Reproduce “Denoising Diffusion Restoration Tackles Forward and Inverse Problems for the Laplace Operator” on OpenResearch

About this paper

Amartya Mukherjee, Melissa M. Stadt, Lena Podina, Mohammad Kohandel, Jun Liu

This University of Waterloo research demonstrates how integrating the physical properties of the Laplace operator, specifically its eigenfunctions and eigenvalues, into Denoising Diffusion Restoration Models (DDRMs) can address challenges in solving partial differential equations. The method shows improved accuracy for both forward (estimating solutions from parameters) and inverse (estimating parameters from solutions) problems compared to purely data-driven diffusion models and other deep learning benchmarks, achieving MAEs of 1.175 x 10^-6 for forward and 3.215 x 10^-2 for inverse problems.

Key insights

• Integrating the known eigenfunctions and eigenvalues of the Laplace operator into Denoising Diffusion Restoration Models (DDRMs) significantly enhances their ability to solve partial differential equations.
• Explicitly leveraging the spectral properties (via Discrete Sine Transform) allows DDRMs to effectively address the challenges of noise amplification in inverse problems for the Laplace operator.
• Physics-informed guidance in the diffusion sampling process, rather than purely data-driven approaches, leads to more accurate and robust solutions for both forward and inverse PDE problems.

Summary by alphaXiv.