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

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 boldsymbolPhi42 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 boldsymbolPhi42, as measured by relative L2 loss, out-of-distribution L2 loss, and autocorrelation score; all without the help of renormalisation factor. In addition, we also show the potential of simulating boldsymbolPhi43 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 boldsymbolPhi43 model.