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 $\boldsymbolΦ^4_2$ 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.

💡 Research Summary

This paper introduces a novel neural operator architecture, called WCE‑FiLM‑NO, designed to learn data‑driven surrogates for singular stochastic partial differential equations (SPDEs) such as the dynamic Φ⁴₂ and Φ⁴₃ models. The authors build on two complementary ideas: (i) the Wiener Chaos Expansion (WCE) theory, which expresses the solution of a semi‑linear SPDE as a finite linear combination of deterministic propagators multiplied by Wick‑Hermite basis functions (the “Wiener features”), and (ii) Feature‑wise Linear Modulation (FiLM), a conditioning mechanism originally developed for computer vision that applies learned affine transformations to intermediate feature maps.

In the context of the dynamic Φ⁴₂ equation, the solution can be decomposed via the Da Prato‑Debussche (DPDD) approach into a rough stochastic component X (the linear stochastic heat solution) and a smooth remainder v that satisfies a deterministic “Shift Equation”. Crucially, X, its Wick square X⊙², and its Wick cube X⊙³ can each be written exactly as linear combinations of Wiener features of order 1, 2, and 3 respectively. The authors therefore feed all Wiener features up to chaos order K = 3 into the model.

The architecture consists of three stages: (1) a Fourier Neural Operator (FNO) receives the Wiener features and predicts an initial approximation b_v(x,t) of the smooth remainder v; (2) a lightweight conditioning network g_θ (implemented as a 2‑D convolution) processes the same Wiener features together with the spatio‑temporal coordinates (x,t) and outputs FiLM parameters γ_θ(x,t) and τ_θ(x,t); (3) the final field prediction is obtained by an affine transformation b_u(x,t) = (1 + γ_θ) ⊙ b_v + τ_θ, which effectively adds the stochastic component X back to the corrected remainder. This FiLM‑based affine modulation mirrors the DPDD reconstruction step while allowing the network to learn the nonlinear interaction between the stochastic driving terms and the smooth remainder.

Training is performed on synthetic datasets generated by the recent renormalized Φ⁴₂ pipeline (Li et al., 2025). Two noise spectral truncation levels, ε = 2 and ε = 128, are used to create distinct training regimes. The model is evaluated both on the same truncation (in‑distribution) and on the opposite truncation (cross‑distribution) to assess robustness to changes in the underlying noise resolution. Table 1 reports relative L₂ errors compared to the benchmark NSPDE model under four information settings: (i) only the noise W is given, (ii) W plus the renormalization constant a_ε, (iii) W plus varying initial condition u₀, and (iv) full information (W, u₀, a_ε). WCE‑FiLM‑NO consistently outperforms NSPDE in all settings, achieving errors as low as 0.004 (ε = 2) and 0.017 (ε = 128) without ever using the renormalization constant a_ε. In contrast, NSPDE’s best cross‑truncation performance relies heavily on a_ε (≈ 0.998), highlighting the superior generalization of the proposed method.

Beyond Φ⁴₂, the authors outline a preliminary simulation pipeline for the more challenging dynamic Φ⁴₃ model. Following the lattice renormalization scheme of Zhu & Zhu (2018), the continuous SPDE is replaced by a finite‑dimensional SDE on a periodic three‑dimensional lattice. The linear diffusion term is treated implicitly in Fourier space via an FFT‑diagonalized operator, while the cubic drift and the mass‑shift term m_ε = 3C₀ − 9C₁ are applied pointwise in physical space. The stochastic forcing is sampled as i.i.d. Gaussian lattice noise with variance Δt ε⁻³ per site. Only the principal part of the counterterm C₁ is approximated; the bounded side‑band contribution is set to zero, making the current implementation a “partial” renormalization. The authors plan to incorporate the missing side‑band term and adopt a stochastic exponential Runge–Kutta integrator to improve numerical fidelity. Figure 2 visualizes the field at t = 0 and t = 1, demonstrating that the pipeline can generate plausible Φ⁴₃ trajectories suitable for future learning experiments.

The paper concludes by emphasizing that WCE‑FiLM‑NO respects the analytic structure of singular SPDEs: it leverages the exact representation of stochastic driving terms via Wiener chaos, and it uses FiLM to learn the affine relationship dictated by DPDD. This results in a surrogate that achieves high accuracy without explicit renormalization constants, exhibits strong cross‑resolution robustness, and provides a solid foundation for tackling even higher‑dimensional singular SPDEs.

Future directions identified include (i) completing the full renormalization for Φ⁴₃ and testing long‑time stability, (ii) replacing the simple Conv2D FiLM generator with more expressive architectures such as Transformers or graph neural networks to capture richer space‑time dependencies, (iii) benchmarking against other recent SPDE‑specific operators (e.g., NORS, DLR‑Net) under identical data regimes, and (iv) exploring multi‑scale noise conditioning and hierarchical operator designs to further improve scalability to three‑dimensional quantum field models.