An Adaptive Random Fourier Features approach Applied to Learning Stochastic Differential Equations

TL;DR

This paper introduces an adaptive random Fourier features algorithm with Metropolis sampling for efficiently learning stochastic differential equations from data, outperforming traditional methods in various benchmark problems.

Contribution

It presents a novel ARFF-based training algorithm for stochastic differential equations, demonstrating improved performance over Adam optimization in diverse stochastic models.

Findings

ARFF matches or surpasses Adam in loss minimization

Faster convergence with ARFF in benchmark problems

Effective for diverse stochastic dynamical systems

Abstract

This work proposes a training algorithm based on adaptive random Fourier features (ARFF) with Metropolis sampling and resampling \cite{kammonen2024adaptiverandomfourierfeatures} for learning drift and diffusion components of stochastic differential equations from snapshot data. Specifically, this study considers It\^{o} diffusion processes and a likelihood-based loss function derived from the Euler-Maruyama integration introduced in \cite{Dietrich2023} and \cite{dridi2021learningstochasticdynamicalsystems}. This work evaluates the proposed method against benchmark problems presented in \cite{Dietrich2023}, including polynomial examples, underdamped Langevin dynamics, a stochastic susceptible-infected-recovered model, and a stochastic wave equation. Across all cases, the ARFF-based approach matches or surpasses the performance of conventional Adam-based optimization in both loss minimization and convergence speed. These results highlight the potential of ARFF as a compelling alternative for data-driven modeling of stochastic dynamics.