Learning reaction coordinates via cross-entropy minimization: Application to alanine dipeptide

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.