VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Models

TL;DR

VideoPDE introduces a unified transformer-based diffusion model that treats PDE solving as a video inpainting task, enabling flexible, high-fidelity, and efficient solutions for various PDE problems.

Contribution

It unifies PDE solving tasks under a single generative framework using video inpainting diffusion models, a novel approach compared to specialized existing methods.

Findings

Outperforms state-of-the-art PDE solvers across multiple tasks

Provides high-fidelity, flexible PDE solutions with a unified model

Enhances computational efficiency through hierarchical modeling

Abstract

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.