Towards Deep Physics-Informed Kolmogorov-Arnold Networks
This addresses a specific bottleneck for researchers using physics-informed machine learning on PDE problems, though it appears incremental as it builds on existing KAN and PirateNet architectures.
The paper tackles training instability in deep physics-informed Kolmogorov-Arnold Networks (cPIKANs) by proposing a new initialization scheme and Residual-Gated Adaptive KANs (RGA KANs), which outperform baseline methods by several orders of magnitude on seven PDE benchmarks while maintaining stability.
Since their introduction, Kolmogorov-Arnold Networks (KANs) have been successfully applied across several domains, with physics-informed machine learning (PIML) emerging as one of the areas where they have thrived. In the PIML setting, Chebyshev-based physics-informed KANs (cPIKANs) have become the standard due to their computational efficiency. However, like their multilayer perceptron-based counterparts, cPIKANs face significant challenges when scaled to depth, leading to training instabilities that limit their applicability to several PDE problems. To address this, we propose a basis-agnostic, Glorot-like initialization scheme that preserves activation variance and yields substantial improvements in stability and accuracy over the default initialization of cPIKANs. Inspired by the PirateNet architecture, we further introduce Residual-Gated Adaptive KANs (RGA KANs), designed to mitigate divergence in deep cPIKANs where initialization alone is not sufficient. Through empirical tests and information bottleneck analysis, we show that RGA KANs successfully traverse all training phases, unlike baseline cPIKANs, which stagnate in the diffusion phase in specific PDE settings. Evaluations on seven standard forward PDE benchmarks under a fixed training pipeline with adaptive components demonstrate that RGA KANs consistently outperform parameter-matched cPIKANs and PirateNets - often by several orders of magnitude - while remaining stable in settings where the others diverge.