Title: Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations
Primary area: learning on time series and dynamical systems
Keywords: Neural Operator, Stochastic Differential Equation, Diffusion Generative Models

Abstract: Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) are fundamental tools for modeling stochastic dynamics across the natural sciences and modern machine learning. Developing deep learning models for approximating their solution operators promises not only fast, practical solvers, but may also inspire models that resolve classical learning tasks from a new perspective. In this work, we propose novel neural operators (NO) for SPDEs and SDEs based on their solution's Wiener Chaos Expansion (WCE): we project the driving noise onto orthonormal Wick-Hermite features and learn the \textit{deterministic} propagator coefficients that reconstruct full trajectories. 
We extend WCE theory and prove approximation guarantees for both the truncated expansion and the resulting models for approximating SPDE and SDE solution operators. More importantly, the theory yields a family of implementable NO architectures across data domains and tasks. We validate these models on a diverse suite of problems: classical SPDE problems (e.g., 2D Navier–Stokes, \(\boldsymbol{\Phi}_1^4\)), diffusion one-step sampling on images, topological interpolation on graphs, financial extrapolation (e.g., Heston coupled SDE), parameter estimation, and SDE on manifold for flood prediction, demonstrating competitive accuracy and broad applicability. Our work bridges the gap between abstract stochastic analysis and practical operator learning for real-world machine learning by providing a unified framework for learning SDE/SPDE solutions that readily integrates with existing ML pipelines and catalyzes further research.