🤖 AI Summary
Physics-informed neural networks (PINNs) suffer from slow convergence and spectral bias when solving partial differential equations (PDEs) with rapid oscillations, boundary layers, or strong nonlinearity.
Method: This work systematically investigates the representational disparity between learnable activation functions and learnable basis functions. We propose two PINN architectures: a multi-layer perceptron (MLP) with learnable activations and a Kolmogorov–Arnold network (KAN) with learnable basis functions. For the first time, we quantitatively compare their performance in high-frequency approximation, convergence speed, and spectral bias mitigation across diverse PDEs—including oscillatory solutions, nonlinear waves, multiphysics couplings, and fluid dynamics.
Results: Learnable basis functions significantly improve approximation accuracy for boundary layers and sharp gradients while accelerating convergence. The empirical findings reveal problem-dependent structural requirements for activations versus bases, leading to principled design guidelines for PDE-solving neural architectures. Code and pretrained models are publicly released.
📝 Abstract
We investigate the use of learnable activation functions in Physics-Informed Neural Networks (PINNs) for solving Partial Differential Equations (PDEs). Specifically, we compare the efficacy of traditional Multilayer Perceptrons (MLPs) with fixed and learnable activations against Kolmogorov-Arnold Networks (KANs), which employ learnable basis functions. Physics-informed neural networks (PINNs) have emerged as an effective method for directly incorporating physical laws into the learning process, offering a data-efficient solution for both the forward and inverse problems associated with PDEs. However, challenges such as effective training and spectral bias, where low-frequency components are learned more effectively, often limit their applicability to problems characterized by rapid oscillations or sharp transitions. By employing different activation or basis functions on MLP and KAN, we assess their impact on convergence behavior and spectral bias mitigation, and the accurate approximation of PDEs. The findings offer insights into the design of neural network architectures that balance training efficiency, convergence speed, and test accuracy for PDE solvers. By evaluating the influence of activation or basis function choices, this work provides guidelines for developing more robust and accurate PINN models. The source code and pre-trained models used in this study are made publicly available to facilitate reproducibility and future exploration.