Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems

TL;DR

This work develops Finite Basis Kolmogorov–Arnold Networks (FBKANs), a domain-decomposition framework that couples multiple small KANs via a partition of unity to tackle multiscale and physics-informed problems efficiently. FBKANs extend prior FBPINN ideas to KAN architectures, enabling parallel training and improved accuracy without enforcing inter-domain transmission conditions through penalties. Data-driven results show robust performance under noise and clear scaling benefits with more subdomains, while physics-informed tests demonstrate superior handling of multiscale and wave-like problems. The paper also introduces multilevel FBKANs (MLFBKANs), which further enhance accuracy for high-frequency and multiscale scenarios, suggesting broad applicability and compatibility with other PINN and KAN enhancements.

Abstract

Kolmogorov-Arnold networks (KANs) have attracted attention recently as an alternative to multilayer perceptrons (MLPs) for scientific machine learning. However, KANs can be expensive to train, even for relatively small networks. Inspired by finite basis physics-informed neural networks (FBPINNs), in this work, we develop a domain decomposition method for KANs that allows for several small KANs to be trained in parallel to give accurate solutions for multiscale problems. We show that finite basis KANs (FBKANs) can provide accurate results with noisy data and for physics-informed training.

Figures (17)

• Figure 1: Graphical abstract: The computational domain is decomposed into overlapping subdomains, and an individual KAN model is defined on each subdomain. Then, the global model output is obtained by scaling the outputs of the local KAN models with partition of unity functions and summing up the local contributions.
• Figure 2: Example of the partition of unity functions on the domain $\Omega = [0, 2]$ with $L=4$ subdomains.
• Figure 3: (Left) Example domain decomposition on the domain $\Omega = [-1, 1]\times[-1, 1]$ with $L=4$ subdomains. (Right) One example partition of unity function $\omega_{11}(x, y).$
• Figure 4: Results with $L=2, 8,$ and $32$ subdomains for \ref{['eq:Case_1']}. (a) Training data and plot of exact $f(x)$. (b) Loss curves (\ref{['eq:loss_KAN']}). (c) Plot of the outputs $f(x)$ and $\sum_{j=1}^L \omega_j(x)\mathcal{K}_j(x)$. (d) Pointwise errors $f(x) - \sum_{j=1}^L \omega_j(x)\mathcal{K}_j(x)$. (e) Scaling results for Test 1 with $L$ subdomains.
• Figure 5: Results for Test 1 with noisy training data; cf. \ref{['eq:Case_1']}. (a) Example training data and plot of exact $f(x)$ with $9.6\%$ mean relative noise. (b) Loss curves (\ref{['eq:loss_KAN']}) for an example training with $9.6\%$ mean relative noise. (c) Plot of the outputs $f(x)$ and $\sum_{j=1}^L \omega_j(x)\mathcal{K}_j(x)$ with $9.6\%$ mean relative noise. (d) Pointwise errors $f(x) - \sum_{j=1}^L \omega_j(x)\mathcal{K}_j(x)$ with $9.6\%$ mean relative noise. (e) Relative $\ell_2$ error of the KANs and FBKANs with respect to the magnitude of the noise added to the training data.