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.

翻译：本研究探讨了在用于求解偏微分方程的物理信息神经网络中使用可学习激活函数的效果。具体而言，我们将采用固定激活函数和可学习激活函数的传统多层感知机，与采用可学习基函数的Kolmogorov-Arnold网络进行了效能比较。物理信息神经网络已成为一种将物理定律直接融入学习过程的有效方法，为偏微分方程相关的正问题和反问题提供了数据高效的解决方案。然而，诸如有效训练和频谱偏差等挑战——即神经网络更倾向于学习低频分量——常常限制了其在具有快速振荡或急剧变化特征问题上的适用性。通过在多层感知机和KAN上采用不同的激活函数或基函数，我们评估了它们对收敛行为、频谱偏差缓解以及偏微分方程精确逼近的影响。研究结果为设计能够平衡训练效率、收敛速度和测试精度的偏微分方程求解器神经网络架构提供了见解。通过评估激活函数或基函数选择的影响，本工作为开发更鲁棒、更精确的PINN模型提供了指导。本研究中使用的源代码和预训练模型已公开，以促进可重复性和未来的探索。