Adaptive Interface-PINNs (AdaI-PINNs) for inverse problems: Determining material properties for heterogeneous systems

We determine spatially varying discontinuous material properties using a domain-decomposition based physics-informed neural networks (PINNs) framework named the Adaptive Interface-PINNs or AdaI-PINNs (Roy et al., 2024). We propose the use of distinct neural networks for the field variables and material properties within each material, utilizing adaptive activation functions. While the neural networks across different materials share the same weights and biases, their activation functions are uniquely tailored using a hyperparameter that influences the slope of the activation function. The proposed framework is tested on several one-dimensional and two-dimensional benchmark examples, and its performance is compared with conventional PINNs and existing domain-decomposition PINNs frameworks, namely, the Multi-domain physics-informed neural network (M-PINN), and the eXtended physics-informed neural networks (XPINNs). The results demonstrate that the proposed approach can determine randomly distributed discontinuous material properties with an $L_2$ error of $O\left(10^{-3}\right)$ for the material property and the root-mean-square error of $O\left(10^{-3}\right)$ for the primary variable while the other approaches yield errors that are approximately two orders of magnitude larger (that is, $O\left(10^{-1}\right)$). Moreover, the spatial distribution of material properties obtained using the proposed framework is in close agreement with the true distribution, whereas the other approaches fare much worse. Additionally, the proposed approach is approximately 40% faster than its competitors, indicating its potential as a robust alternative for solving inverse problems in heterogeneous materials.