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

• URL: http://arxiv.org/abs/2603.08219v1
• Date: Mon, 09 Mar 2026 10:50:30 GMT
• Title: Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations
• Abstract summary: This paper explores how our recently developed Wiener Chaos Expansion (WCE)-based neural operator (NO) can be applied to singular partial differential equations.<n>We leverage feature-wise linear modulation (FiLM) to appropriately capture the dependency between the solution of singular SPDE and its smooth remainder.<n>The resulting WCE-FiLM-NO shows excellent performance on $boldsymbol4$, as measured by relative $L$ loss, out-of-distribution and autocorrelation score.
• Score: 58.188204772101756
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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