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：扩散模型作为一类将含噪输入映射为逼真图像的生成模型，已展现出巨大潜力。近年来，它们被用于生成偏微分方程的解，但面对拉普拉斯算子相关的逆问题（如泊松方程）时仍存在局限——这是因为大特征值会放大测量噪声。本文提出一种创新方法，通过去噪扩散修复模型实现偏微分方程的逆问题与正问题求解。该模型在线性逆问题中利用线性算子的奇异值分解恢复原始干净信号。等效地，我们通过拉普拉斯算子的特征值与特征函数，提出了恢复泊松方程解与参数的方法。实验结果表明，采用去噪扩散修复显著提升了解与参数的估计精度。本研究开创性地将扩散模型与基础物理原理相融合，为偏微分方程求解开辟了新途径。