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

2025/07/21 by Douglas, Owen, Kammonen, Aku, Pandey, Anamika +1
#FOS: Computer and information sciences #Machine Learning (cs.LG)

paper · doi:10.48550/arxiv.2507.15442

Abstract

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 Itô 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.

Citations

• Adaptive Random Fourier Features Training Stabilized By Resampling With Applications in Image Regression
• Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning
• Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme
• Adaptive random Fourier features with Metropolis sampling
• SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimates
• Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Neural Ordinary Differential Equations
• PDE-Net: Learning PDEs from Data
• ImageNet classification with deep convolutional neural networks
• A general method for numerically simulating the stochastic time evolution of coupled chemical reactions

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