"auto","created_at":"2026-02-20 14:39:54","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":287659,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eOverview of the deep learning-driven design framework.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e(a)\u003c/strong\u003e Multi-material structures with anisotropic characteristics. \u003cstrong\u003e(b)\u003c/strong\u003e Quasi-static compressive stress-strain curves of multi-material structures in X, Y and Z directions. \u003cstrong\u003e(c)\u003c/strong\u003e Deep learning-driven inverse design model. \u003cstrong\u003e(d)\u003c/strong\u003e Binary spatial vectors for characterizing spatial material distribution. \u003cstrong\u003e(e)\u003c/strong\u003e The 3D printed optimal sample. Scale bar, 3 cm.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-8760116/v1/fc67f9ea01681a94e3619be8.png"},{"id":103081119,"identity":"f0a97c1c-2fa3-43e1-ad81-449e6e5bd331","added_by":"auto","created_at":"2026-02-20 14:40:11","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":385475,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eArchitecture and performance validation of the deep learning framework.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e(a)\u003c/strong\u003e The two core models in the proposed framework. \u003cstrong\u003e(b)\u003c/strong\u003e Schematic diagrams of the forward prediction model, which maps voxel-based material distributions to stress-strain responses, and the inverse design model, which predicts material distributions from target stress characteristic points. \u003cstrong\u003e(c)\u003c/strong\u003e Training and validation loss curves of the forward model. \u003cstrong\u003e(d)\u003c/strong\u003e Correlation plot comparing the forward model-predicted stress values against the truth for 2,000 test samples. \u003cstrong\u003e(e)\u003c/strong\u003e Training and validation loss curves of the inverse model. \u003cstrong\u003e(f)\u003c/strong\u003e Correlation plot comparing the inverse model-predicted stress values against the truth for 2,000 test samples. \u003cstrong\u003e(g)\u003c/strong\u003e The workflow for the practical invocation of the forward and inverse models. By taking the stress characteristics as input data, the inverse model of our machine learning approach predicts five sets of design candidates. These design candidates are subsequently passed to the forward model to evaluate their mechanical responses. Each response is then compared with the target curve features to select the optimal design.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-8760116/v1/218095572590514a868e54d0.png"},{"id":103080809,"identity":"49dfa77f-b68a-45ed-b79f-ad5e8089f287","added_by":"auto","created_at":"2026-02-20 14:38:49","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":335870,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eInverse Design with Material Ratio Constraints.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e(a)\u003c/strong\u003e Illustration of the \"overlap zones\" in the design space. \u003cstrong\u003e(b)\u003c/strong\u003e Representative examples of topological configurations explaining the overlap: high soft-material ratio structures with localized rigid supports vs. low soft-material ratio structures with deformation-dominant soft zones. \u003cstrong\u003e(c)\u003c/strong\u003e Schematic of the proposed joint loss function incorporating a material ratio constraint term weighted by coefficient \u003cem\u003en\u003c/em\u003e. \u003cstrong\u003e(d)\u003c/strong\u003e Comparison showing the model's ability to bias designs toward specific material constraints when trained with the joint loss function. \u003cstrong\u003e(e)\u003c/strong\u003e Evolution trends of fitting accuracy (R\u003csup\u003e2\u003c/sup\u003e) and material ratio compliance rate as the weighting coefficient \u003cem\u003en\u003c/em\u003e increases, before and after introducing the Hard Constraint Check (HCC) strategy. \u003cstrong\u003e(f)\u003c/strong\u003e The generation of distinct, high-fidelity optimal designs for the same target curve within different material ratio regimes.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-8760116/v1/07061c33d60248770cf2425d.png"},{"id":103081386,"identity":"db6fac4f-23d9-4357-96d6-536ff7b61db5","added_by":"auto","created_at":"2026-02-20 14:40:43","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":257794,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eAnisotropic inverse design strategy based on weighted multi-objective optimization.\u003c/strong\u003e \u003cstrong\u003e(a)\u003c/strong\u003e Architecture of the inverse model incorporating a \u003cstrong\u003eCoordinate Transformation\u003c/strong\u003e module and a weighted loss function (\u003cem\u003e\u003cstrong\u003eLoss = αloss\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cem\u003e\u003cstrong\u003eX\u003c/strong\u003e\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e\u003cstrong\u003e + βloss\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cem\u003e\u003cstrong\u003eY\u003c/strong\u003e\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e\u003cstrong\u003e + γloss\u003c/strong\u003e\u003c/em\u003e\u003csub\u003e\u003cem\u003e\u003cstrong\u003eZ\u003c/strong\u003e\u003c/em\u003e\u003c/sub\u003e). This setup allows for the simultaneous optimization of stress features in three orthogonal directions. \u003cstrong\u003e(b–d)\u003c/strong\u003e Model performance metrics (R\u003csup\u003e2\u003c/sup\u003e and MSE) under three distinct weighting schemes: \u003cstrong\u003e(b)\u003c/strong\u003e Uniform weights (\u003cem\u003e\u003cstrong\u003eα\u003c/strong\u003e\u003c/em\u003e = \u003cem\u003e\u003cstrong\u003eβ\u003c/strong\u003e\u003c/em\u003e = \u003cem\u003e\u003cstrong\u003eγ\u003c/strong\u003e\u003c/em\u003e), \u003cstrong\u003e(c)\u003c/strong\u003e Z-priority weights (\u003cem\u003e\u003cstrong\u003eγ \u0026gt;\u0026gt; α\u003c/strong\u003e\u003c/em\u003e, \u003cem\u003e\u003cstrong\u003eβ \u003c/strong\u003e\u003c/em\u003e), and \u003cstrong\u003e(d)\u003c/strong\u003e Gradient weights (\u003cem\u003e\u003cstrong\u003eγ\u003c/strong\u003e\u003c/em\u003e \u0026gt; \u003cem\u003e\u003cstrong\u003eβ\u003c/strong\u003e\u003c/em\u003e \u0026gt; \u003cem\u003e\u003cstrong\u003eα\u003c/strong\u003e\u003c/em\u003e). \u003cstrong\u003e(e–g)\u003c/strong\u003e Representative comparisons of target vs. predicted stress-strain curves corresponding to the three weighting strategies.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-8760116/v1/2dc5c177632be6643ea27bb9.png"},{"id":103080914,"identity":"e1201bf4-75f2-46d5-8393-410aea289053","added_by":"auto","created_at":"2026-02-20 14:39:19","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":717678,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eApplication of the designed anisotropic metamaterials for UAV impact protection.\u003c/strong\u003e \u003cstrong\u003e(a)\u003c/strong\u003eSchematic representation of two typical collision scenarios for a quadrotor UAV: horizontal impact with a wall and vertical free-fall to the ground. \u003cstrong\u003e(b)\u003c/strong\u003eDesign of the propeller protective cage integrated with the proposed anisotropic multi-material structure. \u003cstrong\u003e(c–e)\u003c/strong\u003e Dynamic analysis showing the evolution of the UAV’s kinetic energy over time during collision events. \u003cstrong\u003e(f–h)\u003c/strong\u003eMechanical response of the optimized protective structure under compression in the \u003cstrong\u003e(f)\u003c/strong\u003e X-direction, \u003cstrong\u003e(g)\u003c/strong\u003e Y-direction, and \u003cstrong\u003e(h)\u003c/strong\u003e Z-direction. The structure is designed with a specific weight ratio (\u003cem\u003e\u003cstrong\u003eα\u003c/strong\u003e\u003c/em\u003e : \u003cem\u003e\u003cstrong\u003eβ\u003c/strong\u003e\u003c/em\u003e: \u003cem\u003e\u003cstrong\u003eγ\u003c/strong\u003e\u003c/em\u003e = 1:1:1000) to prioritize energy absorption in the Z-direction (fall protection) while maintaining lateral impact resistance. \u003cstrong\u003e(i–k)\u003c/strong\u003eTwo kind of collision processes of the UAV equipped with the protect cage along the \u003cstrong\u003e(i)\u003c/strong\u003e X-direction, \u003cstrong\u003e(j)\u003c/strong\u003e Y-direction, and \u003cstrong\u003e(k)\u003c/strong\u003e Z-direction.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-8760116/v1/ff6afa85472496deed6292c2.png"},{"id":103081424,"identity":"cb1c8865-882a-4290-b539-7b0d3a64e759","added_by":"auto","created_at":"2026-02-20 14:41:00","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2602339,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8760116/v1/9b9a6f1b-9db2-437b-b96a-eae215b3c349.pdf"},{"id":103081154,"identity":"ce2d0615-3460-40da-9de4-6364ea2cce87","added_by":"auto","created_at":"2026-02-20 14:40:15","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":1990343,"visible":true,"origin":"","legend":"Supplementary_Information","description":"","filename":"SupplementaryInformation.docx","url":"https://assets-eu.researchsquare.com/files/rs-8760116/v1/2e0d0f2d18e381494e58bf75.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Deep learning driven inverse design of multi-material structures with tailored anisotropic mechanical responses","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eAnisotropic structures, characterized by direction-dependent mechanical properties, serve as the foundation for the development of advanced functional devices\u003csup\u003e[1-6]\u003c/sup\u003e. Structures exhibiting significant anisotropic features hold broad application prospects in fields such as acoustics\u003csup\u003e[7-9]\u003c/sup\u003e, biomedicine\u003csup\u003e[10-17]\u003c/sup\u003e, and aerospace\u003csup\u003e[18-22]\u003c/sup\u003e. However, designing components that exhibit precise, pre-defined stress-strain behaviors in different directions presents a significant complexity. This challenge is compounded not only by the non-linear constitutive relationships of the materials involved but also by the non-intuitive mapping between microstructural geometry and macroscopic performance. Multi-material architectures offer a powerful pathway to achieve such complex anisotropy. People can create digital materials\u003csup\u003e[23-26]\u003c/sup\u003e or hierarchical structures\u003csup\u003e[27-29]\u003c/sup\u003e by strategically distributing materials with distinct moduli. These voxel-based designs provide immense freedom to tune local stiffness and global deformation mechanisms. However, this flexibility comes at a cost: as the resolution of the material distribution increases, the combinatorial design space explodes exponentially\u003csup\u003e[30, 31]\u003c/sup\u003e. Current design methodologies predominantly rely on forward trial-and-error approaches or physics-based topology optimization (TO). Empirical designs, dependent on human intuition, are time-consuming and often fail to exploit the full potential of the design space. While topology optimization has proven effective for maximizing stiffness or compliance, it often incurs prohibitive computational costs\u003csup\u003e[32]\u003c/sup\u003e. Furthermore, as the demand for \u0026quot;customized\u0026quot; mechanical responses grows\u0026mdash;requiring the matching of entire stress-strain curves rather than single scalar values like modulus\u0026mdash;traditional methods struggle to meet the requirements for both accuracy and design efficiency.\u003c/p\u003e\n\u003cp\u003eIn recent years, Machine Learning (ML) has emerged as a transformative paradigm for solving inverse problems in mechanics and materials science\u003csup\u003e[33-39]\u003c/sup\u003e. Zheng et al.\u003csup\u003e[40]\u003c/sup\u003e introduced a generative ML pipeline capable of creating mechanical metamaterials that replicate nearly all possible uniaxial compressive stress-strain curves, achieving almost 90% fidelity between target and experimental results. Similarly, Zong et al.\u003csup\u003e[41]\u003c/sup\u003e proposed a machine-learning-powered approach for multi-target inverse design based on graded triply periodic minimal surface (TPMS) architectures, achieving rapid and accurate results within seconds for complex applications such as customized soles. Moving beyond static metamaterials, Matthews et al.\u003csup\u003e[42]\u003c/sup\u003e developed a gradient-based optimization method capable of designing functional robots from scratch in under 30 seconds, demonstrating that complex physical interdependencies can be co-optimized far more efficiently than through traditional evolutionary algorithms. Furthermore, the capability to design for precise spatial responses has been explored by Ge et al.\u003csup\u003e[27]\u003c/sup\u003e, who pioneered the inverse design of strain fields in 2D multi-material structures using recurrent neural networks (RNNs) and evolutionary algorithms. Although these works have leveraged machine learning to significantly enhance the accuracy and efficiency of inverse design, they still lack sufficient degrees of freedom regarding spatial distribution.\u003c/p\u003e\n\u003cp\u003eIn this work, we present a deep learning-driven inverse design framework capable of instantly generating multi-material microstructures based on user-defined anisotropic stress-strain curves. Unlike iterative optimization methods, our data-driven model learns the intrinsic structure-property relationships, reducing the design cycle from hours to mere seconds while achieving a curve-fitting accuracy of over 95%. We address the \u0026quot;non-uniqueness\u0026quot; or \u0026quot;overlap\u0026quot; problem inherent in multi-material design, where different material distributions might yield similar mechanical responses. We propose customized joint loss function that integrates a Mean Squared Error (MSE) term for performance matching with a material ratio constraint term. This approach, combined with \u0026quot;hard constraint check,\u0026quot; enables the explicit control of the volume fraction of soft and hard materials without compromising the target mechanical performance. By adjusting the loss weights across different loading directions, our model enables the flexible customization of anisotropic behaviors, paving the way for the efficient design of high-performance metamaterials.\u003c/p\u003e"},{"header":"2. Experimental section","content":"\u003cp\u003e2.1. Material Selection and Additive Manufacturing\u003c/p\u003e\n\u003cp\u003eWe selected commercial polylactic acid (PLA Basic) and thermoplastic polyurethane (TPU for AMS) produced by Bambu Lab (Shenzhen, China) as the constituent materials. PLA Basic serves as the rigid phase, providing structural support and high stiffness, with a density of 1.24g/cm\u003csup\u003e3\u003c/sup\u003e and a Young\u0026apos;s modulus of 2750MPa. Conversely, TPU for AMS serves as the soft phase to induce large deformations and energy absorption. The significant contrast in mechanical properties between these two materials enables a vast design space for tuning the global mechanical behavior of the composite structures.\u003c/p\u003e\n\u003cp\u003eThe physical specimens were fabricated using the Bambu Lab X1 Carbon Fused Deposition Modeling (FDM) 3D printer. This printer features a build volume of 256 mm\u0026nbsp;\u0026times;\u0026nbsp;256 mm\u0026nbsp;\u0026times;\u0026nbsp;256 mm and is equipped with an Automatic Material System (AMS), which allows for the seamless printing of up to four different materials within a single layer. To ensure optimal interlayer adhesion and interface bonding between the PLA and TPU phases, the nozzle temperature was set to 220℃ for PLA and 230℃ for TPU, with a heated bed temperature of 50℃. The layer thickness was set to 0.2mm with 100% infill density to replicate the solid voxel structure used in the simulations. More details about the materials used in this work are provided in Table S1.\u003c/p\u003e\n\u003cp\u003e2.2. Finite Element Simulation and Dataset Generation\u003c/p\u003e\n\u003cp\u003eTo train the deep learning model, a large-scale dataset linking material distributions to their mechanical responses was generated via Finite Element Analysis (FEA). We utilized a custom Python script to randomly generate 8,000 unique material distribution combinations. Each sample was modeled as a cubic structure with overall dimensions of 36 \u0026times; 36 \u0026times; 36 mm\u003csup\u003e3\u003c/sup\u003e. The structure was discretized into a 3 \u0026times; 3 \u0026times; 3 voxel array, consisting of 27 individual unit cells, where each unit cell (voxel) had dimensions of 12 \u0026times; 12 \u0026times; 12 mm\u003csup\u003e3\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eThe numerical simulations were performed using the commercial FEA software Abaqus (Dassault Systems SIMULIA Established Products 2022, Johnston, United States). In the simulation environment, each voxel was assigned the material properties of either PLA or TPU based on the generated binary matrix (0 for hard material, 1 for soft material). A uniaxial compression test was simulated for each case. The boundary conditions were defined as follows: the bottom surface of the cubic structure was fully fixed (Ux = Uy = Uz = 0), while a uniform vertical displacement was applied to the top surface to achieve a global compressive strain of 20% (equivalent to a displacement of 7.2 mm). An implicit solver was employed to ensure convergence. The resulting reaction forces and displacements were extracted to compute the stress-strain curves, which served as the ground truth labels for the neural network training. A more detailed description about this part is provided in Note S1.\u003c/p\u003e\n\u003cp\u003e2.3. Deep Learning Framework Implementation\u003c/p\u003e\n\u003cp\u003eThe deep learning framework, including the forward prediction network and the inverse design model, was implemented using the Tensorflow library. All model training and computational tasks were conducted on a desktop workstation. The hardware configuration consisted of an Intel Core i5-10500 CPU, 16 GB of RAM, and an NVIDIA GeForce RTX 3060 Ti graphics processing unit (GPU) to accelerate the tensor operations.\u003c/p\u003e\n\u003cp\u003eThe dataset of 8,000 samples was randomly split into a training set (70%), a validation set (30%). The input to the network was the 1 \u0026times; 27 binary matrix representing the material distribution, and the output was the corresponding stress-strain sequence. The model was optimized using the Adam optimizer. To prevent overfitting, we monitored the validation loss and employed early stopping techniques.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e2.4 Experiment methods\u003c/p\u003e\n\u003cp\u003eCompression tests were performed at room temperature with a universal testing machine (Instron 5967) and the testing speed was set to be 10 mm/min. We adopted a quasi-static testing method the strain rate was set to be 0.01/s.\u003c/p\u003e"},{"header":"3. Results and discussion","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n \u003ch2\u003e3.1 The deep learning-driven design framework.\u003c/h2\u003e\n \u003cp\u003eThe proposed framework integrates feature extraction, deep learning-driven inverse prediction, and multi-material additive manufacturing into a closed-loop system, enabling the precise translation of complex stress-strain targets into physical realities (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e). The design process initiates with the definition of target mechanical behaviors. The mechanical performance of the anisotropic structure is defined by distinct quasi-static compression stress-strain curves in orthogonal directions (X, Y, and Z-axis. Figure \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ea). To process these complex non-linear responses, we employ a feature extraction mechanism that condenses the continuous stress-strain data into discrete characteristic points (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003eb). This feature extraction is beneficial for ensuring the efficiency of subsequent neural network processing while retaining key mechanical characteristics, such as modulus and yield strength. These extracted features are then fed into our pre-trained deep learning driven inverse design model (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ec). This model identifies the intrinsic mapping between macroscopic mechanical responses and microscopic material configurations, with its output being a digital material representation in the form of a 3 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\times\\:\\:\\)\u003c/span\u003e\u003c/span\u003e3 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\times\\:\\)\u003c/span\u003e\u003c/span\u003e 3 spatial vector matrix. In this binary matrix, \u0026quot;0\u0026quot; designates the hard phase (PLA), which provides structural rigidity and load-bearing capacity, while \u0026quot;1\u0026quot; designates the soft phase (TPU), which facilitates large deformation and energy absorption (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ed). This discrete, voxel-based representation allows for the generation of complex, heterogeneous architectures that would be counter-intuitive to design using traditional trial-and-error methods. Fig. S2 provide more details about the base unit used in this work. The final optimized design is fabricated using a multi-material 3D printer. Upon testing, the resulting multi-material structure will demonstrate the ability to accurately replicate prescribed stress-strain responses under quasi-static compressive loading.\u003c/p\u003e\n \u003cp\u003eThe proposed deep learning framework is bifurcated into two primary modules: the forward prediction model and the inverse design model. The forward model serves as a high-fidelity surrogate framework, effectively bypassing the computationally prohibitive Finite Element Analysis (FEA) simulations. It takes the voxel-based material distribution matrix as input, executes sequential processing across three hidden layers, and yields discrete stress characteristic values which are further reconstructed into complete macroscopic stress-strain curves. To train this model, a comprehensive dataset comprising 8,000 unique samples was generated via automated FEA scripts (Fig. S3). This dataset covers a diverse spectrum of material volume fractions to ensure the model captures the full complexity of the design space (Fig. S4). As illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ec, the loss values for both training and validation sets converge to a remarkably low level (10\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e) as the number of training epochs increases. Notably, the close alignment between training and validation loss curves indicates the absence of overfitting, confirming the model\u0026rsquo;s robust generalization capabilities across unseen material configurations. To rigorously validate the predictive accuracy, the model was tested against 2,000 novel datasets. As shown in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ed, the predicted stress values exhibit high consistency with the FEA ground truth, with data points tightly clustered around the y\u0026thinsp;=\u0026thinsp;x identity line. This correlation is quantitatively confirmed by a coefficient of determination (R\u003csup\u003e2\u003c/sup\u003e) of 0.9874, underscoring the model\u0026apos;s precision and reliability as a surrogate for physical simulations. Furthermore, we systematically investigated the influence of various hyperparameters on model efficacy (Fig. S3 and Table S2), including the training set size, batch size, learning rate and hidden layer dimensions.\u003c/p\u003e\n \u003cp\u003eBy fine-tuning these parameters, we optimized the forward model to achieve a balance between computational efficiency and predictive accuracy, ensuring stable performance across diverse structural scenarios. Inverse Model Training is trained by utilizing stress characteristic values as inputs to generate corresponding material distribution matrices. During the training phase, the pre-trained (frozen) forward model is employed as a rapid evaluator to generate predicted stress responses from the inverse model\u0026rsquo;s output. The error between these predicted responses and the ground truth labels is then backpropagated to update the inverse network\u0026rsquo;s weights. Similar to the forward module, the inverse model exhibits excellent convergence performance, with the convergence error reaching 10\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e (Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ee), and achieves a high R\u003csup\u003e2\u003c/sup\u003e value of 0.9707 on 2,000 new test cases (Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ef). The influences of various hyperparameters on inverse model efficacy are demonstrated in Fig. S4 and Table S3.\u003c/p\u003e\n \u003cp\u003eIn the deployment of the complete framework, the system takes user-defined stress characteristic points as the target inputs. The inverse model generates potential design candidates, which are subsequently fed into the forward model for the estimation of their respective mechanical behaviors. The optimal design is identified by evaluating the discrepancy between these estimated curves and the predefined target. Since the final selection is based on a direct comparison of the holistic curve features, the proposed framework ensures the uniqueness of the derived solution. This strategy effectively circumvents the \u0026quot;one-to-many\u0026quot; mapping challenge inherent in inverse problems, where multiple distinct microstructures may yield nearly identical macroscopic mechanical responses\u003csup\u003e[\u003cspan class=\"CitationRef\"\u003e40\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n \u003ch2\u003e3.2 Inverse Design with Material Ratio Constraints\u003c/h2\u003e\n \u003cp\u003eThe mechanical performance of multi-material structures bears no simple linear relationship to the proportion of constituent materials; instead, prominent \u0026quot;overlap zones\u0026quot; emerge within the design space (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ea). Taking the proposed multi-material architecture as an example, a higher volume fraction of soft material generally leads to reduced structural stiffness, yet certain configurations with high soft-material ratios maintain remarkably high mechanical performance due to the formation of localized support zones through strategic spatial distribution. Conversely, designs with lower soft-material ratios exhibit macroscopic flexibility because their deformation zones\u0026mdash;comprised of soft phases\u0026mdash;absorb the majority of the strain during compression (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eb). Consequently, inverse design tasks targeting stress-strain curves within these overlap zones face a non-uniqueness challenge, where the solution may fall into either a high or low soft-material proportion region. To explicitly regulate the material usage in the final design, we propose a joint loss function that simultaneously considers curve fitting accuracy and material volume fractions. This function incorporates a material ratio constraint term into the original objective, with a weighting coefficient \u003cem\u003en\u003c/em\u003e used to modulate the priority of the ratio constraint (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ec):\u003c/p\u003e\n \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$\\:L={nloss}_{constraint}+MSE$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eBy training the model with this joint loss, the framework can generate spatial distributions that satisfy a predefined material ratio range while maintaining high fidelity to the target stress-strain curve. In scenarios where multiple distinct microstructures could potentially yield the same target response, the model is biased toward designs that comply with the specified material constraints (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ed).\u003c/p\u003e\n \u003cp\u003eWe further investigated the influence of the weighting coefficient \u003cem\u003en\u003c/em\u003e on the training efficacy. Figure \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ee illustrates the evolution of the coefficient of determination (R\u003csup\u003e2\u003c/sup\u003e) and the ratio accuracy\u0026mdash;defined as the proportion of samples satisfying the predefined material limits\u0026mdash;as \u003cem\u003en\u003c/em\u003e increases. As \u003cem\u003en\u003c/em\u003e rises, the ratio accuracy improves significantly from less than 30% to 100%, demonstrating that the weighting coefficient effectively drives the model toward compliant designs. However, we observed a trade-off: the R\u003csup\u003e2\u003c/sup\u003e values decrease from over 90% to below 60% at excessive \u003cem\u003en\u003c/em\u003e values, indicating that the model becomes overly focused on material proportions at the expense of performance matching. To resolve this conflict, we introduced a Hard Constraint Check (HCC) strategy. The core of this strategy involves a post-processing step before the final design output: for results that do not satisfy the material ratio requirements, the \u003cem\u003em\u003c/em\u003e voxels with the lowest probability of belonging to their assigned phase are identified as \u0026quot;replacement points.\u0026quot; These points are then toggled (e.g., from soft to hard or vice-versa) until the material ratio strictly meets the constraint. This mechanism ensures absolute adherence to manufacturing requirements while preventing the performance degradation associated with overly high \u003cem\u003en\u003c/em\u003e values. Figure \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ef demonstrates an application of this approach within the 0\u0026ndash;50% and 50\u0026ndash;100% overlap zones, where the inverse model successfully identified two distinct, highly accurate optimal designs (NRMSE(A)\u0026thinsp;=\u0026thinsp;0.03150, NRMSE(B)\u0026thinsp;=\u0026thinsp;0.04655) for the same target curve across different material ratio regimes.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n \u003ch2\u003e3.3 Anisotropic Inverse Design with Weighted Multi-Objective Optimization\u003c/h2\u003e\n \u003cp\u003eTo comprehensively address the anisotropy of multi-material structures and enable the inverse design of customized mechanical responses over all spatial directions, we significantly expanded the training dataset for the proposed model. Beyond the original Z-direction compression simulations, we conducted supplementary simulations for compression along the X and Y axes, resulting in a compiled dataset of three sets of stress characteristic arrays (totaling 63 feature points). The anisotropy of the structure is thus explicitly characterized by the stress-strain features across these three orthogonal directions. We retrained the deep learning model using this augmented dataset comprising the new multi-directional stress features and material distributions. As illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003ea, the network architecture was modified to handle directional dependencies: while the material distribution generated by the inverse model is directly fed into the forward model to predict Z-direction stress features, predictions for the X and Y directions are obtained by passing the material distribution through a Coordinate Transformation module before entering the shared forward model. Furthermore, to guide the optimization process effectively, we introduced a weighted joint loss function:\u003c/p\u003e\n \u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e$$\\:Loss={\\alpha\\:loss}_{X}+{\\beta\\:loss}_{Y}+{\\gamma\\:loss}_{Z}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere weights \u003cstrong\u003e\u0026alpha;\u003c/strong\u003e, \u003cstrong\u003e\u0026beta;\u003c/strong\u003e, and \u003cstrong\u003e\u0026gamma;\u003c/strong\u003e assign priority to the stress features of specific directions.\u003c/p\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003eb presents the performance metrics (R\u003csup\u003e2\u003c/sup\u003e and MSE) of the model trained with uniform weights (\u003cstrong\u003e\u0026alpha;\u003c/strong\u003e\u0026thinsp;=\u0026thinsp;\u003cstrong\u003e\u0026beta;\u003c/strong\u003e\u0026thinsp;=\u0026thinsp;\u003cstrong\u003e\u0026gamma;\u003c/strong\u003e). Compared to the previous inverse design tasks targeting a single direction, the design efficacy of the multi-directional model declined, with R\u003csup\u003e2\u003c/sup\u003e values across all three directions falling below 0.9. Figure \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003ee displays the target versus actual stress-strain curves for a representative sample under this uniform weighting scheme. A consistent but mediocre fitting performance is observed across all three directions (X, Y, and Z). This performance degradation is likely attributed to the increased complexity of the optimization landscape, where the simultaneous fitting of excessive feature points (63 points) without prioritization hinders the model\u0026apos;s convergence.\u003c/p\u003e\n \u003cp\u003eBy significantly increasing the weighting coefficient for the Z-direction (\u003cstrong\u003e\u0026alpha;\u003c/strong\u003e : \u003cstrong\u003e\u0026beta;\u003c/strong\u003e : \u003cstrong\u003e\u0026gamma;\u003c/strong\u003e\u0026thinsp;=\u0026thinsp;1:1:1000), the trained model exhibited a marked prioritization for satisfying Z-direction compression features. As shown in Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003ec, the R\u003csup\u003e2\u003c/sup\u003e and MSE for the Z-direction improved significantly compared to the X and Y directions during prediction. We observed a trade-off mechanism: while the biased weights successfully guided the model to meet the high-priority Z-target, this came at the cost of increased MSE in the X and Y directions. Figure \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003ef visualizes a typical sample predicted by this Z-weighted model, clearly demonstrating a high degree of fidelity between the predicted and target curves in the Z-direction, while deviations remain in the secondary directions.\u003c/p\u003e\n \u003cp\u003eTo further refine this control, we adjusted the weights to a hierarchical distribution (\u003cstrong\u003e\u0026alpha;\u003c/strong\u003e : \u003cstrong\u003e\u0026beta;\u003c/strong\u003e : \u003cstrong\u003e\u0026gamma;\u003c/strong\u003e\u0026thinsp;=\u0026thinsp;1:1000:1000000). The resulting model demonstrated a fitting priority order of Z\u0026thinsp;\u0026gt;\u0026thinsp;Y \u0026gt; X, satisfying stress features in the Z-direction first, followed by Y, and finally X. This characteristic is quantitatively reflected in the MSE distribution shown in Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003ed. Figure \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003eg presents a representative sample generated by this gradient-weight model, where the fitting accuracy clearly degrades sequentially from Z to Y to X, aligning perfectly with the imposed weight hierarchy. This strategy of utilizing deep learning to inversely design anisotropic multi-material structures, coupled with a weighted loss function to prioritize mechanical performance in specific directions, represents a novel approach not previously reported in existing literature. Our methodology provides a new paradigm for designing metamaterials with complex, directionally tailored mechanical properties.\u003c/p\u003e\n \u003cp\u003eQuadrotor Unmanned Aerial Vehicles (UAVs) typically feature four exposed propellers, rendering them highly vulnerable to structural damage upon impact with rigid surfaces such as walls. Furthermore, free-fall incidents from significant altitudes may result in catastrophic fuselage damage (Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003ea). To address these challenges, we categorized the collision dynamics of a UAV into two distinct scenarios: horizontal impact and vertical fall. Horizontal impact refers to collisions within the X-Y plane, where the UAV strikes a vertical obstacle (e.g., a wall) with a specific initial velocity, resulting in the rapid dissipation of horizontal kinetic energy. Conversely, a vertical fall represents a collision along the Z-axis, where the drone undergoes free-fall from a specific height, impacting the ground and dissipating potential energy converted into kinetic energy. Both scenarios pose significant risks of irreversible structural failure.\u003c/p\u003e\n \u003cp\u003eTo mitigate these risks, we developed a specialized propeller protective cage for quadrotors. This assembly consists of an external supporting frame integrated with the proposed anisotropic multi-material structures (Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003eb). The kinetic energy generated during impact is dissipated through the deformation of multi-material lattices. To quantify the energy absorption requirements, we conducted a dynamic analysis of a commercial UAV under both collision scenarios (Table. S3). The resulting relationships between impact time and the evolution of the drone\u0026apos;s kinetic energy are presented in Figs. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003ec-e.\u003c/p\u003e\n \u003cp\u003eTo ensure the UAV\u0026apos;s kinetic energy is entirely absorbed through the deformation of the protective cage, we denote the initial kinetic energy of the drone as E\u003csub\u003e0\u003c/sub\u003e and the final kinetic energy as E\u003csub\u003e1\u003c/sub\u003e:\u003c/p\u003e\n \u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e$$\\:{E}_{0}=\\frac{1}{2}m{v}^{2}\\:;\\:\\:{E}_{1}=0$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eWhere m is the mass of the UAV and v is the velocity of the UAV.\u003c/p\u003e\n \u003cp\u003eThe entire energy absorption process is determined by calculating the integral of the stress-strain curve of the protective cage:\u003c/p\u003e\n \u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e$$\\:{W}_{total}=V{\\int\\:}_{0}^{{\\epsilon\\:}_{max}}\\delta\\:{d}_{\\epsilon\\:}=\\:{E}_{0}-{E}_{1}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eWhere V is the volume of the energy absorption module, \u0026epsilon; is the strain, and \u0026epsilon; is the stress.\u003c/p\u003e\n \u003cp\u003eThe energy absorbed by each individual energy absorption unit within the protective cage is:\u003c/p\u003e\n \u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e$$\\:{W}_{1}=\\frac{{W}_{total}}{n}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eWhere n is the number of units that participate in energy absorption during the impact process and W\u003csub\u003e1\u003c/sub\u003e was used as the basis to design the target stress-strain curve.\u003c/p\u003e\n \u003cp\u003eTo maximize energy absorption efficiency, we tailored the target stress-strain characteristics of the metamaterial in each spatial direction to match the specific energy dissipation profiles required by the different collision modes. Given that vertical drop incidents occur with higher frequency than horizontal collisions in practical operation, we prioritized energy absorption in the vertical direction. Consequently, the weighting coefficients for the inverse design model were assigned as \u003cstrong\u003e\u0026alpha;\u003c/strong\u003e : \u003cstrong\u003e\u0026beta;\u003c/strong\u003e : \u003cstrong\u003e\u0026gamma;\u003c/strong\u003e\u0026thinsp;=\u0026thinsp;1:1:1000 to bias the optimization toward the Z-axis performance while maintaining sufficient protection in lateral directions. Utilizing the trained weighted inverse model, we generated an optimized multi-material microstructure. As illustrated in Figs. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003ef-h, the resulting structure exhibits excellent agreement with the target response in the Z-direction compression test, demonstrating high fidelity in the primary load-bearing direction. Simultaneously, the structure satisfies the energy absorption criteria for the X and Y directions, balancing anisotropy with functional requirements. Figures \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003ei-k illustrate the two collision processes of the UAV equipped with the protective shell along the three orthogonal directions. The protective assembly effectively safeguards the core components of the UAV, including the propellers.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eIn this study, we established an efficient deep learning-driven framework for the rapid inverse design of anisotropic multi-material structures. By integrating a high-fidelity forward surrogate model with an inverse generation network, we effectively mapped the complex relationship between mechanical responses and voxelized material distributions. A key innovation of this work lies in the introduction of a weighted multi-objective optimization strategy and a material ratio constraint mechanism. These strategies endow the model with specific design preferences, allowing it to prioritize directional mechanical performance weights or specific material composition ratios within the vast solution space, thereby achieving highly customized structural generation according to practical requirements. Our framework significantly enhances design efficiency, reducing the cycle from hours to seconds while maintaining a prediction accuracy exceeding 90%. We demonstrated the practical application value of the proposed method through the actual design of a protective cage for a quadrotor UAV. This data-driven paradigm overcomes the computational limitations of traditional approaches and provides a flexible, efficient tool for the design of advanced metamaterials in fields such as soft robotics, aerospace, and personalized protective equipment.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work was supported by the National Natural Science Foundation of China (grant No. 52305398) and Key Research and Development Project of Sichuan Province (No. 2024YFFK0043).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eZhengda Chen: Investigation, Methodology, Software, Validation. Haocheng Yang: Visualization, Investigation. Xiangjun Zha: Data curation, Methodology, Writing-Original draft preparation. Lin Chen: Methodology. Xianglei Li: Data curation, Software. Jigang Huang: Conceptualization, Supervision, Resources, Writing- Reviewing and Editing.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe raw data required to reproduce these findings are available upon request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDisclosure statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNo potential conflict of interest was reported by the author(s).\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eCheng J, Yu K L, Xu J Y, et al. Two-dimensional anisotropic semiconductors: from structure and properties to device applications [J]. Nanoscale, 2025, 17(25): 15086-109.\u003c/li\u003e\n \u003cli\u003eGao Y, Zhao X Q, Han X Y, et al. Soft Actuator Based on Metal/Hydrogel Nanocomposites with Anisotropic Structure [J]. 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Proceedings of the National Academy of Sciences of the United States of America, 2023, 120(41).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"multi-material 3D printing, inverse design, anisotropic responses, deep learning","lastPublishedDoi":"10.21203/rs.3.rs-8760116/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8760116/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMulti-material structures are complex architectures formed by the selective distribution of diverse materials at the microstructural level, enabling the anisotropic designs with customized mechanical responses. Although these structures offer significant advantages via their immense design freedom, the vast design space and the non-linear coupling of multi-phase mechanical properties pose formidable challenges for inverse design of such structures. This study develops a deep learning-driven framework for the rapid design of voxelized multi-material structures. By integrating a high-fidelity forward surrogate model with an inverse neural network, our method can generate spatial material distributions that satisfy targeted stress-strain curves within seconds. Validated through multi-material 3D printing and quasi-static compression tests, the framework achieves an accuracy of over 95%. Notably, we introduce a joint loss function coupled with a Hard Constraint Check (HCC) strategy, allowing the model to selectively bias designs toward specific soft-to-hard material ratios without compromising mechanical performance. Furthermore, a weighted multi-objective optimization scheme is implemented to incorporate priorities when managing anisotropic responses. Experimental results demonstrate the immense potential of deep learning in the spatial distribution design of multi-material structures, paving the way for advanced material applications in fields such as medicine, robotics, aerospace, civil engineering, and vehicle engineering.\u003c/p\u003e","manuscriptTitle":"Deep learning driven inverse design of multi-material structures with tailored anisotropic mechanical responses","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-20 14:38:10","doi":"10.21203/rs.3.rs-8760116/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nature-communications","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"NCOMMS","sideBox":"Learn more about [Nature Communications](http://www.nature.com/ncomms/)","snPcode":"","submissionUrl":"https://mts-ncomms.nature.com/","title":"Nature Communications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Nature Communications","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"c32db5b6-df9c-49e2-846b-e38e8e3d54d8","owner":[],"postedDate":"February 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":63154914,"name":"Physical sciences/Engineering/Mechanical engineering"},{"id":63154915,"name":"Physical sciences/Materials science/Structural materials/Composites"}],"tags":[],"updatedAt":"2026-03-30T19:30:39+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-20 14:38:10","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8760116","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8760116","identity":"rs-8760116","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}