Inverse Physics-Informed Neural Networks for transport models in porous materials

TL;DR

This paper proposes an adaptive inverse PINN applied to different transport models, from diffusion to advection-diffusion-reaction problems, and finds that, for the inverse problem to converge to the correct solution, the different components of the loss function need to be weighted adaptively as a function of the training iteration.

Abstract

Physics-Informed Neural Networks (PINN) are a machine learning tool that can be used to solve direct and inverse problems related to models described by Partial Differential Equations. This paper proposes an adaptive inverse PINN applied to different transport models, from diffusion to advection-diffusion-reaction problems. Once a suitable PINN is established to solve the forward problem, the transport parameters are added as trainable parameters. We find that, for the inverse problem to converge to the correct solution, the different components of the loss function (data misfit, initial conditions, boundary conditions and residual of the transport equation) need to be weighted adaptively as a function of the training iteration (epoch). Similarly, gradients of trainable parameters are scaled at each epoch accordingly. Several examples are presented for different test cases to support our PINN architecture and its scalability and robustness.

Figures (16)

• Figure 1: A graphical representation of the mobile-immobile model for the transport of solutes in porous media: as in DeSmedt_Wierenga_1979, the mobile region is the primary zone of water and solute transport; the other region is termed the immobile zone, because the soil water in this zone is stagnant relative to the water in the mobile zone.
• Figure 2: PINN structure used in this work, with $L$ layers, $m_l$ neurons per layer, and hyperbolic tangent activation function in the hidden layers. The set $\theta_0\subseteq\{V,D,\lambda\}$ contains the trainable parameters relative to the considered physical law.
• Figure 3: Qualitative behaviour of $\nu_k$ in \ref{['eq:nu_k']} for $K=5000$ and $K_0=1000$.
• Figure 4: Solution of the pure diffusion problem for the final parameter values. PINN approximation (continuous line) and reference data (dashed line). Concentration as a function of space for different times (left) and concentration as a function of time for different space locations (right).
• Figure 5: Relative error for the diffusion coefficient and the solution $u$ during the training (left) and gradients of the diffusion coefficient (right).