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

2025-07-21

AI Summary

This work addresses the challenge of learning drift and diffusion terms of Itô stochastic differential equations (SDEs) from sparse snapshot data. We propose an end-to-end learning framework based on adaptive random Fourier features (ARFF). The method integrates Metropolis sampling with resampling to enhance sample quality and employs Euler–Maruyama discretization to construct a differentiable likelihood-based loss function, enabling efficient and stable optimization of the SDE coefficient functions. Compared to standard Adam optimization, our approach achieves significantly faster convergence and higher modeling accuracy. Extensive experiments on benchmark systems—including polynomial SDEs, Langevin dynamics, the stochastic SIR epidemiological model, and the stochastic wave equation—demonstrate that the method matches or surpasses state-of-the-art approaches in accuracy, generalizability, and robustness.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 Itô 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.

Problem

• Learning drift and diffusion in stochastic differential equations
• Evaluating ARFF method against benchmark stochastic dynamics
• Comparing ARFF performance with Adam-based optimization

Innovation

• Adaptive random Fourier features for SDE learning
• Metropolis sampling and resampling technique
• Likelihood-based loss from Euler-Maruyama integration