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

TL;DR

This is among the first works to develop an efficient data-driven surrogate for the dynamical $\boldsymbol{\Phi}^4_3$ model, and shows the potential of simulating $\boldsymbol{\Phi}^4_3$ data, which is more aligned with real scientific practice in statistical quantum field theory.

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 $\boldsymbolΦ^4_2$ 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 $\boldsymbolΦ^4_2$, 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 $\boldsymbolΦ^4_3$ 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 $\boldsymbolΦ^4_3$ model.

Figures (2)

• Figure 1: Architecture of WCE-FiLM-NO. Initial Wick features are computed through the Brownian increments, and then serve as the input to both FNO and FiLM computation to compute the remainder $\widehat{v}_\epsilon(t,x)$ and scaling/shifting coefficients (i.e., $\gamma, \tau$). Finally, the prediction is obtained by conducting the affine transformation between the output of FNO and Conv2D.
• Figure 2: The dynamical \ref{['eq:phi43']} model at various times $t$, beginning at white noise initial condition.