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

TL;DR

This work extends physics-informed neural networks to forward uncertainty quantification in PDEs by embedding uncertain parameters into the input of the neural solver (PINN-UU). Using a transfer-learning, stage-wise training strategy, the framework handles high-dimensional parametric spaces without dense observational data, demonstrated on a 1D advective–dispersive transport problem with Langmuir sorption. PINN-UU accurately reproduces mean, variance, and PDFs of outlet concentrations relative to a reference finite-difference solver and supports both global and local sensitivity analyses (Sobol and DELSA) with fast post-processing. The results suggest PINN-UU as an efficient, physics-driven alternative for robust UQ in groundwater systems, with potential extensions to heterogeneous media and more complex reaction terms.

Abstract

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.

Figures (9)

• Figure 1: PINN-UU architecture. Here, $\xi$ denotes a random number generator associated with a desired distribution (e.g., uniform) and employed to sample training points in the $\Omega_{\boldsymbol{\lambda}}$ space.
• Figure 2: Model tuning: distribution of the relative percentage errors $RE_{L2}$ (Eq. \ref{['RE']}) between the reference FDM solution and the examined PINN-UU architectures. Shaded areas with different colors, blue and grey, correspond to the two activation functions employed, i.e., Tanh and SiLU, respectively. Results are shown for three different regimes corresponding to a) low, b) medium, and c) high Péclet, as well as d) overall. Dashed lines correspond to the first, second, and third quartiles. The loosely dashed line and the densely dashed lines represent the median, and $(25^{th},75^{th})$ percentiles, respectively.
• Figure 3: Space-time distribution of the results of the best performing PINN-UU, $u_{pre}$ ($1^\text{st}$ column), reference Finite Difference solution, $u_{ref}$ ($2^\text{nd}$ column), and their corresponding point-wise error [$u_{pre} - u_{ref}$] ($3^\text{rd}$ column), for a) low, b) medium, and c) high Péclet.
• Figure 4: Space-time distribution of the results of the worst performing PINN-UU ($1^\text{st}$ column), to reference Finite Difference solution ($2^\text{nd}$ column), and their corresponding point-wise error ($3^\text{rd}$ column) for a) low, b) medium, and c) high Péclet.
• Figure 5: PINN-UU blind spots. Relative percentage errors $RE_{L2}$ (Eq. \ref{['RE']}) between the reference FDM solution and the examined PINN-UU architectures in the $K_l$ and $Q$ parameter space. Results are evaluated on the basis of $10^6$ Monte Carlo parameter combination samples and correspond to a) Low, b) Medium, and c) High Péclet.