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

This work proposes a training algorithm based on adaptive random Fourier features (ARFF) with Metropolis sampling and resampling citekammonen2024adaptiverandomfourierfeatures for learning drift and diffusion components of stochastic differential equations from snapshot data. Specifically, this study considers Ito diffusion processes and a likelihood-based loss function derived from the Euler-Maruyama integration introduced in citeDietrich2023 and citedridi2021learningstochasticdynamicalsystems. This work evaluates the proposed method against benchmark problems presented in citeDietrich2023, 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.