Modelling parametric uncertainty in PDEs models via Physics-Informed Neural Networks

Résumé : We provide an approach enabling one to employ physics-informed neural networks (PINNs) for uncertainty quantification. Our approach is applicable to systems where observations are scarce (or even lacking), these being typical situations associated with subsurface water bodies. Our novel physics-informed neural network under uncertainty (PINN-UU) integrates the space-time domain across which processes take place and uncertain parameter spaces within a unique computational domain. PINN-UU is then trained to satisfy the relevant physical principles (e.g., mass conservation) in the defined input domain. We employ a stage training approach via transfer learning to accommodate high-dimensional solution spaces. We demonstrate the effectiveness of PINN-UU in a scenario associated with reactive transport in porous media, showcasing its reliability, efficiency, and applicability to sensitivity analysis. PINN-UU emerges as a promising tool for robust uncertainty quantification, with broad applicability to groundwater systems. As such, it can be considered as a valuable alternative to traditional methods such as multi-realization Monte Carlo simulations based on direct solvers or black-box surrogate models.