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Identifying reaction coordinates (RCs) from many collective variable candidates have been of great challenge in understanding reaction mechanisms in complex systems. Machine learning approaches, especially the deep neural network (DNN), have become a powerful tool in this field, and have actively been applied. On the other hand, the structure of the DNN model is highly flexible, and the hyperparameters that determine the structure is often selected intuitively or in a highly non-trivial and tedious manner. Furthermore, how the choice of hyperparameter affects the quality of the DNN model remains obscure. In this work, we explore the hyperparameter space by developing the hyperparameter tuning approach for the DNN model in RC optimization, and investigate how the choice of parameter sets affect the quality of the RC. The DNN model is constructed from a large number of collective variables to predict the changes of committor along the RC by minimizing the cross-entropy function, and the hyperparameters are determined in an automatic manner using the Bayesian optimization method. The approach is applied to study the isomerization of alanine dipeptide in vacuum and in water, and the features that characterize the RC are extracted using the explainable AI (XAI) tools. The results show that the DNN models with notably different structures can describe the RC with similar accuracy. Furthermore, despite the difference in the hyperparameters, the features analyzed by XAI are highly similar, indicating that the hyperparameter space is multimodal. By studying the reaction in water, it is found that the electrostatic potential from the solvent to the hydrogen (H${}_{18}$), in addition to the dihedral angles $\phi$ and $\theta$, plays an important role in characterizing the RC. The DNN model thus effectively accounts for the torque character suggested previously., Comment: 10 pages, 10 figures, 2 table for main text, 9 pages for supplementary material

The present work shows that the free energy landscape associated with alanine dipeptide isomerization can be effectively represented by specific interatomic distances without explicit reference to dihedral angles. Conventionally, two stable states of alanine dipeptide in vacuum, i.e., C$7_{\mathrm{eq}}$ ($\beta$-sheet structure) and C$7_{\mathrm{ax}}$ (left handed $\alpha$-helix structure), have been primarily characterized using the main chain dihedral angles, $\varphi$ (C-N-C$_\alpha$-C) and $\psi$ (N-C$_\alpha$-C-N). However, our recent deep learning combined with "Explainable AI" (XAI) framework has shown that the transition state can be adequately captured by a free energy landscape using $\varphi$ and $\theta$ (O-C-N-C$_\alpha$) [T. Kikutsuji, et al. J. Chem. Phys. 156, 154108 (2022)]. In perspective of extending these insights to other collective variables, a more detailed characterization of transition state is required. In this work, we employ the interatomic distances and bond angles as input variables for deep learning, rather than the conventional and more elaborate dihedral angles. Our approach utilizes deep learning to investigate whether changes in the main chain dihedral angle can be expressed in terms of interatomic distances and bond angles. Furthermore, by incorporating XAI into our predictive analysis, we quantified the importance of each input variable and succeeded in clarifying the specific interatomic distance that affects the transition state. The results indicate that constructing a free energy landscape based on using the identified interatomic distance can clearly distinguish between the two stable states and provide a comprehensive explanation for the energy barrier crossing., Comment: 8 pages, 5 figures, 1 table for main text, 4 pages for supplementary material, to appear in J. Chem. Phys

A method for obtaining appropriate reaction coordinates is required to identify transition states distinguishing product and reactant in complex molecular systems. Recently, abundant research has been devoted to obtaining reaction coordinates using artificial neural networks from deep learning literature, where many collective variables are typically utilized in the input layer. However, it is difficult to explain the details of which collective variables contribute to the predicted reaction coordinates owing to the complexity of the nonlinear functions in deep neural networks. To overcome this limitation, we used Explainable Artificial Intelligence (XAI) methods of the Local Interpretable Model-agnostic Explanation (LIME) and the game theory-based framework known as Shapley Additive exPlanations (SHAP). We demonstrated that XAI enables us to obtain the degree of contribution of each collective variable to reaction coordinates that is determined by nonlinear regressions with deep learning for the committor of the alanine dipeptide isomerization in vacuum. In particular, both LIME and SHAP provide important features to the predicted reaction coordinates, which are characterized by appropriate dihedral angles consistent with those previously reported from the committor test analysis. The present study offers an AI-aided framework to explain the appropriate reaction coordinates, which acquires considerable significance when the number of degrees of freedom increases., Comment: 9 pages, 5 figures, 2 tables, to appear in J. Chem. Phys

We propose a cross-entropy minimization method for finding the reaction coordinate from a large number of collective variables in complex molecular systems. This method is an extension of the likelihood maximization approach describing the committor function with a sigmoid. By design, the reaction coordinate as a function of various collective variables is optimized such that the distribution of the committor $p_\mathrm{B}^*$ values generated from molecular dynamics simulations can be described in a sigmoidal manner. We also introduce the $L_2$-norm regularization used in the machine learning field to prevent overfitting when the number of considered collective variables is large. The current method is applied to study the isomerization of alanine dipeptide in vacuum, where 45 dihedral angles are used as candidate variables. The regularization parameter is determined by cross-validation using training and test datasets. It is demonstrated that the optimal reaction coordinate involves important dihedral angles, which are consistent with the previously reported results. Furthermore, the points with $p_\mathrm{B}^*\sim 0.5$ clearly indicate a separatrix distinguishing reactant and product states on the potential of mean force using the extracted dihedral angles., Comment: 8 pages, 5 figures, 1 table for main text, 5 pages for supplementary materials, to appear in J. Chem. Phys. for "Classical Molecular Dynamics (MD) Simulations: Codes, Algorithms, Force Fields, and Applications"