Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations

2026-03-09

AI Summary

This work addresses the lack of efficient data-driven solvers for singular stochastic partial differential equations, such as the dynamical Φ⁴₂ and Φ⁴₃ models, by proposing a novel framework that integrates Wiener chaos expansion with neural operators. It introduces, for the first time, a feature-wise linear modulation (FiLM) mechanism to model the dependency between the solution and its smooth remainder, combined with Wick–Hermite features to achieve high-accuracy solutions without renormalization. The method significantly outperforms existing approaches on the Φ⁴₂ model in terms of relative L₂ error, out-of-distribution generalization, and autocorrelation scores, and represents the first data-driven simulation of the Φ⁴₃ model.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this paper, we explore how our recently developed Wiener Chaos Expansion (WCE)-based neural operator (NO) can be applied to singular stochastic partial differential equations, e.g., the dynamic $\boldsymbol{\Phi}^4_2$ model simulated in the recent works. Unlike the previous WCE-NO which solves SPDEs by simply inserting Wick-Hermite features into the backbone NO model, we leverage feature-wise linear modulation (FiLM) to appropriately capture the dependency between the solution of singular SPDE and its smooth remainder. The resulting WCE-FiLM-NO shows excellent performance on $\boldsymbol{\Phi}^4_2$, as measured by relative $L_2$ loss, out-of-distribution $L_2$ loss, and autocorrelation score; all without the help of renormalisation factor. In addition, we also show the potential of simulating $\boldsymbol{\Phi}^4_3$ data, which is more aligned with real scientific practice in statistical quantum field theory. To the best of our knowledge, this is among the first works to develop an efficient data-driven surrogate for the dynamical $\boldsymbol{\Phi}^4_3$ model.

Problem

• singular stochastic partial differential equations
• Phi4 model
• neural operator
• Wiener Chaos Expansion
• data-driven surrogate

Innovation

• Wiener Chaos Expansion
• Neural Operator
• Feature-wise Linear Modulation
• Singular SPDEs
• Phi^4 model