Neural network approaches for variance reduction in fluctuation formulas

TL;DR

This paper introduces a physics-informed neural network method to solve Poisson equations, serving as control variates to significantly reduce variance in transport coefficient estimations via fluctuation formulas, especially in high-dimensional problems.

Contribution

The paper presents a novel neural network-based approach to efficiently approximate solutions of Poisson equations for variance reduction in fluctuation formula computations.

Findings

Neural network solutions effectively reduce estimator variance.

Method is suitable for moderately high-dimensional problems.

Extensive numerical analysis validates the approach.

Abstract

We propose a method utilizing physics-informed neural networks (PINNs) to solve Poisson equations that serve as control variates in the computation of transport coefficients via fluctuation formulas, such as the Green--Kubo and generalized Einstein-like formulas. By leveraging approximate solutions to the Poisson equation constructed through neural networks, our approach significantly reduces the variance of the estimator at hand. We provide an extensive numerical analysis of the estimators and detail a methodology for training neural networks to solve these Poisson equations. The approximate solutions are then incorporated into Monte Carlo simulations as effective control variates, demonstrating the suitability of the method for moderately high-dimensional problems where fully deterministic solutions are computationally infeasible.