VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Models
This provides a flexible solution for researchers and engineers dealing with diverse PDE problems, though it is incremental as it builds on existing diffusion and inpainting methods.
The paper tackles solving partial differential equations (PDEs) by unifying forward and inverse problems into a single generative framework using video-inpainting diffusion models, achieving accurate and versatile results that outperform state-of-the-art baselines.
We present a unified framework for solving partial differential equations (PDEs) using video-inpainting diffusion transformer models. Unlike existing methods that devise specialized strategies for either forward or inverse problems under full or partial observation, our approach unifies these tasks under a single, flexible generative framework. Specifically, we recast PDE-solving as a generalized inpainting problem, e.g., treating forward prediction as inferring missing spatiotemporal information of future states from initial conditions. To this end, we design a transformer-based architecture that conditions on arbitrary patterns of known data to infer missing values across time and space. Our method proposes pixel-space video diffusion models for fine-grained, high-fidelity inpainting and conditioning, while enhancing computational efficiency through hierarchical modeling. Extensive experiments show that our video inpainting-based diffusion model offers an accurate and versatile solution across a wide range of PDEs and problem setups, outperforming state-of-the-art baselines.