Neural network approaches for variance reduction in fluctuation formulas
Grigorios Pavliotis, Renato Spacek, Gabriel Stoltz, Urbain Vaes
TL;DR
The paper tackles variance reduction in transport-coefficient estimation, recasting estimators as inner products with Poisson-solution operators $G$ of the form $-\mathcal{L}G=g$. It introduces physics-informed neural networks (PINNs) to approximate the Poisson solutions and builds three control variate frameworks (forward, adjoint, and combined) within Green--Kubo and Half-Einstein estimators, plus a zero-cost adjoint variant. It provides rigorous bias and variance analyses and demonstrates substantial variance reduction in underdamped Langevin dynamics and multiscale SDEs, achieving favorable cost-variance trade-offs even when approximations are rough. The results suggest PINNs as a practical tool for variance reduction in moderately high-dimensional stochastic systems where exact deterministic Poisson solves are infeasible, enabling efficient computation of mobility and other transport coefficients via fluctuation formulas.
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.