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.

翻译：扩散模型已成为一类有前景的生成模型，能够将带噪声的输入映射为真实图像。近期，它们被用于生成偏微分方程的解。然而，这类模型在处理拉普拉斯算子的逆问题（例如泊松方程）时仍面临困难，因为幅值较大的特征值会放大测量噪声。本文提出了一种新颖方法，通过去噪扩散恢复模型求解偏微分方程的逆问题与正问题。去噪扩散恢复模型通过利用线性算子的奇异值分解恢复原始干净信号，在求解线性逆问题中已有应用。类比于此，我们提出利用拉普拉斯算子的特征值与特征函数来恢复泊松方程的解与参数。实验结果表明，采用去噪扩散恢复策略显著提升了解与参数的估计精度。因此，本研究开创性地将扩散模型与基础物理原理相结合，用于求解偏微分方程。