Fully probabilistic deep models for forward and inverse problems in parametric PDEs

Arnaud Vadeboncoeur, Ömer Deniz Akyildiz, Ieva Kazlauskaite, Mark Girolami, Fehmi Cirak

Abstract: Highlights•We introduce a fully probabilistic approach to simultaneously learn the forward and inverse maps of parametric PDEs.•In the probabilistic model, the PDE residual is an observed random variable with zero mean and user-prescribed variance.•The marginal likelihood for observing a zero valued residual is maximized by considering the evidence lower bound (ELBO).•The forward and inverse maps are approximated as Gaussians with a mean and covariance parameterized by neural networks.•The proposed model is, after training, suitable for real-time solutions of the forward and inverse problems.