Generative diffusion learning for parametric partial differential equations

We develop a class of data-driven generative models that approximate the solution operator for parameter-dependent partial differential equations (PDEs). We introduce a probabilistic formulation of the operator learning problem based on denoising diffusion probabilistic models (DDPM), which enables learning the input-output mapping between problem parameters and PDE solutions. We generalize DDPM to a supervised setting, where the solution operator is represented by a family of conditional distributions. This probabilistic formulation, combined with DDPM, naturally provides uncertainty quantification through confidence intervals for the learned solutions. Moreover, the framework is directly applicable to learning solution operators from noisy data sets. We evaluate the computational performance of our method against Fourier Neural Operators and show that it achieves comparable accuracy while additionally recovering the noise magnitude in data sets corrupted by additive noise.