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

TL;DR

This paper introduces a novel Wiener Chaos Expansion-based neural operator enhanced with FiLM for efficiently solving singular stochastic PDEs like the $oldsymbol{oldsymbol{ extPhi}}^4_2$ and $oldsymbol{ extPhi}^4_3$ models, outperforming previous methods.

Contribution

It develops a new WCE-FiLM neural operator that effectively captures dependencies in singular SPDEs without renormalization, advancing data-driven simulation of complex quantum field models.

Findings

WCE-FiLM-NO outperforms previous models on $oldsymbol{ extPhi}^4_2$.

The method accurately simulates $oldsymbol{ extPhi}^4_3$ data.

Achieves excellent performance without renormalization.

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.