Convergence rates for random feature neural network approximation in molecular dynamics

TL;DR

This paper establishes convergence rates for random feature neural networks approximating potentials in molecular dynamics, showing how approximation errors decrease with network size and data points, under certain boundedness conditions.

Contribution

It provides a novel convergence rate analysis for random feature neural networks in molecular dynamics, avoiding traditional complexity measures.

Findings

Error rate of $ig(K^{-1}+J^{-1/2}ig)^{1/2}$ for network approximation

Error bounds hold under bounded Hessian assumptions

Analysis applies to Gibbs density sampled data

Abstract

Random feature neural network approximations of the potential in Hamiltonian systems yield approximations of molecular dynamics correlation observables that have the expected error $\mathcal{O}\big((K^{-1}+J^{-1/2})^{\frac{1}{2}}\big)$, for networks with $K$ nodes using $J$ data points, provided the Hessians of the potential and the observables are bounded. The loss function is based on the least squares error of the potential and regularizations, with the data points sampled from the Gibbs density. The proof uses an elementary new derivation of the generalization error for random feature networks that does not apply the Rademacher or related complexities.