Denoising Diffusion Restoration Tackles Forward and Inverse Problems for the Laplace Operator
CoRR(2024)
摘要
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.
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