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

2024-08-06

arXiv.org

AI Summary

For multiscale dynamical systems—such as turbulent flows and Earth system models—that lack scale separation, conventional deterministic, local closure models suffer from poor generalizability, while direct numerical simulation remains computationally prohibitive. This paper proposes a data-driven, stochastic, nonlocal PDE closure framework: it introduces the first integration of Fourier Neural Operators (FNOs) into conditional fractional diffusion models to enable learnable, nonlocal modeling of stochastic closure terms over continuous spatiotemporal fields; and incorporates accelerated Langevin sampling to enhance inference efficiency. The method significantly improves closure generalizability and uncertainty quantification capability while preserving high fidelity and achieving substantial computational speedup. It constitutes the first generative stochastic closure approach for scale-separation–challenged systems that simultaneously satisfies theoretical consistency and practical applicability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Develop stochastic closure models for multiscale dynamical systems
• Enhance generalization using conditional diffusion and neural operators
• Improve efficiency with accelerated sampling methods for PDEs

Innovation

• Data-driven stochastic closure modeling framework
• Conditional diffusion model with neural operator
• Accelerated sampling for efficient model performance