Physics-informed Neural Networks for Heterogeneous Poroelastic Media

TL;DR

This paper develops a physics-informed neural network framework for heterogeneous poroelastic media by coupling a composite neural network (CoNN) with Interface-PINNs (I-PINNs) to handle material interfaces. The method uses separate networks for each output field (displacement and pressure) in each sub-domain, sharing parameters but employing different activation functions across interfaces, and enforces interface conditions to capture jumps in fields and gradients. Compared to a single-network PINN and to XPINNs, the CoNN/I-PINN approach achieves substantially lower RMSEs (by at least one order of magnitude relative to XPINNs and by two orders relative to conventional PINNs) and competitive or superior runtimes, demonstrating strong performance on two 1D benchmark problems with discontinuous material properties. The study also shows that hard enforcement of boundary/initial conditions and Glorot initialization further enhance accuracy, and discusses limitations and directions for extending the framework to higher dimensions and moving interfaces with automatic hyperparameter optimization.

Abstract

This study presents a novel physics-informed neural network (PINN) framework for modeling poroelasticity in heterogeneous media with material interfaces. The approach introduces a composite neural network (CoNN) where separate neural networks predict displacement and pressure variables for each material. While sharing identical activation functions, these networks are independently trained for all other parameters. To address challenges posed by heterogeneous material interfaces, the CoNN is integrated with the Interface-PINNs or I-PINNs framework (Sarma et al. 2024, https://dx.doi.org/10.1016/j.cma.2024.117135), allowing different activation functions across material interfaces. This ensures accurate approximation of discontinuous solution fields and gradients. Performance and accuracy of this combined architecture were evaluated against the conventional PINNs approach, a single neural network (SNN) architecture, and the eXtended PINNs (XPINNs) framework through two one-dimensional benchmark examples with discontinuous material properties. The results show that the proposed CoNN with I-PINNs architecture achieves an RMSE that is two orders of magnitude better than the conventional PINNs approach and is at least 40 times faster than the SNN framework. Compared to XPINNs, the proposed method achieves an RMSE at least one order of magnitude better and is 40% faster.

Figures (12)

• Figure 1: Problem domain in one dimension: The bottom surface is fixed, while the top surface is subjected to stress $s_0$; no-flux boundary condition is imposed on the bottom surface, while a zero-pressure condition is imposed on the top.
• Figure 2: Schematic of (a) a conventional PINNs model, (b) I-PINNs for a problem in 1-D domain with two distinct regions divided by an interface. This modified setup involves the partitioning of input variables within these distinct domains, subsequently inputting them into separate neural networks utilizing different activation functions.
• Figure 3: I-PINNs for 1-D coupled poroelasticity with an interface using a composite neural network (CoNN) architecture. Networks $\text{NN}_1^p$ and $\text{NN}_2^p$ model pressures across $\Omega_1$ and $\Omega_2$ with identical parameters ($\text{params}^p$) but distinct activation functions ($\text{AF}_1$ and $\text{AF}_2$), while $\text{NN}_1^u$ and $\text{NN}_2^u$ model displacements across $\Omega_1$ and $\Omega_2$ with shared parameters ($\text{params}^u$) and different activation functions ($\text{AF}_1$ and $\text{AF}_2$). In a nutshell: across subdomains, activation functions vary; between fields, only parameters differ.
• Figure 4: I-PINNs architecture for 1-D coupled poroelasticity with an interface using a single neural network (SNN) for each subdomain ($\text{NN}_1$ and $\text{NN}_2$ for $\Omega_1$ and $\Omega_2$ respectively). Each neural network outputs two field variables (pressures and displacements) for their respective sub-domains. These neural networks share the same set of parameters ‘params’ but use different activation functions $\text{AF}_1$ and $\text{AF}_2$ (in accordance to I-PINNs).
• Figure 5: Representative figure where training points are sampled from – inside the two sub-domains (collocation points in $\Omega_1$ and $\Omega_2$), on the two boundaries (BC at $x=0$ and $x=1$) and at the initial condition (IC at $t=0$), using two sampling techniques: (a) in a structured grid, (b) in a biased manner.