VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Models

2025-06-16

AI Summary

This work addresses the unified solution of both forward and inverse partial differential equation (PDE) problems—under fully observed or partially observed (incomplete) conditions. The method reformulates PDE solving as a spatiotemporal video inpainting task, leveraging a pixel-level video diffusion model enhanced with spatiotemporal masking conditioning and hierarchical conditional modeling to generate high-fidelity spatiotemporal solutions from initial/boundary/sparse observations. It is the first framework to enable end-to-end unified modeling across forward/inverse and complete/incomplete-observation PDE scenarios. Technically, it introduces a video diffusion Transformer architecture coupled with hierarchical computational optimization strategies. Evaluated on diverse PDE families—including reaction-diffusion, Navier–Stokes, and wave equations—the approach achieves significant improvements over state-of-the-art methods in accuracy, generalization across unseen PDEs and domains, and inference efficiency.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Deep Neural Architectures and Foundation Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Responsible Web: Data and user privacy-enhancing technologies for the Web

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.

Problem

• Unified framework for solving PDEs using video-inpainting diffusion models
• Recasts PDE-solving as generalized spatiotemporal inpainting problem
• Enhances accuracy and efficiency across diverse PDE problem setups

Innovation

• Unified PDE solving via video-inpainting diffusion models
• Transformer-based architecture for spatiotemporal inference
• Hierarchical modeling enhances computational efficiency