Deep Learning Method for Computing Committor Functions with Adaptive Sampling

TL;DR

This work tackles computing the high-dimensional committor function $q(x)$ for transitions between metastable states under overdamped Langevin dynamics by introducing two adaptive sampling schemes that actively generate data via learned bias potentials. Sampling Scheme I uses metadynamics on the one-dimensional variable $r_{ heta}(x)=R_n(q_{ heta}(x))$, while Sampling Scheme II biases the potential by the free energy of the learned committor to yield data uniformly distributed along the transition tube. The authors formulate the neural-network representation with boundary-condition penalties, derive the data-distribution properties of both schemes, and validate the approach on an extended Mueller potential, alanine dipeptide, and a solvated dimer, achieving accurate committor reconstruction and uniform tube sampling. The method promises efficient exploration of transition pathways in high-dimensional systems and may enable analysis of complex biomolecular transitions such as protein folding.

Abstract

The committor function is a central object for quantifying the transitions between metastable states of dynamical systems. Recently, a number of computational methods based on deep neural networks have been developed for computing the high-dimensional committor function. The success of the methods relies on sampling adequate data for the transition, which still is a challenging task for complex systems at low temperatures. In this work, we propose a deep learning method with two novel adaptive sampling schemes (I and II). In the two schemes, the data are generated actively with a modified potential where the bias potential is constructed from the learned committor function. We theoretically demonstrate the advantages of the sampling schemes and show that the data in sampling scheme II are uniformly distributed along the transition tube. This makes a promising method for studying the transition of complex systems. The efficiency of the method is illustrated in high-dimensional systems including the alanine dipeptide and a solvated dimer system.

Figures (9)

• Figure 1: Schematic illustration of the transition tube between metastable states $A$ and $B$, which is introduced in Section \ref{['Stability']}, and the isosurfaces of the learned committor function $q_{\theta}(x)$ ( Upper), as well as of the variable $r_{\theta}(x)=R_5(q_{\theta}(x))$ ( Lower). In both panels, the isosurfaces represented by bold lines correspond to the function values $(2k-1)/10$, $1\leq k\leq 5$.
• Figure 1: (a) Contour plots of the committor function $q(x)$ computed using the finite element method. (b) Distribution of $5\times 10^4$ data points generated by sampling scheme I in \ref{['alg1']} and contour plots of the learned committor $q_{\theta}$ obtained using the sampled data. (c) Contour plots of the variable $r_{\theta}=R_{10}(q_{\theta})$ where $q_{\theta}$ is obtained using sampling scheme I. (d) Distribution of $5\times 10^4$ data points generated by sampling scheme II in \ref{['alg2']} and contour plots of the learned committor $q_{\theta}$ obtained using the sampled data. The data and functions are projected on the $(x_1,x_2)$-plane. The dashed contour lines indicate the potential energy \ref{['example1']} of the system.
• Figure 2: Distribution of the sampled data points and contour plots of the learned committor $q_{\theta}$, projected on the $(x_1,x_2)$-plane, over the all iterations of \ref{['alg2']} with sampling scheme II. The first panel shows contour plots of the initial network for $q_{\theta}$. The last panel shows plots of the two errors for $q_{\theta}$ versus the iteration, which is defined by Eq. \ref{['error']} and \ref{['error_sets']}.
• Figure 3: Stick and ball plot of the alanine dipeptide (C$\text{H}_3$-CONH-CHC$\text{H}_3$-CONH-C$\text{H}_3$). In the neural network architecture of $q_{\theta}(x)$, the input configuration $x$ is transformed through translation and rotation with respect to the four atoms $C_{\alpha}$, $C_{z}$, $N_{x}$ and $C_{y}$, as described in Eq. \ref{['ala_q']}.
• Figure 4: Distribution of $5\times 10^4$ data points generated using sampling scheme I ( Left) or II ( Right) in the algorithms and $200$ states sampled on the $1/2$-isosurface of $q_{\theta}$, projected on the $(\phi(x),\psi(x))$-plane. The isosurface consists of two separate sets $S_1$ and $S_2$, each set with $100$ states sampled. The contour lines indicate the adiabatic energy landscape of the alanine dipeptide which is computed by energy minimization with $\phi$ and $\psi$ fixed.