Application of Denoising Diffusion Probabilistic Model in solving the Cauchy Problem for the Laplace Equation on a unit disk

Yuriy Muzychuk, Rostyslav Solopatych


DOI: http://dx.doi.org/10.30970/vam.2026.36.14084

Анотація


In this paper we study the numerical solution of the Cauchy problem for the Laplace equation in a unit disk using Denoising Diffusion Probabilistic Model (DDPM). The Dirichlet and Neumann data are prescribed on the top half of the boundary while the bottom half and the interior are considered unknown. The inverse problem is formulated as conditional generation of a harmonic field represented as a multi-channel image encoding of the domain, boundary geometry, observed boundary, observed Dirichlet and Neumann values. The model is trained to generate a single-channel image that represents the numerical solution within the interior of the domain, discretized on a grid at a prescribed pixel resolution. A synthetic training dataset is generated by sampling the Fourier coefficients associated with the closed-form analytical solution of the well-posed problem, using as Neumann boundary data a truncated Fourier series up to mode K. Empirical evaluations on datasets comprising up to 50,000 samples at resolution of 64×64 indicate that the proposed model degrades gracefully under the tested Dirichlet noise levels. These findings suggest that conditional diffusion models may serve as learned surrogates for certain PDE inverse problems, though broader validation is still needed.

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Посилання


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