Title: Probabilistic DiffusionNet: A geometry informed probabilistic generative surrogate for PDEs
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: SPDE, DiffusionNet, PDE, CFD, UQ, Variational Inference, VAE, Generative Models, Stochastic Process, Gaussian Process, Neural Operators

Abstract: We propose a probabilistic generative extension of the DiffusionNet architecture, widely used for surface learning tasks, by introducing latent random variables derived from a stochastic reformulation of the underlying diffusion process. The resulting probabilistic model can be used as a resolution-invariant and uncertainty-aware surrogate for the trace solution map of PDEs whose boundary conditions are determined by surface geometry. Such a surrogate can expedite and inform typical engineering design and optimisation processes that require computationally burdensome computational fluid dynamics (CFD) analysis pipelines. We demonstrate that the proposed architecture produces superior uncertainty quantification (UQ) performance on standard CFD datasets without sacrificing predictive accuracy, while enjoying lower computational cost compared to other prevalent geometry-informed PDE surrogates endowed with UQ capabilities.

TL;DR: We reformulate DiffusionNet as a probabilistic model to learn PDE solutions with UQ on 3D geometries