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.
Abstract (translated)
我们提出了一种使用视频修复扩散变换器模型解决偏微分方程（PDE）的统一框架。与现有方法不同，后者为完整或部分观测下的正向问题或逆向问题设计了专门策略，我们的方法将这些任务统一在一个单一且灵活的生成框架下进行处理。具体来说，我们将求解PDE的问题重新表述为一个广义修复问题，例如，将其视为从初始条件推断未来状态缺失的空间-时间信息的过程。 为此，我们设计了一种基于变换器架构的方法，它可以根据任意已知数据模式，在时间和空间上推测出缺失的值。我们的方法采用像素空间视频扩散模型进行精细化、高保真度的修复和条件设定，并通过分层建模提高计算效率。 大量的实验表明，我们提出的基于视频修复的扩散模型在广泛的PDE问题设置中提供了准确且灵活的解决方案，超过了现有的最先进基准。
URL
https://arxiv.org/abs/2506.13754
