Multilevel domain decomposition-based architectures for physics-informed neural networks

TL;DR

This work extends physics-informed neural networks (PINNs) by integrating multilevel overlapping domain decomposition into finite basis PINNs (FBPINNs), enabling scalable and accurate solutions for high-frequency and multi-scale PDEs. By introducing multiple levels of subdomains with partition-of-unity windows, the multilevel FBPINN architecture facilitates improved global information exchange and reduced spectral bias, outperforming standard PINNs and single-level FBPINNs across Laplace and Helmholtz tests. The study defines strong and weak scaling tests to quantify performance under increasing computational effort and solution complexity, and demonstrates significant gains in accuracy and efficiency, with robust ablations and discussions of limitations and future directions. Taken together, the method offers a scalable, mesh-free alternative for challenging PDE regimes, with practical implications for SciML-enabled forward modeling and inverse problems.

Abstract

Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.

Figures (12)

• Figure 1: Scaling high frequency problems to low frequency problems using domain decomposition. FBPINNs decompose the domain into many subdomains, and use neural networks within each subdomain to learn the local solution. The input coordinates to each network are normalized to the range [-1,1] over their individual subdomains. When solving problems with high frequency solutions, this effectively scales each local problem from a high frequency problem to a lower frequency problem, and helps reduce the network's spectral bias.
• Figure 2: Plot of a square domain $\Omega$ decomposed into four overlapping subdomains, using a uniform rectangular decomposition.
• Figure 3: Example of a multilevel FBPINN solving Laplace's equation in one and two dimensions. For the 1D problem, the multilevel FBPINN uses $L=3$ levels, where each level has $1$, $2$ and $4$ subdomains respectively. The window functions, $\hat{\omega}^{(l)}_j(x)$, used for each level are shown in (a), the individual solutions learned by each subdomain network are shown in (b), and the multilevel FBPINN solution is shown in (c). For the 2D problem, the multilevel FBPINN uses $L=3$ levels, where each level has $1\times1$, $2\times2$ and $4\times4$ subdomains respectively, using a uniform rectangular DD. The DDs for level $2$ and level $3$ are plotted in (d) and (e), and the multilevel FBPINN solution is shown in (f). Note the subdomain boundaries and window functions extend past the problem domain (in this case, $[0,1]^d$). Example collocation points used to train the multilevel FBPINN are plotted in (a), (d) and (e).
• Figure 4: Hierarchy of levels used in the multilevel FBPINN. For all the multilevel FBPINNs tested we use an exponential level structure. This means that the number of subdomains in each level is given by $2^{d(l-1)}$, where $l$ is the level number and $d$ is the dimensionality of the domain. Our hypothesis is that this helps the multilevel FBPINN model solutions with frequency components that span multiple orders of magnitude.
• Figure 5: Ablation tests using the homogeneous Laplacian problem. The convergence curve of a baseline multilevel FBPINN is plotted when changing the number of levels (top right), overlap ratio (bottom left), and number of hidden units for each subdomain network (bottom right). The baseline model has $L=3$ levels, an overlap ratio of $\delta=1.9$, and 16 hidden units for each subdomain network. The exact solution is shown (top left). Convergence curves of two other benchmarks are shown; a PINN (bottom right), and one-level FBPINNs with varying numbers of subdomains (top right). The lists which label each model in the top right plot contain the number of subdomains along each dimension for each level in the model. Filled region edges show the minimum and maximum loss values across 10 random starting seeds and lines show the average.