Beyond Euler: An Explainable Machine Learning Framework for Predicting and Interpreting Buckling Instabilities in Non-Ideal Materials

Beyond Euler: An Explainable Machine Learning Framework for Predicting and Interpreting Buckling Instabilities in Non-Ideal Materials | 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 Research Article Beyond Euler: An Explainable Machine Learning Framework for Predicting and Interpreting Buckling Instabilities in Non-Ideal Materials Pranil Raichura, Abdiel Rivera This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7668574/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Predicting structural failure is a fundamental objective in materials science and mechanical engineering. Euler’s classical formula, the standard for predicting the buckling instability of slender columns for over 250 years, assumes idealized material properties that can lead to unreliable predictions and potentially catastrophic failures in critical infrastructure. This study proposes a solution by introducing a novel framework that synergizes machine learning and modern explainability techniques to model complex physical systems. We used pasta as a model non-ideal material for a comprehensive experimental analysis and a dataset from 147 controlled buckling experiments on four distinct pasta gauges. We then developed a physics-informed XGBoost model, incorporating both raw geometric measurements and a composite feature derived from Euler’s formula (G = d4/L2), and subsequently evaluated the model’s performance using a 5-fold cross-validation scheme. The model demonstrated an outstanding predictive power, achieving an average coefficient of determination (R²) of 0.97 and a Root Mean Squared Error (RMSE) of 0.14 N. We also examined the model’s internal decision-making process by employing SHAP (SHapley Additive exPlanations). The analysis confirmed the primary importance of the theoretically-derived feature but also revealed that the model learned to use raw geometric data as crucial correction factors. This study presents a powerful proof of concept for using interpretable machine learning not only to achieve predictive accuracy but also gain deeper physical insights into complex, non-ideal systems. The framework presented has broad implications for advancing our understanding and design capabilities in materials science, engineering, and advanced manufacturing. Artificial Intelligence and Machine Learning explainable AI (XAI) machine learning structural mechanics Extreme Gradient Boosting (XGBoost) buckling materials science physics-informed ML SHAP Figures Figure 1 Figure 2 Figure 3 I. INTRODUCTION A. The Engineering Imperative of Stability Analysis Civil and mechanical engineering have long focused on predicting the response of materials to applied forces [10, 11]. The reliability of any structure depends on its ability to remain stable under its prescribed operational loads [12, 13]. Accurate predictions of instabilities are essential to prevent catastrophic collapses, which can incur severe economic and human costs [14 ] . Among the various modes of structural failure, the phenomenon of buckling stands out as particularly insidious [15]. Unlike a material slowly yielding under tension, a column undergoing buckling can transition from a state of stable equilibrium to total failure with little to no warning [16]. This is not merely a theoretical concern; it is a real-world failure mode seen in critical infrastructure, from columns in buildings and bridges to sub-sea pipelines [17] and railway tracks that can buckle under the compressive forces of thermal expansion [18]. Accurately predicting the onset of buckling proves itself to be an engineering imperative of the highest order [13]. B. The Classical Framework: Euler’s Buckling Theory Leonhard Euler achieved the first successful mathematical description of column buckling in 1744. His work established a foundational pillar of structural mechanics and provided an equation that remains central to engineering education and practice [1]. Euler’s critical load formula, shown in (1), defines the maximum axial compressive load an ideal column can sustain before it becomes unstable [1]. P_{cr} = \dfrac{\pi^2 EI}{(KL)^2} (1) To fully appreciate this equation, it is necessary to understand its constituent terms: P_{cr} (the critical load), E (Young’s Modulus of Elasticity), I (Area Moment of Inertia), L (Effective Length), and K (Effective Length Factor). C. The Gap Between Ideal Theory and Physical Reality The elegance and enduring power of Euler’s formula lie in its foundation of mathematical idealism; however, this is also its primary limitation in real-world scenarios. A significant and well-documented “reality gap” exists because no physical object perfectly satisfies the formula’s core assumptions [1]. The key sources of this discrepancy are: • Material Heterogeneity and Anisotropy: Euler’s formula assumes the material is homogeneous and isotropic. Many real-world materials, however, do not meet this ideal. For example, natural materials like wood have grain and growth rings, and biological materials like bone have complex, porous internal structures [26]. Similarly, modern engineered materials like 3D-printed polymers exhibit anisotropic properties due to their layered construction [27]. Pasta, as an extruded product, exhibits brittle failure modes inconsistent with the above ideal materials [7, 19]. • Geometric Imperfections: The formula assumes a perfectly straight column. All real objects exhibit minute variations that create eccentricities where the applied load is not perfectly aligned with the column’s central axis, inducing bending moments that the ideal theory does not account for [16, 20]. • Difficulty in Parameter Estimation: The formula’s accuracy depends critically on the value of Young’s Modulus, E. For pasta; this value has been investigated using various methods, yielding a range of results and highlighting the difficulty in characterization [2, 6, 21]. • Non-Linear Material Behavior: The theory assumes linear elasticity. Many materials, including pasta, exhibit non-linear and brittle behavior, failing suddenly without significant prior elastic deformation [7, 22]. These limitations mean that any attempt to use Euler’s formula to precisely predict the failure of a real-world object is fraught with uncertainty; indeed, advanced software tools like SAP2000 or ETABS have to be used in order to account for these imperfections and nuances . Building on prior studies in the field, Our study focuses on this gap, using pasta as an accessible and illustrative example of a non-ideal material [2, 6, 7]. D. A New Paradigm: Data-Driven Modeling with Explainable AI To bridge this reality gap, we turn to a new paradigm: supervised machine learning. The application of machine learning to predict column buckling is an active area of research, with recent studies successfully using models like artificial neural networks for braced columns [23] and corrugated steel girders [9]. Instead of beginning with a theoretical formula, a data-driven approach inverts the process, learning the complex and underlying associations —such as non-linear interactions between features and the subtle effects of material imperfections— directly from experimental observations. We hypothesize that a machine learning model, when presented with measurable geometric features, can learn a highly accurate mapping to the experimentally observed buckling load. High predictive accuracy alone, however, is not sufficient for scientific inquiry. An AI model that acts as an impenetrable black box offers little new physical insight, as we are unable to understand its decision-making process as well as what specific factors affect the outcome. This brings us to the core novelty of this study: the integration of Explainable AI (XAI), a growing field of study in engineering and computer science [8]. By using state-of-the-art XAI techniques, specifically SHAP (SHapley Additive exPlanations) [4], we aim to “open the black box” and understand how the model arrives at its predictions. E. Statement of Contributions This study makes the following novel contributions: • It provides a rigorous, quantitative demonstration of the limitations of Euler’s classical buckling formula when applied to a common, non-ideal material. • It develops and validates a high-performance XGBoost Machine Learning model that predicts the critical buckling load with exceptionally high accuracy (R²=0.97), outperforming the theoretical model. • It successfully integrates a physics-informed feature derived from classical theory into a machine learning pipeline, demonstrating a powerful synergy between the two approaches. • Most importantly, it employs a state-of-the-art XAI technique (SHAP) to provide a deep, mechanistic interpretation of the model’s decision-making process, revealing the subtle, non-linear correction factors that the model learns to apply to the classical framework. II. MATERIALS AND METHODS A. Experimental Apparatus We designed the experimental setup for replicability and precision based on the Johns Hopkins University Exploring Engineering Innovation (EEI) curriculum. The experimental apparatus, as depicted in Fig. 1, consisted of a digital scale (with a precision of ±0.01 g), standard masking tape, a ruler, and a digital caliper (±0.01 mm). This minimalist setup allows for the direct measurement of the critical buckling force, which is recorded as a mass reading on the digital scale. B. Data Collection Protocol We followed a meticulous protocol to ensure data consistency. • Sample Preparation: 147 Individual strands of pasta were selected and visually inspected for major defects, such as fractures or significant curvature; strands with such defects were discarded. • Geometric Measurement: For each selected strand, we took two sets of measurements using a Mitutoyo digital caliper with a precision of ±0.01 mm. The total length (L) of the strand was measured once. The diameter (d) was measured at three distinct points along the strand (approximately at the 25%, 50%, and 75% marks). We then averaged the three measurements to provide a representative diameter for that sample. • Buckling Test: A single pasta strand was placed vertically on the surface of the digital scale, which was subsequently zeroed. We used a small piece of tape to secure the bottom of the strand to the scale’s surface to prevent slipping. Using an index finger, we applied a slow, steady, and vertically aligned compressive force to the top of the pasta strand. • Load Measurement: We applied a downward force on the member and closely observed both the pasta strand and the reading on the digital scale. The critical buckling point was identified as the maximum mass reading displayed on the scale at the precise moment the pasta strand underwent sudden lateral deformation (buckling). • Calculation: The critical load (P_{cr}) in Newtons was calculated by multiplying the measured mass (in kg) by the standard acceleration due to gravity (9.81 m/s²). This entire process was repeated for all 147 valid samples collected across the experimental groups. C. Dataset Characteristics The final dataset comprised 147 distinct observations from four groups and includes four pasta types: Spaghetti, Angel Hair, Thin Spaghetti, and Vermicelli; These four types span a broad diameter range and introduce controlled manufacturing heterogeneity, thereby improving model generalization and enabling explainability analyses Statistical analysis revealed a well-distributed set of experimental conditions and outcomes, summarized in Table I. D. Computational Methodology Feature Engineering: Raw features included length (m), diameter (m), and a categorical variable pasta_type we designated in our code. We one-hot encoded the pasta_type, creating four binary columns. A physics-informed feature, G, was engineered from Euler’s formula, G = d^4/L^2. This approach aligns with the emerging paradigm of theory-guided data science, where scientific principles are used to guide machine learning models, providing them with a variable that encapsulates the core theoretical interactions [5]. Gradient Boosting Machine (XGBoost): We selected the Extreme Gradient Boosting (XGBoost) algorithm [24], a powerful implementation of the gradient boosting machine (GBM) framework. A GBM builds a predictive model as an ensemble of decision trees, which are trained sequentially to correct the errors of the previous ones by combining the predictions of several models. The overall analysis pipeline was implemented in Python using the imported Scikit-learn library [3]. Model Training and Validation: To ensure robust performance, we employed a 5-fold cross-validation scheme. The dataset of 147 samples was randomly partitioned into 5 folds. We then trained and evaluated the model 5 times, with each fold serving as the test set once. Explainable AI (XAI) with SHAP: To interpret the model, we used SHAP, a game theory-based approach that calculates the contribution of each feature to each individual prediction [4]. III. RESULTS A. Predictive Accuracy of the Gradient Boosting Model The 5-fold cross-validation process provided a robust estimate of the model’s performance on unseen data. Cross-validation allows us to evaluate the model’s generalization capability for unseen data points, while metrics provide information about its predictive accuracy [25]. The averaged metrics were exceptional: • Coefficient of Determination (R²): 0.97 • Root Mean Squared Error (RMSE): 0.14 N An R² value of 0.97 indicates that our model can explain 97% of the variance in the experimental buckling loads. The RMSE of 0.14 N signifies a very low average prediction error, given that measured loads ranged up to 3.22 N. This high accuracy is visualized in the Predicted vs. Actual plot (Fig. 2). B. Model Interpretation with Explainable AI A SHAP summary plot (Fig. 3) provides a comprehensive overview of global feature importance, ranking features by their impact on the model’s predictions. The analysis reveals that the physics-informed feature, G, is the most influential predictor. However, the raw ‘diameter‘ and ‘length‘ features remain highly important, suggesting the model is learning subtle correction factors beyond the scope of the classical formula. C . Walkthrough of a Single Prediction To illustrate how the model makes a prediction, we can examine a single sample from our dataset. Consider a spaghetti strand with a length of 12 cm and a diameter of 1.7 mm. The experimentally measured buckling load for this strand was 1.32 N. The model doesn't use a simple formula. Instead, SHAP analysis reveals how each feature contributes a specific value (a "SHAP value") to push the prediction away from the baseline average prediction (which for our dataset is 0.81 N). For this specific spaghetti strand: The baseline average prediction is 0.81 N. The high value of the physics-informed feature (G) is the most important, providing a SHAP value of +0.45 N. The relatively high diameter provides another push, with a SHAP value of +0.12 N. The moderate length has a small negative impact, with a SHAP value of -0.03 N. By adding up all these individual contributions to the baseline (0.81 + 0.45 + 0.12 - 0.03), the model arrives at a final predicted buckling load of 1.35 N, which is very close to the actual experimental value of 1.32 N. This process is repeated for every sample, with the SHAP values changing based on the specific geometry of the pasta strand. D. Comparative Analysis To place the performance of the XGBoost model in context, we compared it against a Random Forest model and the classical Euler formula, for which we estimated an average Young’s Modulus (E) of approximately 2.9 GPa from the dataset. The results are summarized in Table II. IV. DISCUSSION The results validate our central hypothesis that a hybrid data-driven approach can bridge the gap between idealized theory and experimental reality. The outperformance of the XGBoost model demonstrates the limitations of a purely theoretical approach for non-ideal materials. The success of the physics-informed feature, G, shows that classical theory remains invaluable, providing a strong foundation upon which the machine learning model can learn the complex, messy reality on top of it. This synergy aligns with the principles of Theory-Guided Data Science, which advocates for integrating scientific knowledge into data-driven models to improve performance and interpretability [5]. The most novel contribution is using XAI for scientific insight [8]. The independent importance of “diameter” and “length” is the key finding. It implies the true relationships are more complex than simple power laws. For example, the non-linear importance of ‘diameter‘ may suggest the model is learning about the relationship between a column’s thickness and its resistance to localized crushing at the contact points, an effect not present in Euler’s pure bending theory. Similarly, the independent importance of ‘length‘ could be the model’s way of approximating the increased statistical probability of a critical micro-fracture existing in a longer strand. This transforms the model from an engineering tool into a scientific instrument for generating new, testable hypotheses about the underlying mechanics of the system. A. Broader Implications and Future Vision The framework presented here is a generalizable template with broad implications for fields like additive manufacturing, biomedical engineering, and composite materials science. The vision is an automated XAI pipeline that can accelerate materials discovery by rapidly generating accurate, interpretable models from new experimental data. B. Limitations and Avenues for Future Research The primary limitations are the dataset’s scope (147 samples, single brand) and the lack of environmental controls. Future work should focus on creating a larger, more diverse dataset. Variations in operator reaction time may have also led to slight inaccuracies in recording the mass at the exact moment of buckling. Similarly, the manual application of force with a finger may have introduced minor, off-axis loads that could influence the critical buckling value. While the model is designed to learn the general trend from this noisy data, these factors represent a source of experimental uncertainty that could be reduced in future work with automated testing equipment. A particularly promising avenue is the use of computer vision to automate measurement and, crucially, to identify and quantify surface defects as new input features for the model. V. CONCLUSION This research confronted a foundational challenge in structural mechanics. We have demonstrated that a synergistic framework combining classical theory, experimental data, and explainable machine learning can successfully bridge this gap. Our XGBoost model, informed by the principles of Euler’s formula, predicted the critical buckling load of a non-ideal material with substantially lower error than a calibrated Euler baseline (R^2 = 0.97). The application of XAI provided a deep interpretation of the model’s logic, transforming it from an opaque “black box” into a scientific instrument. This study serves as a robust proof-of-concept, illustrating that the future of modeling complex physical systems lies in the intelligent synthesis of classical theory, experimental data, and the interpretive power of modern artificial intelligence. Declarations ACKNOWLEDGMENT The author would like to thank the instructors and staff of the Johns Hopkins University Explore Engineering Innovation (EEI) program for providing the foundational knowledge and experimental framework for this research. The author would like to thank Lexi Deutsch for providing the experimental setup photograph from the EEI program. References S. P. 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Ding, “Explainable Artificial Intelligence: Precepts, Methods, and Opportunities for Research in Construction,” 2022 (review). İ. H. Kalyoncuoğlu, M. Ö. Sevin, and A. İnce, “Critical buckling load prediction of corrugated steel plate girders using machine learning,” Structural Engineering and Mechanics , 2024. G. E. O. Simitses and D. H. Hodges, Fundamentals of Structural Stability . AIAA, 2006. Z. P. Bažant and L. Cedolin, Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories , 3rd ed. World Scientific, 2010. ASCE/SEI 7-22, Minimum Design Loads and Associated Criteria for Buildings and Other Structures . American Society of Civil Engineers, 2022. CEN, EN 1993-1-1 Eurocode 3: Design of steel structures — General rules and rules for buildings . European Committee for Standardization, latest ed. American Society of Civil Engineers (ASCE), Failure to Act: Economic Impacts of Status Quo Investment Across Infrastructure Sectors , 2021–2022 program. J. M. T. 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Hou, “Asian noodles: History, classification, raw materials, and processing—Impacts on quality attributes,” Comprehensive Reviews in Food Science and Food Safety , vol. 9, no. 2, pp. 154–169, 2010 (and related technical bulletins). J. R. Gladden, N. Z. Handzy, A. Belmonte, and E. Villermaux, “Dynamic buckling and fragmentation in brittle rods,” Physical Review Letters , 94, 035503, 2005. A. Chadha, P. R. Maiti, and R. Chatterjee, “Critical buckling load prediction of Y-braced columns using artificial neural networks,” Int. J. Numer. Methods Eng. , 2021. T. Chen and C. Guestrin, “XGBoost: A Scalable Tree Boosting System,” in KDD ’16 , 2016. A. C. C. Lai, and T. C. P. Cheang, "Predictive modeling of buckling in composite tubes: Integrating artificial neural networks for damage detection," Composite Structures , vol. 273, 114275, 2021. S. Timoshenko and J. M. Gere, Forest Products Laboratory, “Mechanical properties of wood,” in Wood Handbook: Wood as an Engineering Material, FPL–GTR–190, ch. 5. Madison, WI, USA: U.S. Dept. of Agriculture, Forest Service, Forest Products Laboratory, 2021. [Online]. Available: https://www.fpl.fs.usda.gov/documnts/fplgtr/fplgtr190/chapter_05.pdf A. Harmon, V. Khilkevich, and K. M. Donnell, “High permittivity anisotropic 3D printed material,” in Proc. Electromagnetic Compatibility Laboratory, Missouri Univ. of Sci. and Technol., Rolla, MO, USA, 2021. Tables Table I and II are available in the Supplementary Files section. Additional Declarations The authors declare no competing interests. Supplementary Files TABLEIandII.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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08:26:37","extension":"html","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":62474,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7668574/v1/397febbb216b13c7f6839cfb.html"},{"id":92062894,"identity":"cf117baa-5da3-4c36-a671-02778bad20ba","added_by":"auto","created_at":"2025-09-24 08:34:39","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":3053546,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eThe experimental setup, showing a pasta strand positioned vertically on a digital scale, ready for compressive force to be applied by finger.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"editedbucklingspaghettipic1.png","url":"https://assets-eu.researchsquare.com/files/rs-7668574/v1/2d59b0fdaeec47185011fce9.png"},{"id":92062681,"identity":"fb8294ad-a75c-439e-83e2-23180bb4adad","added_by":"auto","created_at":"2025-09-24 08:26:37","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":189067,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003ePredicted vs. Actual critical buckling load (N). The tight clustering along the diagonal line (R²=0.97) demonstrates exceptional accuracy from the XGBoost Model vs. the Experimental Data.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"predictedvsactual.png","url":"https://assets-eu.researchsquare.com/files/rs-7668574/v1/0aa415f001790aa6c34096da.png"},{"id":92062684,"identity":"2507ebad-5417-45a6-8275-610aab36531f","added_by":"auto","created_at":"2025-09-24 08:26:37","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":148274,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eSHAP summary plot showing global feature importance.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"shapsummaryplot.png","url":"https://assets-eu.researchsquare.com/files/rs-7668574/v1/af9d83fb1afaf29b5f66ebd6.png"},{"id":92063426,"identity":"5537f7c2-6943-4348-9557-07821f8e8cd0","added_by":"auto","created_at":"2025-09-24 08:38:19","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4674114,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7668574/v1/3358b632-8425-48e1-8032-8a74cb413d4f.pdf"},{"id":92062893,"identity":"b9f3891b-4c6c-4570-8716-8e14cdc299a8","added_by":"auto","created_at":"2025-09-24 08:34:38","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":14190,"visible":true,"origin":"","legend":"","description":"","filename":"TABLEIandII.docx","url":"https://assets-eu.researchsquare.com/files/rs-7668574/v1/7d1ea4575d6e1e25136f85d7.docx"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eBeyond Euler: An Explainable Machine Learning Framework for Predicting and Interpreting Buckling Instabilities in Non-Ideal Materials\u003c/p\u003e","fulltext":[{"header":"I. INTRODUCTION","content":"\u003cp\u003e\u003cstrong\u003eA. The Engineering Imperative of Stability Analysis\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eCivil and mechanical engineering have long focused on predicting the response of materials to applied forces [10, 11]. The reliability of any structure depends on its ability to remain stable under its prescribed operational loads [12, 13]. Accurate predictions of instabilities are essential to prevent catastrophic collapses, which can incur severe economic and human costs [14\u003cstrong\u003e]\u003c/strong\u003e. Among the various modes of structural failure, the phenomenon of buckling stands out as particularly insidious [15]. Unlike a material slowly yielding under tension, a column undergoing buckling can transition from a state of stable equilibrium to total failure with little to no warning [16]. This is not merely a theoretical concern; it is a real-world failure mode seen in critical infrastructure, from columns in buildings and bridges to sub-sea pipelines [17] and railway tracks that can buckle under the compressive forces of thermal expansion [18]. Accurately predicting the onset of buckling proves itself to be an engineering imperative of the highest order [13].\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eB. The Classical Framework: Euler\u0026rsquo;s Buckling Theory\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eLeonhard Euler achieved the first successful mathematical description of column buckling in 1744. His work established a foundational pillar of structural mechanics and provided an equation that remains central to engineering education and practice [1]. Euler\u0026rsquo;s critical load formula, shown in (1), defines the maximum axial compressive load an ideal column can sustain before it becomes unstable [1].\u003c/p\u003e\n\u003cp\u003eP_{cr} = \\dfrac{\\pi^2 EI}{(KL)^2} (1)\u003c/p\u003e\n\u003cp\u003eTo fully appreciate this equation, it is necessary to understand its constituent terms: P_{cr} (the critical load), E (Young\u0026rsquo;s Modulus of Elasticity), I (Area Moment of Inertia), L (Effective Length), and K (Effective Length Factor).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eC. The Gap Between Ideal Theory and Physical Reality\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe elegance and enduring power of Euler\u0026rsquo;s formula lie in its foundation of mathematical idealism; however, this is also its primary limitation in real-world scenarios. A significant and well-documented \u0026ldquo;reality gap\u0026rdquo; exists because no physical object perfectly satisfies the formula\u0026rsquo;s core assumptions [1]. The key sources of this discrepancy are:\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eMaterial Heterogeneity and Anisotropy:\u003c/strong\u003e Euler\u0026rsquo;s formula assumes the material is homogeneous and isotropic. Many real-world materials, however, do not meet this ideal. For example, natural materials like wood have grain and growth rings, and biological materials like bone have complex, porous internal structures [26]. Similarly, modern engineered materials like 3D-printed polymers exhibit anisotropic properties due to their layered construction [27]. Pasta, as an extruded product, exhibits brittle failure modes inconsistent with the above ideal materials [7, 19].\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eGeometric Imperfections:\u003c/strong\u003e The formula assumes a perfectly straight column. All real objects exhibit minute variations that create eccentricities where the applied load is not perfectly aligned with the column\u0026rsquo;s central axis, inducing bending moments that the ideal theory does not account for [16, 20].\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eDifficulty in Parameter Estimation:\u003c/strong\u003e The formula\u0026rsquo;s accuracy depends critically on the value of Young\u0026rsquo;s Modulus, E. For pasta; this value has been investigated using various methods, yielding a range of results and highlighting the difficulty in characterization [2, 6, 21].\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eNon-Linear Material Behavior:\u003c/strong\u003e The theory assumes linear elasticity. Many materials, including pasta, exhibit non-linear and brittle behavior, failing suddenly without significant prior elastic deformation [7, 22].\u003c/p\u003e\n\u003cp\u003eThese limitations mean that any attempt to use Euler\u0026rsquo;s formula to precisely predict the failure of a real-world object is fraught with uncertainty; indeed, \u003cstrong\u003eadvanced software tools like SAP2000 or ETABS have to be used in order to account for these imperfections and nuances\u003c/strong\u003e. Building on prior studies in the field, Our study focuses on this gap, using pasta as an accessible and illustrative example of a non-ideal material [2, 6, 7].\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eD. A New Paradigm: Data-Driven Modeling with Explainable AI\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTo bridge this reality gap, we turn to a new paradigm: supervised machine learning. The application of machine learning to predict column buckling is an active area of research, with recent studies successfully using models like artificial neural networks for braced columns [23] and corrugated steel girders [9]. Instead of beginning with a theoretical formula, a data-driven approach inverts the process, learning the complex and underlying associations \u0026mdash;such as non-linear interactions between features and the subtle effects of material imperfections\u0026mdash; directly from experimental observations. We hypothesize that a machine learning model, when presented with measurable geometric features, can learn a highly accurate mapping to the experimentally observed buckling load.\u003c/p\u003e\n\u003cp\u003eHigh predictive accuracy alone, however, is not sufficient for scientific inquiry. An AI model that acts as an impenetrable black box offers little new physical insight, as we are unable to understand its decision-making process as well as what specific factors affect the outcome. This brings us to the core novelty of this study: the integration of Explainable AI (XAI), a growing field of study in engineering and computer science [8]. By using state-of-the-art XAI techniques, specifically SHAP (SHapley Additive exPlanations) [4], we aim to \u0026ldquo;open the black box\u0026rdquo; and understand how the model arrives at its predictions.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eE. Statement of Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study makes the following novel contributions:\u003c/p\u003e\n\u003cp\u003e\u0026bull; It provides a rigorous, quantitative demonstration of the limitations of Euler\u0026rsquo;s classical buckling formula when applied to a common, non-ideal material.\u003c/p\u003e\n\u003cp\u003e\u0026bull; It develops and validates a high-performance XGBoost Machine Learning model that predicts the critical buckling load with exceptionally high accuracy (R\u0026sup2;=0.97), outperforming the theoretical model.\u003c/p\u003e\n\u003cp\u003e\u0026bull; It successfully integrates a physics-informed feature derived from classical theory into a machine learning pipeline, demonstrating a powerful synergy between the two approaches.\u003c/p\u003e\n\u003cp\u003e\u0026bull; Most importantly, it employs a state-of-the-art XAI technique (SHAP) to provide a deep, mechanistic interpretation of the model\u0026rsquo;s decision-making process, revealing the subtle, non-linear correction factors that the model learns to apply to the classical framework.\u003c/p\u003e"},{"header":"II. MATERIALS AND METHODS","content":"\u003cp\u003e\u003cstrong\u003eA. Experimental Apparatus\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe designed the experimental setup for replicability and precision based on the Johns Hopkins University Exploring Engineering Innovation (EEI) curriculum. The experimental apparatus, as depicted in Fig. 1, consisted of a digital scale (with a precision of \u0026plusmn;0.01 g), standard masking tape, a ruler, and a digital caliper (\u0026plusmn;0.01 mm). This minimalist setup allows for the direct measurement of the critical buckling force, which is recorded as a mass reading on the digital scale.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eB. Data Collection Protocol\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe followed a meticulous protocol to ensure data consistency.\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eSample Preparation:\u003c/strong\u003e 147 Individual strands of pasta were selected and visually inspected for major defects, such as fractures or significant curvature; strands with such defects were discarded.\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eGeometric Measurement:\u003c/strong\u003e For each selected strand, we took two sets of measurements using a Mitutoyo digital caliper with a precision of \u0026plusmn;0.01 mm. The total length (L) of the strand was measured once. The diameter (d) was measured at three distinct points along the strand (approximately at the 25%, 50%, and 75% marks). We then averaged the three measurements to provide a representative diameter for that sample.\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eBuckling Test:\u003c/strong\u003e A single pasta strand was placed vertically on the surface of the digital scale, which was subsequently zeroed. We used a small piece of tape to secure the bottom of the strand to the scale\u0026rsquo;s surface to prevent slipping. Using an index finger, we applied a slow, steady, and vertically aligned compressive force to the top of the pasta strand.\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eLoad Measurement:\u003c/strong\u003e We applied a downward force on the member and closely observed both the pasta strand and the reading on the digital scale. The critical buckling point was identified as the maximum mass reading displayed on the scale at the precise moment the pasta strand underwent sudden lateral deformation (buckling).\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eCalculation:\u003c/strong\u003e The critical load (P_{cr}) in Newtons was calculated by multiplying the measured mass (in kg) by the standard acceleration due to gravity (9.81 m/s\u0026sup2;).\u003c/p\u003e\n\u003cp\u003eThis entire process was repeated for all 147 valid samples collected across the experimental groups.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eC. Dataset Characteristics\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe final dataset comprised 147 distinct observations from four groups and includes four pasta types: Spaghetti, Angel Hair, Thin Spaghetti, and Vermicelli; These four types span a broad diameter range and introduce controlled manufacturing heterogeneity, thereby improving model generalization and enabling explainability analyses \u0026nbsp;Statistical analysis revealed a well-distributed set of experimental conditions and outcomes, summarized in Table I.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eD. Computational Methodology\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFeature Engineering:\u003c/strong\u003e Raw features included length (m), diameter (m), and a categorical variable pasta_type we designated in our code. We one-hot encoded the pasta_type, creating four binary columns. A physics-informed feature, G, was engineered from Euler\u0026rsquo;s formula, G = d^4/L^2. This approach aligns with the emerging paradigm of theory-guided data science, where scientific principles are used to guide machine learning models, providing them with a variable that encapsulates the core theoretical interactions [5].\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eGradient Boosting Machine (XGBoost):\u003c/strong\u003e We selected the Extreme Gradient Boosting (XGBoost) algorithm [24], a powerful implementation of the gradient boosting machine (GBM) framework. A GBM builds a predictive model as an ensemble of decision trees, which are trained sequentially to correct the errors of the previous ones by combining the predictions of several models. The overall analysis pipeline was implemented in Python using the imported Scikit-learn library [3].\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eModel Training and Validation:\u003c/strong\u003e To ensure robust performance, we employed a 5-fold cross-validation scheme. The dataset of 147 samples was randomly partitioned into 5 folds. We then trained and evaluated the model 5 times, with each fold serving as the test set once.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExplainable AI (XAI) with SHAP:\u003c/strong\u003e To interpret the model, we used SHAP, a game theory-based approach that calculates the contribution of each feature to each individual prediction [4].\u003c/p\u003e"},{"header":"III. RESULTS","content":"\u003cp\u003e\u003cstrong\u003eA. Predictive Accuracy of the Gradient Boosting Model\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe 5-fold cross-validation process provided a robust estimate of the model\u0026rsquo;s performance on unseen data. Cross-validation allows us to evaluate the model\u0026rsquo;s generalization capability for unseen data points, while metrics provide information about its predictive accuracy [25]. The averaged metrics were exceptional:\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eCoefficient of Determination (R\u0026sup2;):\u003c/strong\u003e 0.97\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eRoot Mean Squared Error (RMSE):\u003c/strong\u003e 0.14 N\u003c/p\u003e\n\u003cp\u003eAn R\u0026sup2; value of 0.97 indicates that our model can explain 97% of the variance in the experimental buckling loads. The RMSE of 0.14 N signifies a very low average prediction error, given that measured loads ranged up to 3.22 N. This high accuracy is visualized in the Predicted vs. Actual plot (Fig. 2).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eB. Model Interpretation with Explainable AI\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eA SHAP summary plot (Fig. 3) provides a comprehensive overview of global feature importance, ranking features by their impact on the model\u0026rsquo;s predictions. The analysis reveals that the physics-informed feature, G, is the most influential predictor. However, the raw \u0026lsquo;diameter\u0026lsquo; and \u0026lsquo;length\u0026lsquo; features remain highly important, suggesting the model is learning subtle correction factors beyond the scope of the classical formula.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eC\u003c/strong\u003e. Walkthrough of a Single Prediction\u003c/p\u003e\n\u003cp\u003eTo illustrate how the model makes a prediction, we can examine a single sample from our dataset. Consider a spaghetti strand with a length of 12 cm and a diameter of 1.7 mm. The experimentally measured buckling load for this strand was 1.32 N. The model doesn\u0026apos;t use a simple formula. Instead, SHAP analysis reveals how each feature contributes a specific value (a \u0026quot;SHAP value\u0026quot;) to push the prediction away from the baseline average prediction (which for our dataset is 0.81 N). For this specific spaghetti strand:\u003c/p\u003e\n\u003cp\u003eThe baseline average prediction is 0.81 N.\u003c/p\u003e\n\u003cp\u003eThe high value of the physics-informed feature (G) is the most important, providing a SHAP value of +0.45 N.\u003c/p\u003e\n\u003cp\u003eThe relatively high diameter provides another push, with a SHAP value of +0.12 N.\u003c/p\u003e\n\u003cp\u003eThe moderate length has a small negative impact, with a SHAP value of -0.03 N.\u003c/p\u003e\n\u003cp\u003eBy adding up all these individual contributions to the baseline (0.81 + 0.45 + 0.12 - 0.03), the model arrives at a final predicted buckling load of 1.35 N, which is very close to the actual experimental value of 1.32 N. This process is repeated for every sample, with the SHAP values changing based on the specific geometry of the pasta strand.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eD. Comparative Analysis\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTo place the performance of the XGBoost model in context, we compared it against a Random Forest model and the classical Euler formula, for which we estimated an average Young\u0026rsquo;s Modulus (E) of approximately 2.9 GPa from the dataset. The results are summarized in Table II.\u003c/p\u003e"},{"header":"IV. DISCUSSION","content":"\u003cp\u003eThe results validate our central hypothesis that a hybrid data-driven approach can bridge the gap between idealized theory and experimental reality. The outperformance of the XGBoost model demonstrates the limitations of a purely theoretical approach for non-ideal materials. The success of the physics-informed feature, G, shows that classical theory remains invaluable, providing a strong foundation upon which the machine learning model can learn the complex, messy reality on top of it. This synergy aligns with the principles of Theory-Guided Data Science, which advocates for integrating scientific knowledge into data-driven models to improve performance and interpretability [5].\u003c/p\u003e\n\u003cp\u003eThe most novel contribution is using XAI for scientific insight [8]. The independent importance of \u0026ldquo;diameter\u0026rdquo; and \u0026ldquo;length\u0026rdquo; is the key finding. It implies the true relationships are more complex than simple power laws. For example, the non-linear importance of \u0026lsquo;diameter\u0026lsquo; may suggest the model is learning about the relationship between a column\u0026rsquo;s thickness and its resistance to localized crushing at the contact points, an effect not present in Euler\u0026rsquo;s pure bending theory. Similarly, the independent importance of \u0026lsquo;length\u0026lsquo; could be the model\u0026rsquo;s way of approximating the increased statistical probability of a critical micro-fracture existing in a longer strand. This transforms the model from an engineering tool into a scientific instrument for generating new, testable hypotheses about the underlying mechanics of the system.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eA. Broader Implications and Future Vision\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe framework presented here is a generalizable template with broad implications for fields like additive manufacturing, biomedical engineering, and composite materials science. The vision is an automated XAI pipeline that can accelerate materials discovery by rapidly generating accurate, interpretable models from new experimental data.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eB. Limitations and Avenues for Future Research\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe primary limitations are the dataset\u0026rsquo;s scope (147 samples, single brand) and the lack of environmental controls. Future work should focus on creating a larger, more diverse dataset. Variations in operator reaction time may have also led to slight inaccuracies in recording the mass at the exact moment of buckling. Similarly, the manual application of force with a finger may have introduced minor, off-axis loads that could influence the critical buckling value. While the model is designed to learn the general trend from this noisy data, these factors represent a source of experimental uncertainty that could be reduced in future work with automated testing equipment. A particularly promising avenue is the use of computer vision to automate measurement and, crucially, to identify and quantify surface defects as new input features for the model.\u003c/p\u003e"},{"header":"V. CONCLUSION","content":"\u003cp\u003eThis research confronted a foundational challenge in structural mechanics. We have demonstrated that a synergistic framework combining classical theory, experimental data, and explainable machine learning can successfully bridge this gap. Our XGBoost model, informed by the principles of Euler\u0026rsquo;s formula, predicted the critical buckling load of a non-ideal material with substantially lower error than a calibrated Euler baseline (R^2\u0026thinsp;=\u0026thinsp;0.97). The application of XAI provided a deep interpretation of the model\u0026rsquo;s logic, transforming it from an opaque \u0026ldquo;black box\u0026rdquo; into a scientific instrument. This study serves as a robust proof-of-concept, illustrating that the future of modeling complex physical systems lies in the intelligent synthesis of classical theory, experimental data, and the interpretive power of modern artificial intelligence.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eACKNOWLEDGMENT\u003c/h2\u003e\u003cp\u003eThe author would like to thank the instructors and staff of the Johns Hopkins University Explore Engineering Innovation (EEI) program for providing the foundational knowledge and experimental framework for this research. The author would like to thank Lexi Deutsch for providing the experimental setup photograph from the EEI program.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eS. P. Timoshenko and J. M. Gere, \u003cem\u003eTheory of Elastic Stability\u003c/em\u003e, 2nd ed. New York: McGraw\u0026ndash;Hill, 1961.\u003c/li\u003e\n\u003cli\u003eM. Vemareddy, S. Bandhari, and P. 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Hodges, \u003cem\u003eFundamentals of Structural Stability\u003c/em\u003e. AIAA, 2006.\u003c/li\u003e\n\u003cli\u003eZ. P. Bažant and L. Cedolin, \u003cem\u003eStability of Structures: Elastic, Inelastic, Fracture and Damage Theories\u003c/em\u003e, 3rd ed. World Scientific, 2010.\u003c/li\u003e\n\u003cli\u003eASCE/SEI 7-22, \u003cem\u003eMinimum Design Loads and Associated Criteria for Buildings and Other Structures\u003c/em\u003e. American Society of Civil Engineers, 2022.\u003c/li\u003e\n\u003cli\u003eCEN, \u003cem\u003eEN 1993-1-1 Eurocode 3: Design of steel structures \u0026mdash; General rules and rules for buildings\u003c/em\u003e. European Committee for Standardization, latest ed.\u003c/li\u003e\n\u003cli\u003eAmerican Society of Civil Engineers (ASCE), \u003cem\u003eFailure to Act: Economic Impacts of Status Quo Investment Across Infrastructure Sectors\u003c/em\u003e, 2021\u0026ndash;2022 program.\u003c/li\u003e\n\u003cli\u003eJ. M. T. 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Gere, Forest Products Laboratory, \u0026ldquo;Mechanical properties of wood,\u0026rdquo; in Wood Handbook: Wood as an Engineering Material, FPL\u0026ndash;GTR\u0026ndash;190, ch. 5. Madison, WI, USA: U.S. Dept. of Agriculture, Forest Service, Forest Products Laboratory, 2021. [Online]. Available: https://www.fpl.fs.usda.gov/documnts/fplgtr/fplgtr190/chapter_05.pdf\u003c/li\u003e\n\u003cli\u003eA. Harmon, V. Khilkevich, and K. M. Donnell, \u0026ldquo;High permittivity anisotropic 3D printed material,\u0026rdquo; in Proc. Electromagnetic Compatibility Laboratory, Missouri Univ. of Sci. and Technol., Rolla, MO, USA, 2021.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Tables","content":"\u003cp\u003eTable I and II are available in the Supplementary Files section.\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"University of Connecticut","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"explainable AI (XAI), machine learning, structural mechanics, Extreme Gradient Boosting (XGBoost), buckling, materials science, physics-informed ML, SHAP","lastPublishedDoi":"10.21203/rs.3.rs-7668574/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7668574/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003ePredicting structural failure is a fundamental objective in materials science and mechanical engineering. Euler\u0026rsquo;s classical formula, the standard for predicting the buckling instability of slender columns for over 250 years, assumes idealized material properties that can lead to unreliable predictions and potentially catastrophic failures in critical infrastructure. This study proposes a solution by introducing a novel framework that synergizes machine learning and modern explainability techniques to model complex physical systems. We used pasta as a model non-ideal material for a comprehensive experimental analysis and a dataset from 147 controlled buckling experiments on four distinct pasta gauges. We then developed a physics-informed XGBoost model, incorporating both raw geometric measurements and a composite feature derived from Euler\u0026rsquo;s formula (G\u0026thinsp;=\u0026thinsp;d4/L2), and subsequently evaluated the model\u0026rsquo;s performance using a 5-fold cross-validation scheme. The model demonstrated an outstanding predictive power, achieving an average coefficient of determination (R\u0026sup2;) of 0.97 and a Root Mean Squared Error (RMSE) of 0.14 N. We also examined the model\u0026rsquo;s internal decision-making process by employing SHAP (SHapley Additive exPlanations). The analysis confirmed the primary importance of the theoretically-derived feature but also revealed that the model learned to use raw geometric data as crucial correction factors. This study presents a powerful proof of concept for using interpretable machine learning not only to achieve predictive accuracy but also gain deeper physical insights into complex, non-ideal systems. The framework presented has broad implications for advancing our understanding and design capabilities in materials science, engineering, and advanced manufacturing.\u003c/p\u003e","manuscriptTitle":"Beyond Euler: An Explainable Machine Learning Framework for Predicting and Interpreting Buckling Instabilities in Non-Ideal Materials","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-24 08:26:33","doi":"10.21203/rs.3.rs-7668574/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d88b6006-bcfa-4ce6-8807-ad739ed92e90","owner":[],"postedDate":"September 24th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":55080706,"name":"Artificial Intelligence and Machine Learning"}],"tags":[],"updatedAt":"2025-09-24T08:26:33+00:00","versionOfRecord":[],"versionCreatedAt":"2025-09-24 08:26:33","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7668574","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7668574","identity":"rs-7668574","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}