Wavelet Diffusion Neural Operator

• URL: http://arxiv.org/abs/2412.04833v2
• Date: Thu, 17 Apr 2025 12:34:20 GMT
• Title: Wavelet Diffusion Neural Operator
• Authors: Peiyan Hu, Rui Wang, Xiang Zheng, Tao Zhang, Haodong Feng, Ruiqi Feng, Long Wei, Yue Wang, Zhi-Ming Ma, Tailin Wu,
• Abstract summary: We propose Wavelet Neural Diffusion Operator (WDNO), a novel PDE simulation and control framework.<n>WDNO performs diffusion-based generative modeling in the wavelet domain to handle abrupt changes and long-term dependencies effectively.<n>To address the issue of poor generalization across different resolutions, we introduce multi-resolution training.
• Score: 17.617919636212445
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Simulating and controlling physical systems described by partial differential equations (PDEs) are crucial tasks across science and engineering. Recently, diffusion generative models have emerged as a competitive class of methods for these tasks due to their ability to capture long-term dependencies and model high-dimensional states. However, diffusion models typically struggle with handling system states with abrupt changes and generalizing to higher resolutions. In this work, we propose Wavelet Diffusion Neural Operator (WDNO), a novel PDE simulation and control framework that enhances the handling of these complexities. WDNO comprises two key innovations. Firstly, WDNO performs diffusion-based generative modeling in the wavelet domain for the entire trajectory to handle abrupt changes and long-term dependencies effectively. Secondly, to address the issue of poor generalization across different resolutions, which is one of the fundamental tasks in modeling physical systems, we introduce multi-resolution training. We validate WDNO on five physical systems, including 1D advection equation, three challenging physical systems with abrupt changes (1D Burgers' equation, 1D compressible Navier-Stokes equation and 2D incompressible fluid), and a real-world dataset ERA5, which demonstrates superior performance on both simulation and control tasks over state-of-the-art methods, with significant improvements in long-term and detail prediction accuracy. Remarkably, in the challenging context of the 2D high-dimensional and indirect control task aimed at reducing smoke leakage, WDNO reduces the leakage by 33.2% compared to the second-best baseline. The code can be found at https://github.com/AI4Science-WestlakeU/wdno.git.

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