Stochastic Neural Network Approach for Learning High-Dimensional Free Energy Surfaces

The generation of free energy landscapes corresponding to conformational equilibria in complex molecular systems remains a significant computational challenge. Adding to this challenge is the need to represent, store, and manipulate the often high-dimensional surfaces that result from rare-event sampling approaches employed to compute them. In this Letter, we propose the use of artificial neural networks as a solution to these issues. Using specific examples, we discuss network training using enhanced-sampling methods and the use of the networks in the calculation of ensemble averages.

Figure 1

Schematic representation of an artificial neural network that takes in $n$ CGV values and outputs the free energy $A(s)$ and possibly its gradient $\partial A/\partial s_{\alpha }$. The network contains $m_1,m_2,\dots ,m_K$ nodes each of $K$ hidden layers. In each node, $x_j^i$ are the arguments of the activation functions in Eq. (2).

Figure 2

The $L^2$ global error versus the number of training points in the training set for the alanine dipeptide (left) and the alanine tripeptide (right).

Figure 3

FES of the alanine dipeptide in the gas phase generated by a trained ANN based on the free energy gradient using training data of Chen, Cuendet, and Tuckerman [19] (left). Comparison of free energies generated using the gradient-based ANN for the gas-phase alanine tripeptide and free energies obtained directly from the free-energy-based ANN used to generate Fig. 2. The black line represent the identity function $f(x)=x$.

Figure 4

The ensemble average of the RMSD observable defined in Eq. (7) for the alanine dipeptide (left) and alanine tripeptide (right) as a function of the number of data in the training set.

Figure 5

NMR spin-spin $J$ couplings between ${\mathrm{H}}_N$ and ${\mathrm{H}}_{\alpha }$ for each of the five residues in met-enkephalin using the trained ANN together with Monte Carlo simulations in order to evaluate Eq. (5). Dots on the right are converged $NVT$ results.

Figure 6

Left: Projection onto the $Q_4-Q_6$ plane of the FES (in eV) generated by a trained ANN. Thermodynamically inaccessible regions are shown as white. Right: Dependence of the isothermal compressibility on the number of training points.