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

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 Φ^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 Φ^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 Φ^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 Φ^4_3 model.

Reproductions