Training Deep Physics-Informed Kolmogorov-Arnold Networks

Abstract

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 nine 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.

84 pages

References in corpus (14)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks
• A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks
• Deep Learning of Vortex Induced Vibrations
• Efficient training of physics-informed neural networks via importance sampling
• Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks
• Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels
• Non-invasive Inference of Thrombus Material Properties with Physics-informed Neural Networks
• Residual-based attention in physics-informed neural networks
• A physics-informed neural network for quantifying the microstructure properties of polycrystalline Nickel using ultrasound data
• Learning Unknown Physics of non-Newtonian Fluids
• Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networks
• Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks
• Learning in PINNs: Phase transition, diffusion equilibrium, and generalization