We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric 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.

翻译：我们提出了一种将生成式潜在扩散模型与物理信息机器学习相结合的方法，用于生成基于部分观测条件参数化偏微分方程（PDE）的解，特别涵盖正演与反演PDE问题。我们通过在缩放谱表示构成的潜在空间中建立扩散过程，学习PDE参数与解的联合分布，其中高斯噪声对应于具有可控正则性的函数。相较于基于网格的扩散模型，这种谱表示形式实现了显著的维度缩减，并确保函数空间中诱导的过程始终处于PDE算子明确定义的函数类中。基于扩散后验采样方法，我们在推理过程中强制施加物理信息约束与测量条件，在每一步扩散迭代中应用基于Adam优化器的更新策略。我们在泊松方程、亥姆霍兹方程以及不可压缩纳维-斯托克斯方程上评估了所提方法，结果表明相较于现有基于扩散的PDE求解器（在稀疏观测条件下处于领先水平），本方法在精度与计算效率方面均有提升。代码发布于 https://github.com/deeplearningmethods/PISD。