Beyond Тraining Data Volume: Predicting Nonlinear Effects in Catalytic Reactions with Multiple Kinetic Regimes via Machine Learning

Beyond Тraining Data Volume: Predicting Nonlinear Effects in Catalytic Reactions with Multiple Kinetic Regimes via Machine Learning | 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 Beyond Тraining Data Volume: Predicting Nonlinear Effects in Catalytic Reactions with Multiple Kinetic Regimes via Machine Learning Alexander Schmidt, Anna Kurokhtina, Elizaveta Larina, Nadezhda Lagoda This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7755576/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 While data scarcity is often viewed as the primary obstacle to developing predictive machine learning (ML) models in chemistry, we demonstrate that data “sufficiency“ alone is inadequate for reactions exhibiting nonlinear kinetics. These phenomena are common in complex reactions, including catalytic ones. Therefore, such systems can possess multiple stable steady states and distinct kinetic regimes. We show that if training data fail to encompass this full dynamic diversity, ML models cannot achieve high predictive accuracy, as their predictions are fundamentally constrained to the regimes represented in the training set. This work underscores a critical limitation of ML for complex reaction modeling: the imperative for data completeness over mere volume to avoid failures in prediction. Physical sciences/Chemistry/Catalysis/Catalytic mechanisms Physical sciences/Chemistry/Theoretical chemistry/Reaction mechanisms Physical sciences/Chemistry/Physical chemistry/Reaction kinetics and dynamics machine learning big data prediction catalysis kinetics Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 1 Introduction In recent years, machine learning (ML) methods have gained significant traction for describing, analyzing, and predicting behavior in catalytic reaction systems. These approaches have been successfully applied to diverse scientific challenges, including the identification of optimal catalysts and reaction conditions to maximize yield and selectivity [ 1 – 4 ]. A fundamental prerequisite for implementing ML in catalysis is the availability of high-quality training datasets. Although no universal criteria exist to define dataset suitability, it is generally accepted that such datasets must be sufficiently large and encompass a broad spectrum of reaction conditions — whether obtained experimentally or through in silico methods [ 5 – 7 ]. A critical limitation of ML models lies in their constrained predictive capability: they typically deliver reliable accuracy only within the parameter space defined by the training data. While proficient at interpolation, these models offer no guarantees for extrapolation beyond the trained domain [ 5 , 8 , 9 ]. This limitation often arises from shifts in reaction mechanisms under conditions outside the training set [ 10 ]. The situation becomes particularly complex in nonlinear reaction systems exhibiting multiple steady states or multiple kinetic regimes. In such systems, even within the trained parameter space, the system may undergo sudden transitions — or “switches” — between distinct kinetic regimes due to inherent nonlinearities. Such behavior is especially relevant in catalysis, where off-cycle processes like catalyst activation or deactivation often introduce strong nonlinearities [ 8 , 11 – 16 ]. The presence of multiple catalytically active species further amplifies this complexity, consistent with the “catalyst cocktail” concept [ 17 , 18 ]. From a kinetic perspective, these systems can support multiple steady states and exhibit different regimes of kinetic behavior [ 19 – 21 ]. Transitions between them — known as bifurcations [ 19 ] — can lead to fundamentally different kinetic outcomes (e.g., in activity, selectivity, or stability) under specific initial conditions. In this work, we show that if a training dataset fails to represent all accessible steady states and kinetic regimes of a reaction, the resulting ML model will be inherently limited in predictive performance — regardless of whether the dataset is “big” or “small”. 2 Methods 2.1 Kinetic Model of a Catalytic System with Nonlinear Kinetics A well-known example of a nonlinear system exhibiting different kinetic regimes for reaching a steady state was presented in Lotka's original paper [ 22 ] (now commonly referred to as the Lotka-Volterra model). This system described consecutive chemical transformations involving one autocatalytic step. Although the original Lotka system did not include catalytic transformations, its characteristic properties remain unchanged even when steps representing catalytic cycles are incorporated, provided they do not influence the rates of the steps in the Lotka-Volterra model [ 11 ]. As a representative example of a real catalytic system potentially exhibiting two different kinetic regimes for attaining a stable steady state, we considered Pd-catalyzed cross-coupling reactions, which are highly attractive methodologies in fine organic synthesis [ 23 – 25 ]. The catalytic systems for cross-coupling reactions consist of the following processes, which form the corresponding dynamic subsystems (Scheme 1 ): A pseudo-elementary step for the formation of active Pd(0) species from the catalyst precursor (here, Pd(II) complexes [ 26 – 29 ]), labeled as step (I). Catalytic conversion of the substrate S to the reaction product P, represented by the pseudo-elementary step (II) simulating the catalytic cycle. Deactivation of active Pd(0) into inactive colloidal species — soluble Pd nanoclusters Pd(0)ₙ — involving pseudo-elementary steps of particle nucleation and its autocatalytic growth (steps (III) and (IV), respectively) [ 16 , 30 ]. A pseudo-elementary step for the formation of aggregated Pd black as a result of the terminal deactivation of Pd (step (V)) [ 11 , 29 , 30 ]. Here, a pseudo-elementary step is defined as the sum of multiple elementary steps that can, however, be treated kinetically as a single elementary step [ 31 , 32 ]. The key distinctions from the classical Lotka model are: (i) the presence of step (II), which represents the catalytic transformation, and (ii) the Pd nanocluster nucleation step (III). These features make the system under consideration more representative of the real process of nanocluster formation [ 31 , 32 ]. Owing to the nonlinear autocatalytic formation of Pd(0)ₙ particles (Step IV), the reaction system presented in Scheme 1 is characterized by two distinct kinetic regimes for reaching a steady state, with a bifurcation between them. This bistable behavior is a well-known feature of nonlinear systems, regardless of whether they are catalytic or not. The bifurcation point is defined by a specific ratio of the rate constants and the concentrations of the reacting species [ 21 ]. To derive a criterion for predicting which of the two kinetic regimes will be established, the following system of ordinary differential equations (ODEs), corresponding to the reaction steps (I)-(V), should be considered: \(\:\frac{d\left[Pd\left(II\right)\right]}{dt}=\:-{k}_{1}\left[Pd\left(II\right)\right]\) $$\:\frac{d\left[Pd\left(0\right)\right]}{dt}=\:{k}_{1}\left[Pd\left(II\right)\right]-{k}_{3}\left[Pd\left(0\right)\right]-{k}_{4}\left[Pd\left(0\right)\right]\left[{Pd\left(0\right)}_{n}\right]$$ 1 $$\:\frac{d\left[{Pd\left(0\right)}_{n}\right]}{dt}=\:{k}_{3}\left[Pd\left(0\right)\right]+{k}_{4}\left[Pd\left(0\right)\right]\left[{Pd\left(0\right)}_{n}\right]-{k}_{5}\left[{Pd\left(0\right)}_{n}\right]$$ $$\:\frac{d\left[P\right]}{dt}=\:{k}_{2}\left[Pd\left(0\right)\right]$$ The dynamic behavior of the system outlined in Scheme 1 , governed by the differential equation system (1), is expected to closely resemble that of the classical Lotka system. This similarity arises from the hierarchical organization wherein catalyst transformations constitute the "core" system, while the catalytic conversion of S to P occurs within a "subordinate" system. Under the condition where k 3 ≪ k 4 [ Pd(0)ₙ ] (indicating that the nucleation rate of Pd(0)ₙ is significantly slower than its autocatalytic growth rate [ 33 ]), the bifurcation point can be identified through Lyapunov stability analysis of system (1) [ 13 , 19 ]. The critical transition between the two distinct kinetic regimes of steady state attainment is defined by the following expression: $$\:\frac{{k}_{1}\left[Pd\left(II\right)\right]{k}_{4}}{{k}_{5}^{2}}=4$$ 2 The stability of this ratio is sensitive to two key parameters: the initial concentration of the Pd(II) precursor and the rate of its consumption. The latter can be quantitatively characterized by the rate constant k 1 , which describes the reduction to active Pd(0) (Step (I) in Scheme 1 ). To map the kinetic behavior across different steady-state regimes, we performed a parameter scan across two distinct ranges of initial Pd(II) concentrations combined with variations in k 1 values. This approach successfully revealed two well-defined kinetic regimes: 1. Oscillatory regime: When the dimensionless parameter \(\:{k}_{1}\left[Pd\left(II\right)\right]{k}_{4}/{k}_{5}^{2}<4\) , the system exhibited damped oscillations in the concentrations of both active Pd(0) and colloidal Pd(0)ₙ clusters during reaction progress (Fig. 1 a, b). Such oscillatory behavior in catalytic species concentrations — affecting both active and inactive forms — represents a well-established phenomenon in nonlinear catalytic systems [ 34 ]. Notably, recent developments in physics-informed neural networks have demonstrated the capability of ML approaches to successfully predict such complex dynamic behavior [ 35 ], suggesting their potential applicability for modeling oscillatory kinetic data. 2. Non-Oscillatory regime: When \(\:{k}_{1}\left[Pd\left(II\right)\right]{k}_{4}/{k}_{5}^{2}>4\) , the system displayed monotonic convergence to steady state without oscillatory behavior in any Pd species concentration (Fig. 1 d, e). 2.2. Dataset construction To encompass the reaction conditions leading to the two distinct kinetic regimes (oscillatory and non-oscillatory) in the system shown in Scheme 1 , kinetic simulations were conducted across specified ranges of initial Pd(II) concentration ([Pd(II)] 0 ) and the reduction rate constant k 1 . Variation in k 1 serves as a proxy for altering the reducibility of the Pd(II) precursor. The following parameter spaces were explored: Oscillatory regime : [Pd(II)] 0 = 0.01–1 mM; k 1 = 0.01–0.4 s⁻¹ (10 values per parameter, evenly spaced). Non-Oscillatory regime : [Pd(II)] 0 = 1–10 mM; k 1 = 3–10 s⁻¹ (10 values per parameter, evenly spaced). For all simulations, the following initial conditions and rate constants were maintained constant: Initial concentrations: [P] 0 =[Pd(0)] 0 =[Pd(0) n ] 0 =[Pd black ] 0 = 0 mM, [S] 0 = 10 mM; Rate constants: k 2 = 0.5 mM − 1 ·s − 1 , k 3 = 1·10 − 7 s − 1 , k 4 = 100 mM − 1 ·s − 1 , k 5 = 10 s − 1 . The temporal evolution of concentrations for product P and all catalytic species (Pd(II), Pd(0), Pd(0)ₙ, Pd black) was simulated over a 100 s period, with data points recorded at 5 s intervals. All numerical integrations were performed using GEPASI software [ 36 ]. Characteristic kinetic profiles illustrating both dynamical regimes are presented in Fig. 1 . The simulated kinetic data served as the foundation for constructing training and test datasets for the subsequent ML analysis. The training dataset incorporated the initial parameters — namely, the Pd(II) precursor concentration ([Pd(II)] 0 ) and the reduction rate constant k 1 — together with the corresponding product concentration profile [P] t (Scheme 1 ) across 21 time points (including t = 0) per each numerical experiment. To maintain a complete representation of the reaction system's initial state, each entry in the dataset was constructed to include [Pd(II)] 0 and k 1 . Preserving these parameters is essential, as the system's dynamic behavior and its evolution through kinetic parameter space are fundamentally governed by its initial conditions [ 19 ]. Consistent with practical experimental limitations, the dataset excluded real-time concentrations of catalytic intermediates such as [Pd(II)] t , [Pd(0)] t , [Pd(0) n ] t , [Pd black ] t . While advanced analytical techniques theoretically enable tracking of certain catalyst species, achieving reliable in situ monitoring under operational catalytic conditions remains challenging and is rarely accomplished. Furthermore, critical intermediates often exist at concentrations below the detection limits of standard methodologies. To verify the robustness of this approach, supplementary simulations incorporating full intermediate concentration data were conducted. The resulting ML predictions proved consistent with those derived from the practically constrained dataset, affirming that the exclusion of intermediate species does not affect the principal conclusions of this investigation. By systematically sampling the predefined ranges of [Pd(II)] 0 and k 1 , two comprehensive datasets were generated: one consisting of 10,000 numerical experiments corresponding to the oscillatory regime, and another of equal size representing the non-oscillatory regime. Both of these datasets, corresponding to the oscillatory and non-oscillatory regimes, were used to train three distinct ML models (described subsequently). Prior to training, five kinetic experiments were systematically extracted from each dataset. This curated subset, comprising 10 numerical experiments (210 observations total), was assembled into a unified test set. This specific construction was designed to rigorously evaluate the ability of the trained models to generalize across different kinetic regimes — that is, to make accurate predictions on data originating from a regime fundamentally different from that of their training data. 2.3 ML models training and test ML approaches — particularly decision trees, random forest, and neural networks — have proven effective in modeling complex catalytic reaction systems [ 6 , 35 , 37 ]. Tree-based models (decision trees and random forest) often achieve strong predictive performance with computational efficiency, yet their inherent piecewise-constant nature limits their ability to generate high-fidelity continuous predictions. In contrast, neural networks offer greater representational flexibility but require substantially longer training times and larger datasets. Our previous research demonstrated that decision trees and random forest outperform neural networks when trained on limited kinetic data derived from real catalytic experiments [ 38 ]. Motivated by these findings, we selected these three model classes to rigorously evaluate their capability in predicting behavior in complex reaction systems characterized by nonlinear kinetics and bifurcation phenomena. Model training was conducted in MATLAB [ 39 ] utilizing the Regression Learner application with 5-fold cross-validation. The concentration of the reaction product [P] at time t ([P] t ) served as the target output variable for all predictive modeling. Following training, all models underwent rigorous evaluation using a comprehensive test set incorporating kinetic data from both oscillatory and non-oscillatory regimes. To assess model robustness across different dynamical conditions, three distinct training strategies were implemented: exclusive training on oscillatory regime data, exclusive training on non-oscillatory regime data, and training on a combined dataset encompassing both dynamical regimes. Additionally, Principal Component Analysis (PCA) was performed using the ExStatR extension (version 1.2) [ 40 ] for Microsoft Excel, leveraging the computational capabilities of the R statistical environment. 2.4 Use of Large Language Model While preparing this work, the authors used DeepSeek (open AI) to assist in language refinement and editing of this manuscript. The AI tool was used strictly as a language assistant. The authors take full responsibility for the manuscript’s content. 3 Results and Discussion From a kinetic standpoint, it is reasonable to hypothesize that a model trained exclusively on data from a single steady state attainment regime will inherently lack the capacity to predict kinetic behavior under alternative dynamical regimes, even if the training dataset is extensively sampled within its original regime. This fundamental limitation arises from the model's inability to capture bifurcation phenomena and transitions between distinct kinetic regimes. To quantitatively validate this hypothesis, we conducted a systematic comparison of the predictive performance of models trained on three distinct data configurations, evaluating their generalization capabilities across oscillatory and non-oscillatory regimes. The performance metrics for all models following training and testing are comprehensively presented in Table 1 . Models trained exclusively on oscillatory regime data (Section 1 in Table 1 ) demonstrated strong predictive accuracy during validation; however, their performance deteriorated markedly when evaluated against the combined test set containing both oscillatory and non-oscillatory kinetic data. Close examination of prediction parity plots (Fig. 2 ) revealed that all models trained exclusively on oscillatory regime data achieved satisfactory accuracy for test data derived from the oscillatory regime but consistently failed to generate meaningful predictions for non-oscillatory regime data. This asymmetric generalization pattern was also observed in models trained exclusively on non-oscillatory data ((Section 1 in Table 1 and Fig. 3 ), which maintained accuracy within their training domain but exhibited dramatic performance degradation when applied to oscillatory regime data. Table 1 Generalization performance of ML models across kinetic regimes: comparative evaluation of training strategies using oscillatory, non-oscillatory, and combined datasets. Section No. Model type Hyperparameters Training dataset Test dataset R 2 MAE R 2 MAE 1. Training using the oscillatory dataset Random forest Minimum leaf size: 8 Number of learners: 30 0.9996 0.0310 -0.0760 1.5500 Fine Tree Minimum leaf size: 4 Surrogate decision splits: Off 0.9990 0.0440 -0.0807 1.5958 Wide Neural Network Number of fully connected layers: 1 First layer size: 100 Activation: ReLU Iteration limit: 1000 Regularization strength (Lambda): 0 Standardize data: Yes 0.9984 0.06646 -293.40 26.0910 2. Training using the non-oscillatory dataset Random forest Minimum leaf size: 8 Number of learners: 30 0.9997 0.0290 -3.3305 3.2959 Fine Tree Minimum leaf size: 4 Surrogate decision splits: Off 0.9989 0.0542 -3.3293 3.3033 Wide Neural Network Number of fully connected layers: 1 First layer size: 100 Activation: ReLU Iteration limit: 1000 Regularization strength (Lambda): 0 Standardize data: Yes 0.9999 0.0166 -1.5317 2.5107 3. Training using the combined dataset(oscillatory and non-oscillatory data) Random forest Minimum leaf size: 8 Number of learners: 30 0.9996 0.0375 0.9958 0.0979 Fine Tree Minimum leaf size: 4 Surrogate decision splits: Off 0.9988 0.0587 0.9899 0.1523 Wide Neural Network Number of fully connected layers: 1 First layer size: 100 Activation: ReLU Iteration limit: 1000 Regularization strength (Lambda): 0 Standardize data: Yes 0.9942 0.1134 0.7155 0.8060 The results presented in Sections 1 and 2 in Table 1 and Figs. 2 – 3 demonstrate that ML models trained exclusively on either oscillatory or non-oscillatory regime data exhibit strong predictive performance within their training regime but show dramatic failure when applied to the alternative kinetic regime. This fundamental limitation highlights the inherent constraint of models developed within a single kinetic paradigm when confronted with divergent steady state attainment behaviors. These findings logically indicate that robust predictive performance across both regimes requires incorporating training data representing both oscillatory and non-oscillatory conditions. Indeed, when training datasets were combined to include data from both kinetic regimes, all three ML model architectures showed substantially improved generalization capabilities (Section 3 in Table 1 ). Consistent with our previous findings, tree-based methods (random forest and decision trees) significantly outperformed neural network architectures across all performance metrics. Further analysis of the parity plots in Fig. 4 reveals additional insights: while neural networks failed to achieve satisfactory predictive accuracy for either regime, tree-based methods demonstrated superior overall performance. However, close examination of these results reveals important limitations — the decision tree predictions (Fig. 4 b) exhibit extended horizontal segments where [P] t values remain essentially constant despite variations in predictor values. Similar, though less pronounced, patterns are observable in the random forest predictions (Fig. 4 a). Notably, these horizontal segments correspond predominantly to oscillatory regime conditions, suggesting that capturing oscillatory kinetics presents particular challenges even when using combined training data. This observation indicates that while data combination significantly improves model performance, certain nonlinear characteristics of oscillatory behavior remain difficult to capture with current ML approaches. 4 Conclusions The results fundamentally reshape our understanding of machine learning for the reactions possessing nonlinear kinetics, including catalytic ones: satisfactory predictive performance depends not only on dataset size, but on whether training data comprehensively captures all possible kinetic regimes. When models encounter kinetic behavior outside their training distribution — even within the same parameter ranges — predictive accuracy collapses completely. This limitation extends universally across nonlinear systems exhibiting multiple stable steady states and distinct kinetic regimes. The inherent autocatalytic nature of active species formation/deactivation in cross-coupling reactions [ 16 , 29 , 30 , 41 ] creates complex nonlinearities that demand exceptional representation in training data. Merely accumulating more data points proves insufficient; what matters is covering the full behavioral spectrum. Our findings reveal a critical dual challenge: beyond traditional extrapolation concerns [ 5 , 8 , 9 ], we identify an interpolation susceptibility where models fail even within trained parameter ranges. Specifically, when predictor combinations ([Pd(II)] 0 and k 1 in Eq. ( 2 )) yield previously unobserved kinetic regimes, predictions fail despite all individual predictor values falling within training bounds [ 42 ]. This represents a profound limitation—not extrapolation in the conventional sense, but a failure to capture emergent behaviors from predictor interactions. Therefore, for such processes, while using extensive data is necessary, it is insufficient to ensure coverage of all possible reaction steady states and kinetic regimes. Training data needs to capture the entire dynamic landscape of kinetic regimes, going beyond the simple numerical ranges of its input parameters. To overcome the challenges identified in modeling nonlinear catalytic systems, we propose a dual strategy combining unsupervised learning with systematic error analysis: First, unsupervised methods like PCA offer theoretical promise for visualizing nonlinear kinetic patterns due to their sensitivity to data structure [ 43 ] and provide delineating distinct kinetic regimes. However, our application of PCA to the combined dataset (containing both oscillatory and non-oscillatory experiments) yielded insufficient separation between regimes (Fig. 5 ), indicating that reliable identification requires supplementary domain knowledge beyond what PCA alone can provide. As a complementary strategy, we recommend a systematic analysis of prediction anomalies revealed in parity plots. By analyzing clusters of poorly predicted points — particularly those exhibiting systematic patterns — researchers can identify regions in parameter space where specific kinetic regimes emerge due to nonlinear effects. Critical to this approach is preserving temporal continuity: test datasets should incorporate complete reaction profiles (from initiation to completion) rather than isolated measurements. Maintaining chronological integrity reveals coherent patterns in prediction errors that reflect the system’s dynamic evolution, as clearly evidenced in Figs. 2 and 3 . These patterns, when analyzed collectively rather than as random noise (experimental data errors), provide actionable insights into model limitations and guide strategic data collection. 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Supplementary Files kineticmodel.txt kinetic_model Dataset1.xlsx Dataset 1 Scheme1.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. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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18:46:42","extension":"html","order_by":53,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":145564,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/54bf7357fe493183c59950c2.html"},{"id":94134935,"identity":"701e502e-a2e0-4827-b906-537a4ed708c8","added_by":"auto","created_at":"2025-10-22 18:46:41","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":102922,"visible":true,"origin":"","legend":"\u003cp\u003eRepresentative kinetic profiles for the species involved in the reaction system shown in Scheme 1. Panels (a)-(c) correspond to the oscillatory regime ([Pd(II)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e = 1 mM, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e = 0.01 s\u003csup\u003e-1\u003c/sup\u003e) while panels (d)-(f) illustrate the non-oscillatory regime ([Pd(II)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e = 2 mM, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e = 3 s\u003csup\u003e-1\u003c/sup\u003e).\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/c84d1d134e03e55fd9bd42a4.png"},{"id":94134929,"identity":"ea4e7c6a-7dfd-4bcf-9087-dc3d06ecd379","added_by":"auto","created_at":"2025-10-22 18:46:41","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":76205,"visible":true,"origin":"","legend":"\u003cp\u003eParity plots comparing predicted versus actual product concentrations ([P]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) for the test dataset using models trained exclusively on oscillatory regime data: (a) random forest, (b) decision tree, and (c) wide neural network.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/e0cfd56661a02dbbc1c1471e.png"},{"id":94134900,"identity":"3cbbbb9d-575b-4351-8de4-71eca802014b","added_by":"auto","created_at":"2025-10-22 18:46:39","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":75819,"visible":true,"origin":"","legend":"\u003cp\u003eParity plots comparing predicted versus actual product concentrations ([P]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) for the test dataset using models trained exclusively on non-oscillatory regime data: (a) random forest, (b) decision tree, and (c) wide neural network.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/f2025e0be7d8352c1dea332f.png"},{"id":94134905,"identity":"6fc507b8-9c65-43c7-b489-e7b218c09ec2","added_by":"auto","created_at":"2025-10-22 18:46:39","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":71707,"visible":true,"origin":"","legend":"\u003cp\u003eParity plots comparing predicted versus actual product concentrations ([P]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) for the test dataset using models trained on combined dataset including oscillatory and non-oscillatory regime data: (a) random forest, (b) decision tree, and (c) wide neural network.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/505df3598c38404d5874e90b.png"},{"id":94135516,"identity":"14443036-1a6e-4d10-97ff-b8d9d7ed8c34","added_by":"auto","created_at":"2025-10-22 18:54:39","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":91027,"visible":true,"origin":"","legend":"\u003cp\u003ePCA of the combined training dataset including oscillatory and non-oscillatory regime data. Highlighted points represent numerical experiments exhibiting oscillatory behavior.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/866a8087f6aa08a1fc036bdc.png"},{"id":94135521,"identity":"5f95c02e-9406-4197-9f3e-26d3220da454","added_by":"auto","created_at":"2025-10-22 18:54:46","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1086972,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/384a63e4-7176-4b91-9b4d-db06661b4480.pdf"},{"id":94134895,"identity":"70bbdb70-7a05-4cf4-b879-6a62293b45d1","added_by":"auto","created_at":"2025-10-22 18:46:38","extension":"txt","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":3493,"visible":true,"origin":"","legend":"kinetic_model","description":"","filename":"kineticmodel.txt","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/f69d1b1a6980c2a4b0c623a4.txt"},{"id":94134896,"identity":"5f3c50b7-84cb-46e7-b79f-cdd8ea56f2fc","added_by":"auto","created_at":"2025-10-22 18:46:38","extension":"xlsx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":21710016,"visible":true,"origin":"","legend":"Dataset 1","description":"","filename":"Dataset1.xlsx","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/3c574cffee156c626d9a876f.xlsx"},{"id":94135517,"identity":"4c3c688b-2d75-4cf5-85ff-1e4a72a4346a","added_by":"auto","created_at":"2025-10-22 18:54:39","extension":"docx","order_by":3,"title":"","display":"","copyAsset":false,"role":"supplement","size":47141,"visible":true,"origin":"","legend":"","description":"","filename":"Scheme1.docx","url":"https://assets-eu.researchsquare.com/files/rs-7755576/v1/7ed40fbe5c462d92229be000.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Beyond Тraining Data Volume: Predicting Nonlinear Effects in Catalytic Reactions with Multiple Kinetic Regimes via Machine Learning","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eIn recent years, machine learning (ML) methods have gained significant traction for describing, analyzing, and predicting behavior in catalytic reaction systems. These approaches have been successfully applied to diverse scientific challenges, including the identification of optimal catalysts and reaction conditions to maximize yield and selectivity [\u003cspan additionalcitationids=\"CR2 CR3\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eA fundamental prerequisite for implementing ML in catalysis is the availability of high-quality training datasets. Although no universal criteria exist to define dataset suitability, it is generally accepted that such datasets must be sufficiently large and encompass a broad spectrum of reaction conditions \u0026mdash; whether obtained experimentally or through \u003cem\u003ein silico\u003c/em\u003e methods [\u003cspan additionalcitationids=\"CR6\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eA critical limitation of ML models lies in their constrained predictive capability: they typically deliver reliable accuracy only within the parameter space defined by the training data. While proficient at interpolation, these models offer no guarantees for extrapolation beyond the trained domain [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. This limitation often arises from shifts in reaction mechanisms under conditions outside the training set [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe situation becomes particularly complex in nonlinear reaction systems exhibiting multiple steady states or multiple kinetic regimes. In such systems, even within the trained parameter space, the system may undergo sudden transitions \u0026mdash; or \u0026ldquo;switches\u0026rdquo; \u0026mdash; between distinct kinetic regimes due to inherent nonlinearities. Such behavior is especially relevant in catalysis, where off-cycle processes like catalyst activation or deactivation often introduce strong nonlinearities [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan additionalcitationids=\"CR12 CR13 CR14 CR15\" citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. The presence of multiple catalytically active species further amplifies this complexity, consistent with the \u0026ldquo;catalyst cocktail\u0026rdquo; concept [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. From a kinetic perspective, these systems can support multiple steady states and exhibit different regimes of kinetic behavior [\u003cspan additionalcitationids=\"CR20\" citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. Transitions between them \u0026mdash; known as bifurcations [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] \u0026mdash; can lead to fundamentally different kinetic outcomes (e.g., in activity, selectivity, or stability) under specific initial conditions.\u003c/p\u003e\u003cp\u003eIn this work, we show that if a training dataset fails to represent all accessible steady states and kinetic regimes of a reaction, the resulting ML model will be inherently limited in predictive performance \u0026mdash; regardless of whether the dataset is \u0026ldquo;big\u0026rdquo; or \u0026ldquo;small\u0026rdquo;.\u003c/p\u003e"},{"header":"2 Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e2.1 Kinetic Model of a Catalytic System with Nonlinear Kinetics\u003c/h2\u003e\u003cp\u003eA well-known example of a nonlinear system exhibiting different kinetic regimes for reaching a steady state was presented in Lotka's original paper [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e] (now commonly referred to as the Lotka-Volterra model). This system described consecutive chemical transformations involving one autocatalytic step. Although the original Lotka system did not include catalytic transformations, its characteristic properties remain unchanged even when steps representing catalytic cycles are incorporated, provided they do not influence the rates of the steps in the Lotka-Volterra model [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eAs a representative example of a real catalytic system potentially exhibiting two different kinetic regimes for attaining a stable steady state, we considered Pd-catalyzed cross-coupling reactions, which are highly attractive methodologies in fine organic synthesis [\u003cspan additionalcitationids=\"CR24\" citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. The catalytic systems for cross-coupling reactions consist of the following processes, which form the corresponding dynamic subsystems (Scheme \u003cspan refid=\"Sch1\" class=\"InternalRef\"\u003e1\u003c/span\u003e):\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eA pseudo-elementary step for the formation of active Pd(0) species from the catalyst precursor (here, Pd(II) complexes [\u003cspan additionalcitationids=\"CR27 CR28\" citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]), labeled as step (I).\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eCatalytic conversion of the substrate S to the reaction product P, represented by the pseudo-elementary step (II) simulating the catalytic cycle.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eDeactivation of active Pd(0) into inactive colloidal species \u0026mdash; soluble Pd nanoclusters Pd(0)ₙ \u0026mdash; involving pseudo-elementary steps of particle nucleation and its autocatalytic growth (steps (III) and (IV), respectively) [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e].\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eA pseudo-elementary step for the formation of aggregated Pd black as a result of the terminal deactivation of Pd (step (V)) [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e].\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003eHere, a pseudo-elementary step is defined as the sum of multiple elementary steps that can, however, be treated kinetically as a single elementary step [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe key distinctions from the classical Lotka model are: (i) the presence of step (II), which represents the catalytic transformation, and (ii) the Pd nanocluster nucleation step (III). These features make the system under consideration more representative of the real process of nanocluster formation [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e].\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eOwing to the nonlinear autocatalytic formation of Pd(0)ₙ particles (Step IV), the reaction system presented in Scheme \u003cspan refid=\"Sch1\" class=\"InternalRef\"\u003e1\u003c/span\u003e is characterized by two distinct kinetic regimes for reaching a steady state, with a bifurcation between them. This bistable behavior is a well-known feature of nonlinear systems, regardless of whether they are catalytic or not. The bifurcation point is defined by a specific ratio of the rate constants and the concentrations of the reacting species [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eTo derive a criterion for predicting which of the two kinetic regimes will be established, the following system of ordinary differential equations (ODEs), corresponding to the reaction steps (I)-(V), should be considered:\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{d\\left[Pd\\left(II\\right)\\right]}{dt}=\\:-{k}_{1}\\left[Pd\\left(II\\right)\\right]\\)\u003c/span\u003e\u003c/span\u003e\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\frac{d\\left[Pd\\left(0\\right)\\right]}{dt}=\\:{k}_{1}\\left[Pd\\left(II\\right)\\right]-{k}_{3}\\left[Pd\\left(0\\right)\\right]-{k}_{4}\\left[Pd\\left(0\\right)\\right]\\left[{Pd\\left(0\\right)}_{n}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:\\frac{d\\left[{Pd\\left(0\\right)}_{n}\\right]}{dt}=\\:{k}_{3}\\left[Pd\\left(0\\right)\\right]+{k}_{4}\\left[Pd\\left(0\\right)\\right]\\left[{Pd\\left(0\\right)}_{n}\\right]-{k}_{5}\\left[{Pd\\left(0\\right)}_{n}\\right]$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:\\frac{d\\left[P\\right]}{dt}=\\:{k}_{2}\\left[Pd\\left(0\\right)\\right]$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe dynamic behavior of the system outlined in Scheme \u003cspan refid=\"Sch1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, governed by the differential equation system (1), is expected to closely resemble that of the classical Lotka system. This similarity arises from the hierarchical organization wherein catalyst transformations constitute the \"core\" system, while the catalytic conversion of S to P occurs within a \"subordinate\" system. Under the condition where \u003cem\u003ek\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e ≪ \u003cem\u003ek\u003c/em\u003e\u003csub\u003e4\u003c/sub\u003e[\u003cem\u003ePd(0)ₙ\u003c/em\u003e] (indicating that the nucleation rate of Pd(0)ₙ is significantly slower than its autocatalytic growth rate [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]), the bifurcation point can be identified through Lyapunov stability analysis of system (1) [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. The critical transition between the two distinct kinetic regimes of steady state attainment is defined by the following expression:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\frac{{k}_{1}\\left[Pd\\left(II\\right)\\right]{k}_{4}}{{k}_{5}^{2}}=4$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe stability of this ratio is sensitive to two key parameters: the initial concentration of the Pd(II) precursor and the rate of its consumption. The latter can be quantitatively characterized by the rate constant \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, which describes the reduction to active Pd(0) (Step (I) in Scheme \u003cspan refid=\"Sch1\" class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eTo map the kinetic behavior across different steady-state regimes, we performed a parameter scan across two distinct ranges of initial Pd(II) concentrations combined with variations in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e values. This approach successfully revealed two well-defined kinetic regimes:\u003c/p\u003e\u003c/div\u003e\n\u003ch3\u003e1. Oscillatory regime:\u003c/h3\u003e\n\u003cp\u003eWhen the dimensionless parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{1}\\left[Pd\\left(II\\right)\\right]{k}_{4}/{k}_{5}^{2}\u0026lt;4\\)\u003c/span\u003e\u003c/span\u003e, the system exhibited damped oscillations in the concentrations of both active Pd(0) and colloidal Pd(0)ₙ clusters during reaction progress (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea, b). Such oscillatory behavior in catalytic species concentrations \u0026mdash; affecting both active and inactive forms \u0026mdash; represents a well-established phenomenon in nonlinear catalytic systems [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e]. Notably, recent developments in physics-informed neural networks have demonstrated the capability of ML approaches to successfully predict such complex dynamic behavior [\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e], suggesting their potential applicability for modeling oscillatory kinetic data.\u003c/p\u003e\n\u003ch3\u003e2. Non-Oscillatory regime:\u003c/h3\u003e\n\u003cp\u003eWhen \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{k}_{1}\\left[Pd\\left(II\\right)\\right]{k}_{4}/{k}_{5}^{2}\u0026gt;4\\)\u003c/span\u003e\u003c/span\u003e, the system displayed monotonic convergence to steady state without oscillatory behavior in any Pd species concentration (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ed, e).\u003c/p\u003e\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003e2.2. Dataset construction\u003c/h2\u003e\u003cp\u003eTo encompass the reaction conditions leading to the two distinct kinetic regimes (oscillatory and non-oscillatory) in the system shown in Scheme \u003cspan refid=\"Sch1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, kinetic simulations were conducted across specified ranges of initial Pd(II) concentration ([Pd(II)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) and the reduction rate constant \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e. Variation in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e serves as a proxy for altering the reducibility of the Pd(II) precursor. The following parameter spaces were explored:\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eOscillatory regime\u003c/b\u003e: [Pd(II)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e = 0.01\u0026ndash;1 mM; \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.01\u0026ndash;0.4 s⁻\u0026sup1; (10 values per parameter, evenly spaced).\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eNon-Oscillatory regime\u003c/b\u003e: [Pd(II)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e = 1\u0026ndash;10 mM; \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;3\u0026ndash;10 s⁻\u0026sup1; (10 values per parameter, evenly spaced).\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003eFor all simulations, the following initial conditions and rate constants were maintained constant:\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eInitial concentrations: [P]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e=[Pd(0)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e=[Pd(0)\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e=[Pd\u003csub\u003eblack\u003c/sub\u003e]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e = 0 mM, [S]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e = 10 mM;\u003c/p\u003e\u003cp\u003eRate constants: \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.5 mM\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u0026middot;s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e4\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;100 mM\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u0026middot;s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e5\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;10 s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe temporal evolution of concentrations for product P and all catalytic species (Pd(II), Pd(0), Pd(0)ₙ, Pd black) was simulated over a 100 s period, with data points recorded at 5 s intervals. All numerical integrations were performed using GEPASI software [\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e]. Characteristic kinetic profiles illustrating both dynamical regimes are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe simulated kinetic data served as the foundation for constructing training and test datasets for the subsequent ML analysis. The training dataset incorporated the initial parameters \u0026mdash; namely, the Pd(II) precursor concentration ([Pd(II)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e) and the reduction rate constant \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e \u0026mdash; together with the corresponding product concentration profile [P]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e (Scheme \u003cspan refid=\"Sch1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) across 21 time points (including \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0) per each numerical experiment. To maintain a complete representation of the reaction system's initial state, \u003cem\u003eeach entry\u003c/em\u003e in the dataset was constructed to include [Pd(II)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e. Preserving these parameters is essential, as the system's dynamic behavior and its evolution through kinetic parameter space are fundamentally governed by its initial conditions [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eConsistent with practical experimental limitations, the dataset excluded real-time concentrations of catalytic intermediates such as [Pd(II)]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e, [Pd(0)]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e, [Pd(0)\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e, [Pd\u003csub\u003eblack\u003c/sub\u003e]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e. While advanced analytical techniques theoretically enable tracking of certain catalyst species, achieving reliable \u003cem\u003ein situ\u003c/em\u003e monitoring under operational catalytic conditions remains challenging and is rarely accomplished. Furthermore, critical intermediates often exist at concentrations below the detection limits of standard methodologies. To verify the robustness of this approach, supplementary simulations incorporating full intermediate concentration data were conducted. The resulting ML predictions proved consistent with those derived from the practically constrained dataset, affirming that the exclusion of intermediate species does not affect the principal conclusions of this investigation.\u003c/p\u003e\u003cp\u003eBy systematically sampling the predefined ranges of [Pd(II)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e, two comprehensive datasets were generated: one consisting of 10,000 numerical experiments corresponding to the oscillatory regime, and another of equal size representing the non-oscillatory regime.\u003c/p\u003e\u003cp\u003eBoth of these datasets, corresponding to the oscillatory and non-oscillatory regimes, were used to train three distinct ML models (described subsequently). Prior to training, five kinetic experiments were systematically extracted from each dataset. This curated subset, comprising 10 numerical experiments (210 observations total), was assembled into a unified test set. This specific construction was designed to rigorously evaluate the ability of the trained models to generalize across different kinetic regimes \u0026mdash; that is, to make accurate predictions on data originating from a regime fundamentally different from that of their training data.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003e2.3 ML models training and test\u003c/h2\u003e\u003cp\u003eML approaches \u0026mdash; particularly decision trees, random forest, and neural networks \u0026mdash; have proven effective in modeling complex catalytic reaction systems [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e]. Tree-based models (decision trees and random forest) often achieve strong predictive performance with computational efficiency, yet their inherent piecewise-constant nature limits their ability to generate high-fidelity continuous predictions. In contrast, neural networks offer greater representational flexibility but require substantially longer training times and larger datasets.\u003c/p\u003e\u003cp\u003eOur previous research demonstrated that decision trees and random forest outperform neural networks when trained on limited kinetic data derived from real catalytic experiments [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e]. Motivated by these findings, we selected these three model classes to rigorously evaluate their capability in predicting behavior in complex reaction systems characterized by nonlinear kinetics and bifurcation phenomena.\u003c/p\u003e\u003cp\u003eModel training was conducted in MATLAB [\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e] utilizing the Regression Learner application with 5-fold cross-validation. The concentration of the reaction product [P] at time \u003cem\u003et\u003c/em\u003e ([P]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) served as the target output variable for all predictive modeling. Following training, all models underwent rigorous evaluation using a comprehensive test set incorporating kinetic data from both oscillatory and non-oscillatory regimes. To assess model robustness across different dynamical conditions, three distinct training strategies were implemented: exclusive training on oscillatory regime data, exclusive training on non-oscillatory regime data, and training on a combined dataset encompassing both dynamical regimes. Additionally, Principal Component Analysis (PCA) was performed using the ExStatR extension (version 1.2) [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e] for Microsoft Excel, leveraging the computational capabilities of the R statistical environment.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e2.4 Use of Large Language Model\u003c/h2\u003e\u003cp\u003eWhile preparing this work, the authors used DeepSeek (open AI) to assist in language refinement and editing of this manuscript. The AI tool was used strictly as a language assistant. The authors take full responsibility for the manuscript\u0026rsquo;s content.\u003c/p\u003e\u003c/div\u003e"},{"header":"3 Results and Discussion","content":"\u003cp\u003eFrom a kinetic standpoint, it is reasonable to hypothesize that a model trained exclusively on data from a single steady state attainment regime will inherently lack the capacity to predict kinetic behavior under alternative dynamical regimes, even if the training dataset is extensively sampled within its original regime. This fundamental limitation arises from the model's inability to capture bifurcation phenomena and transitions between distinct kinetic regimes. To quantitatively validate this hypothesis, we conducted a systematic comparison of the predictive performance of models trained on three distinct data configurations, evaluating their generalization capabilities across oscillatory and non-oscillatory regimes.\u003c/p\u003e\u003cp\u003eThe performance metrics for all models following training and testing are comprehensively presented in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Models trained exclusively on oscillatory regime data (Section \u003cspan refid=\"Sec1\" class=\"InternalRef\"\u003e1\u003c/span\u003e in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) demonstrated strong predictive accuracy during validation; however, their performance deteriorated markedly when evaluated against the combined test set containing both oscillatory and non-oscillatory kinetic data. Close examination of prediction parity plots (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) revealed that all models trained exclusively on oscillatory regime data achieved satisfactory accuracy for test data derived from the oscillatory regime but consistently failed to generate meaningful predictions for non-oscillatory regime data. This asymmetric generalization pattern was also observed in models trained exclusively on non-oscillatory data ((Section \u003cspan refid=\"Sec1\" class=\"InternalRef\"\u003e1\u003c/span\u003e in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), which maintained accuracy within their training domain but exhibited dramatic performance degradation when applied to oscillatory regime data.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eGeneralization performance of ML models across kinetic regimes: comparative evaluation of training strategies using oscillatory, non-oscillatory, and combined datasets.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSection No.\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eModel type\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eHyperparameters\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e\u003cp\u003eTraining dataset\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e\u003cp\u003eTest dataset\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003eMAE\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003eMAE\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"2\" rowspan=\"3\"\u003e\u003cp\u003e\u003cem\u003e1. Training using the oscillatory dataset\u003c/em\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eRandom forest\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMinimum leaf size: 8\u003c/p\u003e\u003cp\u003eNumber of learners: 30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9996\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.0310\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-0.0760\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e1.5500\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eFine Tree\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMinimum leaf size: 4\u003c/p\u003e\u003cp\u003eSurrogate decision splits: Off\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9990\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.0440\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-0.0807\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e1.5958\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eWide Neural Network\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eNumber of fully connected layers: 1\u003c/p\u003e\u003cp\u003eFirst layer size: 100\u003c/p\u003e\u003cp\u003eActivation: ReLU\u003c/p\u003e\u003cp\u003eIteration limit: 1000\u003c/p\u003e\u003cp\u003eRegularization strength (Lambda): 0\u003c/p\u003e\u003cp\u003eStandardize data: Yes\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9984\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.06646\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-293.40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e26.0910\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"2\" rowspan=\"3\"\u003e\u003cp\u003e\u003cem\u003e2. Training using the non-oscillatory dataset\u003c/em\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eRandom forest\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMinimum leaf size: 8\u003c/p\u003e\u003cp\u003eNumber of learners: 30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9997\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.0290\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-3.3305\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e3.2959\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eFine Tree\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMinimum leaf size: 4\u003c/p\u003e\u003cp\u003eSurrogate decision splits: Off\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9989\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.0542\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-3.3293\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e3.3033\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eWide Neural Network\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eNumber of fully connected layers: 1\u003c/p\u003e\u003cp\u003eFirst layer size: 100\u003c/p\u003e\u003cp\u003eActivation: ReLU\u003c/p\u003e\u003cp\u003eIteration limit: 1000\u003c/p\u003e\u003cp\u003eRegularization strength (Lambda): 0\u003c/p\u003e\u003cp\u003eStandardize data: Yes\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9999\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.0166\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-1.5317\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e2.5107\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"2\" rowspan=\"3\"\u003e\u003cp\u003e3. \u003cem\u003eTraining using the combined dataset(oscillatory and non-oscillatory data)\u003c/em\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eRandom forest\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMinimum leaf size: 8\u003c/p\u003e\u003cp\u003eNumber of learners: 30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9996\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.0375\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.9958\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e0.0979\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eFine Tree\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMinimum leaf size: 4\u003c/p\u003e\u003cp\u003eSurrogate decision splits: Off\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9988\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.0587\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.9899\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e0.1523\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eWide Neural Network\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eNumber of fully connected layers: 1\u003c/p\u003e\u003cp\u003eFirst layer size: 100\u003c/p\u003e\u003cp\u003eActivation: ReLU\u003c/p\u003e\u003cp\u003eIteration limit: 1000\u003c/p\u003e\u003cp\u003eRegularization strength (Lambda): 0\u003c/p\u003e\u003cp\u003eStandardize data: Yes\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.9942\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e0.1134\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e0.7155\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e0.8060\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe results presented in Sections \u003cspan refid=\"Sec1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Figs.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e demonstrate that ML models trained exclusively on either oscillatory or non-oscillatory regime data exhibit strong predictive performance within their training regime but show dramatic failure when applied to the alternative kinetic regime. This fundamental limitation highlights the inherent constraint of models developed within a single kinetic paradigm when confronted with divergent steady state attainment behaviors.\u003c/p\u003e\u003cp\u003eThese findings logically indicate that robust predictive performance across both regimes requires incorporating training data representing both oscillatory and non-oscillatory conditions. Indeed, when training datasets were combined to include data from both kinetic regimes, all three ML model architectures showed substantially improved generalization capabilities (Section \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e3\u003c/span\u003e in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). Consistent with our previous findings, tree-based methods (random forest and decision trees) significantly outperformed neural network architectures across all performance metrics.\u003c/p\u003e\u003cp\u003eFurther analysis of the parity plots in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e reveals additional insights: while neural networks failed to achieve satisfactory predictive accuracy for either regime, tree-based methods demonstrated superior overall performance. However, close examination of these results reveals important limitations \u0026mdash; the decision tree predictions (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb) exhibit extended horizontal segments where [P]\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e values remain essentially constant despite variations in predictor values. Similar, though less pronounced, patterns are observable in the random forest predictions (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea).\u003c/p\u003e\u003cp\u003eNotably, these horizontal segments correspond predominantly to oscillatory regime conditions, suggesting that capturing oscillatory kinetics presents particular challenges even when using combined training data. This observation indicates that while data combination significantly improves model performance, certain nonlinear characteristics of oscillatory behavior remain difficult to capture with current ML approaches.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"4 Conclusions","content":"\u003cp\u003eThe results fundamentally reshape our understanding of machine learning for the reactions possessing nonlinear kinetics, including catalytic ones: satisfactory predictive performance depends not only on dataset size, but on whether training data comprehensively captures all possible kinetic regimes. When models encounter kinetic behavior outside their training distribution \u0026mdash; even within the same parameter ranges \u0026mdash; predictive accuracy collapses completely. This limitation extends universally across nonlinear systems exhibiting multiple stable steady states and distinct kinetic regimes. The inherent autocatalytic nature of active species formation/deactivation in cross-coupling reactions [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e, \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e] creates complex nonlinearities that demand exceptional representation in training data. Merely accumulating more data points proves insufficient; what matters is covering the full behavioral spectrum.\u003c/p\u003e\u003cp\u003eOur findings reveal a critical dual challenge: beyond traditional extrapolation concerns [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], we identify an interpolation susceptibility where models fail even within trained parameter ranges. Specifically, when predictor combinations ([Pd(II)]\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e in Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e)) yield previously unobserved kinetic regimes, predictions fail despite all individual predictor values falling within training bounds [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]. This represents a profound limitation\u0026mdash;not extrapolation in the conventional sense, but a failure to capture emergent behaviors from predictor interactions.\u003c/p\u003e\u003cp\u003eTherefore, for such processes, while using extensive data is necessary, it is insufficient to ensure coverage of all possible reaction steady states and kinetic regimes. Training data needs to capture the entire dynamic landscape of kinetic regimes, going beyond the simple numerical ranges of its input parameters.\u003c/p\u003e\u003cp\u003eTo overcome the challenges identified in modeling nonlinear catalytic systems, we propose a dual strategy combining unsupervised learning with systematic error analysis:\u003c/p\u003e\u003cp\u003eFirst, unsupervised methods like PCA offer theoretical promise for visualizing nonlinear kinetic patterns due to their sensitivity to data structure [\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e] and provide delineating distinct kinetic regimes. However, our application of PCA to the combined dataset (containing both oscillatory and non-oscillatory experiments) yielded insufficient separation between regimes (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), indicating that reliable identification requires supplementary domain knowledge beyond what PCA alone can provide.\u003c/p\u003e\u003cp\u003eAs a complementary strategy, we recommend a systematic analysis of prediction anomalies revealed in parity plots. By analyzing clusters of poorly predicted points \u0026mdash; particularly those exhibiting systematic patterns \u0026mdash; researchers can identify regions in parameter space where specific kinetic regimes emerge due to nonlinear effects. Critical to this approach is preserving temporal continuity: test datasets should incorporate complete reaction profiles (from initiation to completion) rather than isolated measurements. Maintaining chronological integrity reveals coherent patterns in prediction errors that reflect the system\u0026rsquo;s dynamic evolution, as clearly evidenced in Figs.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. These patterns, when analyzed collectively rather than as random noise (experimental data errors), provide actionable insights into model limitations and guide strategic data collection.\u003c/p\u003e\u003cp\u003eThis integrated approach \u0026mdash; combining dimensionality reduction with thorough error analysis \u0026mdash; enables targeted refinement of both data composition and model architecture, ultimately enhancing predictive robustness across complex kinetic landscapes.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAcknowledgements\u003c/h2\u003e\u003cp\u003eThis work was supported by the Russian Science Foundation (project 24-23-00382). The studies were carried out using the equipment of the Center for Collective Use of Analytical Equipment of the Irkutsk State University (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttp://ckp-rf.ru/ckp/3264/\u003c/span\u003e\u003cspan address=\"http://ckp-rf.ru/ckp/3264/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eSmith, S. C.; Horbaczewskyj, C. S.; Tanner, T. F. N.; Walder, J. J.; Fairlamb, I. J. S. Automated Approaches, Reaction Parameterisation, and Data Science in Organometallic Chemistry and Catalysis: Towards Improving Synthetic Chemistry and Accelerating Mechanistic Understanding. \u003cem\u003eDigital Discovery\u003c/em\u003e\u003cstrong\u003e2024\u003c/strong\u003e, \u003cem\u003e3\u003c/em\u003e (8), 1467\u0026ndash;1495. https://doi.org/10.1039/D3DD00249G.\u003c/li\u003e\n\u003cli\u003eLong, D.; Finke, R. G.; Bangerth, W. 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Commun. \u003c/em\u003e\u003cstrong\u003e2024\u003c/strong\u003e, \u003cem\u003e15\u003c/em\u003e (1), 3968. https://doi.org/10.1038/s41467-024-47939-5. \u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Scheme","content":"\u003cp\u003eScheme 1 is 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":"","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":"machine learning, big data, prediction, catalysis, kinetics","lastPublishedDoi":"10.21203/rs.3.rs-7755576/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7755576/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWhile data scarcity is often viewed as the primary obstacle to developing predictive machine learning (ML) models in chemistry, we demonstrate that data \u0026ldquo;sufficiency\u0026ldquo; alone is inadequate for reactions exhibiting nonlinear kinetics. These phenomena are common in complex reactions, including catalytic ones. Therefore, such systems can possess multiple stable steady states and distinct kinetic regimes. We show that if training data fail to encompass this full dynamic diversity, ML models cannot achieve high predictive accuracy, as their predictions are fundamentally constrained to the regimes represented in the training set. This work underscores a critical limitation of ML for complex reaction modeling: the imperative for data completeness over mere volume to avoid failures in prediction.\u003c/p\u003e","manuscriptTitle":"Beyond Тraining Data Volume: Predicting Nonlinear Effects in Catalytic Reactions with Multiple Kinetic Regimes via Machine Learning","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-22 18:46:31","doi":"10.21203/rs.3.rs-7755576/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":"d478f79c-a8d8-4e7a-9262-971a52869154","owner":[],"postedDate":"October 22nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":56113606,"name":"Physical sciences/Chemistry/Catalysis/Catalytic mechanisms"},{"id":56113607,"name":"Physical sciences/Chemistry/Theoretical chemistry/Reaction mechanisms"},{"id":56113608,"name":"Physical sciences/Chemistry/Physical chemistry/Reaction kinetics and dynamics"}],"tags":[],"updatedAt":"2025-10-22T18:46:31+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-22 18:46:31","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7755576","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7755576","identity":"rs-7755576","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}