用谱空间扩散模型生成满足物理方程的解，精度和效率都更高。
Physics-informed diffusion models in spectral space
- 在缩放谱表示的隐空间中用扩散过程联合学习方程参数与解的分布
- 对泊松、赫姆霍兹和不可压缩纳维-斯托克斯方程测试，优于现有最优扩散求解器
- 适合需要高精度物理一致解的科学计算场景，如逆问题和稀疏观测
我们提出物理信息谱扩散（PISD），一种将生成式隐空间扩散模型与物理信息机器学习结合的方法，用于生成偏微分方程（PDE）的解，条件为部分观测数据，涵盖前向和逆PDE问题。通过在缩放谱表示的隐空间中进行扩散过程，学习方程参数与解的联合分布，其中高斯噪声对应具有可控正则性的函数。该谱形式实现相比网格基扩散模型显著的降维，并确保函数空间中的过程始终处于使PDE算子有定义的函数类中。基于扩散后验采样，在推断时每一步使用基于Adam的更新施加物理约束和测量条件。我们在泊松方程、赫姆霍兹方程和不可压缩纳维-斯托克斯方程上评估该方法，结果表明其在稀疏观测下相较现有基于扩散的PDE求解器具有更高的精度和计算效率。代码已公开于https://github.com/deeplearningmethods/PISD。
原文摘要 · Abstract (English)
We propose physics-informed spectral diffusion (PISD), a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems. We learn the joint distribution of PDE parameters and solutions via a diffusion process in a latent space of scaled spectral representations, where Gaussian noise corresponds to functions with controlled regularity. This spectral formulation enables significant dimensionality reduction compared to grid-based diffusion models and ensures that the induced process in function space remains within a class of functions for which the PDE operators are well defined. Building on diffusion posterior sampling, we enforce physics-informed constraints and measurement conditions during inference, applying Adam-based updates at each diffusion step. We evaluate the proposed approach on Poisson, Helmholtz, and incompressible Navier-Stokes equations, demonstrating improved accuracy and computational efficiency compared with existing diffusion-based PDE solvers, which are state of the art for sparse observations. Code is available at https://github.com/deeplearningmethods/PISD.
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