Applying Convex Weighting Physics-Informed Neural Networks to Subsurface Modeling and Characterization Problems

The physics-informed neural network (PINN) approach that enforces the approximating functions to satisfy physical constraints and conservation laws for subsurface applications has received increasing attention, as it provides a promising alternative to conduct data assimilation for heterogeneous porous media with sparse measurement [1]. However, training the PINN model remains difficult due to the complex interplay between different terms (e.g., the residuals of different governing equations, the mismatch to the measurements, etc.) in the loss function and even the conflict arising from the uncertain measurements and partially known physical models. To remediate this fundamental issue, we introduce a convex weighting scheme to the PINN approach inspired by the Pareto optimal solution, called convex weighting physics-informed neural networks (CwPINNs). In this scheme, the gradients associated with each loss term are used to construct a convex hull to a given query point, from which the "optimal" gradient direction is chosen adaptively during model training. This scheme can be equivalently viewed as applying the Pareto optimal weights to scale the loss terms to balance their contributions in driving the learning process. The proposed CwPINN method is compared numerically to the original methods on a variety of subsurface transport problems, including steady and time-dependent forward modeling and parameter estimation. The numerical experiments show that the proposed method significantly enhances the predictive accuracy and robustness of the trained PINN.

References:

[1] He, Q., Barajas-Solano, D., Tartakovsky, G., & Tartakovsky, A. M. (2020). Physics-informed neural networks for multiphysics data assimilation with application to subsurface transport. Advances in Water Resources, 103610.