Adaptive Interface-PINNs (AdaI-PINNs): An Efficient Physics-informed Neural Networks Framework for Interface Problems

TL;DR

The paper tackles solving elliptic PDEs with discontinuous coefficients across interfaces using PINNs. It introduces AdaI-PINNs, which learn the slopes of activation functions per subdomain via trainable parameters $a_m$, while keeping a shared network structure, thereby automating AF selection. Compared to I-PINNs, AdaI-PINNs demonstrate substantially lower training costs (2–6x faster) with equal or improved accuracy across 1D, 2D, and 3D benchmark problems, and exhibit rapid convergence of the adaptive parameters. This approach enhances robustness and efficiency for multi-material/interface problems and shows promise for extending to transient and moving-interface settings.

Abstract

We present an efficient physics-informed neural networks (PINNs) framework, termed Adaptive Interface-PINNs (AdaI-PINNs), to improve the modeling of interface problems with discontinuous coefficients and/or interfacial jumps. This framework is an enhanced version of its predecessor, Interface PINNs or I-PINNs (Sarma et al.; https://dx.doi.org/10.2139/ssrn.4766623), which involves domain decomposition and assignment of different predefined activation functions to the neural networks in each subdomain across a sharp interface, while keeping all other parameters of the neural networks identical. In AdaI-PINNs, the activation functions vary solely in their slopes, which are trained along with the other parameters of the neural networks. This makes the AdaI-PINNs framework fully automated without requiring preset activation functions. Comparative studies on one-dimensional, two-dimensional, and three-dimensional benchmark elliptic interface problems reveal that AdaI-PINNs outperform I-PINNs, reducing computational costs by 2-6 times while producing similar or better accuracy.

Figures (8)

• Figure 1: Schematic of the problem domain with two regions $\Omega_1$ and $\Omega_2$ separated by an interface $\Gamma_{\text{int}}$.
• Figure 2: Schematic of the architecture of Adaptive Interface-PINNs (AdaI-PINNs) for a model problem consisting of a domain having two regions separated by an interface.
• Figure 3: (a) Approximation to Eq. \ref{['eq_1d_possions_four_int']} by the AdaI-PINNs method compared to the closed-form analytical solution, (b) the iterative variation of the parameters $a_\text{m}$ corresponding to each sub-domain $\Omega_\text{m}$ within the adaptive tanh activation function.
• Figure 4: The iterative variation of total loss by both AdaI-PINNs and I-PINNs, for the 1D problem outlined in Eq. \ref{['eq_1d_possions_four_int']}. The training of the AdaI-PINNs model was clipped at 60,000 iterations as it reached convergence very quickly, while I-PINNs was left to train for 200000 iterations.
• Figure 5: (a) Contour plot of the analytical solution to Eq \ref{['eq:2d_iitm_equation']}, (b) approximation to the solution by AdaI-PINNs, and, (c) absolute error plot of the approximation obtained AdaI-PINNs (RMSE = $3.48 \times 10^{-6}$). The interfaces are demarcated by the white dash-dot lines.