Denoising Diffusion Restoration Tackles Forward and Inverse Problems for the Laplace Operator
This work addresses inverse problems in PDEs for computational physics and engineering, representing a novel integration of diffusion models with physical principles rather than an incremental improvement.
The paper tackles the challenge of solving inverse and forward problems for the Laplace operator, such as the Poisson equation, where noise amplification hinders performance, and demonstrates that using denoising diffusion restoration models (DDRM) significantly improves estimation of solutions and parameters.
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.