Diffusion models have emerged as a promising class of generative models that map noisy inputs to realistic images. More recently, they have been employed to generate solutions to partial differential equations (PDEs). However, they still struggle with inverse problems in the Laplacian operator, for instance, the Poisson equation, because the eigenvalues that are large in magnitude amplify the measurement noise. This paper presents a novel approach for the inverse and forward solution of PDEs through the use of denoising diffusion restoration models (DDRM). DDRMs were used in linear inverse problems to restore original clean signals by exploiting the singular value decomposition (SVD) of the linear operator. Equivalently, we present an approach to restore the solution and the parameters in the Poisson equation by exploiting the eigenvalues and the eigenfunctions of the Laplacian operator. Our results show that using denoising diffusion restoration significantly improves the estimation of the solution and parameters. Our research, as a result, pioneers the integration of diffusion models with the principles of underlying physics to solve PDEs.

基于扩散模型和物理原理，本研究提出了一种新的方法来解决偏微分方程的逆问题和正问题，通过利用去噪扩散恢复模型（DDRM）来恢复解和参数，从而显著改进了解和参数的估计。

去噪扩散还原处理拉普拉斯算子的正向和反向问题