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

Dai Shi; Luke Thompson; Andi Han; Peiyan Hu; Junbin Gao; José Miguel Hernández-Lobato

cs.LGMar 9, 2026

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

Authors: Dai Shi, Luke Thompson, Andi Han, Peiyan Hu, Junbin Gao, José Miguel Hernández-Lobato

Organizations: University of Cambridge · University of Sydney · Chinese Academy of Science

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 & tables

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.

Train set/Test set	$W\mapsto u$		$(W,a_{\varepsilon})\mapsto u$		$(W,u_{0})\mapsto u$		$(W,u_{0},a_{\varepsilon})\mapsto u$
Train set/Test set	$\mathcal{D}^{\mathrm{re}}_{J}$	$\mathcal{D}^{\mathrm{re}}_{128}$	$\mathcal{D}^{\mathrm{re}}_{J}$	$\mathcal{D}^{\mathrm{re}}_{128}$	$\mathcal{D}^{\mathrm{re}}_{J}$	$\mathcal{D}^{\mathrm{re}}_{128}$	$\mathcal{D}^{\mathrm{re}}_{J}$	$\mathcal{D}^{\mathrm{re}}_{128}$
$\mathcal{D}^{\mathrm{re}}_{2}(\mathrm{NSPDE)}$	0.010	4.087	0.034	0.998	0.010	2.805	0.042	0.998
$\mathcal{D}^{\mathrm{re}}_{128}$	0.230	0.230	0.028	0.028	0.223	0.223	0.029	0.029
$\mathcal{D}^{\mathrm{re}}_{2}(\text{WCE-FiLM-NO})$	0.004	0.948	–	–	0.009	0.915	–	–
$\mathcal{D}^{\mathrm{re}}_{128}$	0.017	0.017	–	–	0.025	0.025	–	–

Table 1: Results in relative

L_{2}

error in

\boldsymbol{\Phi}^{4}_{2}

Panel of Figure 2 — Figure 2: The dynamical ( Φ 3 4 ) model at various times $t$ , beginning at white noise initial condition.

Stochastic PDEs

See all 11 papers

Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization
May 15, 2026Kai HidajatUncertainty Quantification Stochastic PDEs
Department of Applied Mathematics University of Washington Seattle, WA, USA
Stochastic Adaptive Fourier Decomposition for Operator Learning
Oct 3, 2026Pengqing Shi, Liming Zhang, Tao Qian +3PDE Operator Learning Neural PDE Solvers
University of Sydney · University of Macau · Macau University of Science and Technology
Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations
Oct 13, 2026Wonjae Lee, Jiuk Jang, Hyungbin ParkNeural Operators Fourier Neural Operator
InnoCore AI Meta-Scientist Research Group, KAIST, Daejeon 34141, South Korea. · Department of Mathematical Sciences, Seoul National University, Seoul 08826, South Korea. · Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, 1, Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea.

May 15, 2026J/K move · Enter open · S save

Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization

Kai Hidajat

Department of Applied Mathematics University of Washington Seattle, WA, USA

Uncertainty Quantification Stochastic PDEs Neural Operators

Open paper AskPDF arXiv

Neural operators excel as deterministic surrogates, but inevitably collapse to the conditional mean when applied to stochastic PDEs, discarding the variance and tail structure upon which uncertainty quantification depends. Recovering this structure typically requires Monte Carlo rollouts or grafted generative models, both of which surrender the one-shot efficiency and resolution invariance that define the operator paradigm. To resolve this, we draw on the Doob-Meyer theorem, which establishes that any semimartingale fundamentally decomposes into a predictable drift and an unpredictable, zero-mean martingale. Translating this theorem into an architectural prior, we introduce the Martingale Neural Operator (MNO). MNO maps an initial condition directly to the conditional mean and covariance of the terminal law, parameterized by a drift-like mean and a low-rank factor

B_φ

with

B_φ^\top B_φ

positive semi-definite by construction. For our experiments, we use a Gaussian residual instantiation. Across 1D SPDEs, rough volatility, and 2D operator tasks, MNO reduces Wasserstein distance by up to

120\times

φ^4

field theory and

68\times

on stochastic Burgers, evaluating

\sim 3\times

faster than a conditional diffusion baseline at matched wall-clock training budgets. On 2D tasks, MNO is comparable to FNO on zero-shot resolution transfer and turbulent flow, while quasi-deterministic systems such as Gray-Scott remain a failure mode.

Abstract

Stochastic PDEs

Explore similar work

Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization

Stochastic Adaptive Fourier Decomposition for Operator Learning

Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations