Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator

TL;DR

This work tackles the lack of generalizable closures for multiscale PDEs without clear scale separation. It proposes a data-driven framework that couples a conditional score-based diffusion model with Fourier neural operators to learn the conditional distribution $p(U|y)$ of a closure term given the resolved state and sparse measurements. It introduces fast sampling strategies to make diffusion-based closures practical in time-stepping simulations and demonstrates resolution-invariance through the Fourier operator learning. Applied to a stochastic 2-D Navier–Stokes vorticity problem, the approach yields accurate stochastic closures for viscous diffusion and advection, enabling reliable surrogate simulations with substantial speedups.

Abstract

Closure models are widely used in simulating complex multiscale dynamical systems such as turbulence and the earth system, for which direct numerical simulation that resolves all scales is often too expensive. For those systems without a clear scale separation, deterministic and local closure models often lack enough generalization capability, which limits their performance in many real-world applications. In this work, we propose a data-driven modeling framework for constructing stochastic and non-local closure models via conditional diffusion model and neural operator. Specifically, the Fourier neural operator is incorporated into a score-based diffusion model, which serves as a data-driven stochastic closure model for complex dynamical systems governed by partial differential equations (PDEs). We also demonstrate how accelerated sampling methods can improve the efficiency of the data-driven stochastic closure model. The results show that the proposed methodology provides a systematic approach via generative machine learning techniques to construct data-driven stochastic closure models for multiscale dynamical systems with continuous spatiotemporal fields.

Figures (18)

• Figure 1: A schematic diagram of the proposed framework for stochastic closure modeling via score-based generative model and Fourier neural operator. The score function with multimodal inputs is constructed by FNOs and deployed in a conditional diffusion model, with which samples can be generated to serve as a data-driven closure term for a complex dynamical system whose dynamics are only partially known or even completely unknown.
• Figure 2: Left: spatial correlation map $C_{G, \omega}(\mathrm{x})$ calculated between $G(\mathrm{x}_c, t)$ and $\omega(\mathrm{x}, t)$. Right: spatial correlation map $C_{H, \omega}(\mathrm{x})$ calculated between $H(\mathrm{x}_c, t)$ and $\omega(\mathrm{x}, t)$.
• Figure 3: Generated $G(\mathrm{x}, t)$ without sparse information and conditioned only on current vorticity $\omega$. Training data resolution: $64 \times 64$. Test data resolution: $64 \times 64$. First row: random samples of the ground truth $G^\dag$. Second row: corresponding generated samples $G$. Third row: absolute error fields between $G^\dag$ and $G$. The indices $1$ to $4$ correspond to different snapshots randomly sampled from the test dataset.
• Figure 4: Generated $G(\mathrm{x},t)$ conditioned on current vorticity $\omega$ and sparse information of $G^\dagger_{\textrm{sparse}}$. Sparse information is upscaled using bicubic interpolation. Training data resolution: $64 \times 64$. Test data resolution: $64 \times 64$. First row: random samples of the ground truth $G^\dag$. Second row: corresponding generated samples $G$. Third row: absolute error fields between the ground truth $G^\dag$ and generated $G$.
• Figure 5: Generated $G(\mathrm{x},t)$ conditioned on current vorticity $\omega$ and sparse information of $G^\dagger_{\textrm{sparse}}$. Sparse information is upscaled using 2-D convolution. Training data resolution: $64 \times 64$. Test data resolution: $64 \times 64$. First row: random samples of the ground truth $G^\dag$. Second row: corresponding generated samples $G$. Third row: absolute error fields between the ground truth $G^\dag$ and generated $G$.