# Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equations

arXiv: https://arxiv.org/abs/2411.15111 · HTML: https://mlsanity.com/paper/2411.15111/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 79 references that are in the map.
- First posted on arXiv 22 Nov 2024.

## Builds on (most cited first, 30 of 50)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [Wav-KAN: Wavelet Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.12832/) (2024)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [A survey on modern trainable activation functions](https://mlsanity.com/paper/2005.00817/) (2020)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics](https://mlsanity.com/paper/2407.04192/) (2024)
- [On the Maximum Hessian Eigenvalue and Generalization](https://mlsanity.com/paper/2206.10654/) (2022)
- [BSRBF-KAN: A combination of B-splines and Radial Basis Functions in Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.11173/) (2024)
- [Enhancing Graph Collaborative Filtering with FourierKAN Feature Transformation](https://mlsanity.com/paper/2406.01034/) (2024)

## Built on by (most cited first, 7 of 7)

- [A Practitioner's Guide to Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2510.25781/) (2025)
- [QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2503.16678/) (2025)
- [Making Gaussian Kolmogorov-Arnold Networks Reliable and Accurate](https://mlsanity.com/paper/2604.21174/) (2026)
- [Learning Fluid-Structure Interaction with Physics-Informed Machine Learning and Immersed Boundary Methods](https://mlsanity.com/paper/2505.18565/) (2025)
- [High-Fidelity Modeling of Stochastic Chemical Dynamics on Complex Manifolds: A Multi-Scale SIREN-PINN Framework for the Curvature-Perturbed Ginzburg-Landau Equation](https://mlsanity.com/paper/2601.08104/) (2026)
- [A Family of Adaptive Activation Functions for Mitigating Failure Modes in Physics-Informed Neural Networks](https://mlsanity.com/paper/2603.18328/) (2026)
- [Residual Attention Physics-Informed Neural Networks for Robust Multiphysics Simulation of Steady-State Electrothermal Energy Systems](https://mlsanity.com/paper/2603.23578/) (2026)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving Partial Differential Equations (PDEs). However, they face challenges related to spectral bias (the tendency to learn low-frequency components while struggling with high-frequency features) and unstable convergence dynamics (mainly stemming from the multi-objective nature of the PINN loss function). These limitations impact their accuracy for problems involving rapid oscillations, sharp gradients, and complex boundary behaviors. We systematically investigate learnable activation functions as a solution to these challenges, comparing Multilayer Perceptrons (MLPs) using fixed and learnable activation functions against Kolmogorov-Arnold Networks (KANs) that employ learnable basis functions. Our evaluation spans diverse PDE types, including linear and non-linear wave problems, mixed-physics systems, and fluid dynamics. Using empirical Neural Tangent Kernel (NTK) analysis and Hessian eigenvalue decomposition, we assess spectral bias and convergence stability of the models. Our results reveal a trade-off between expressivity and training convergence stability. While learnable activation functions work well in simpler architectures, they encounter scalability issues in complex networks due to the higher functional dimensionality. Counterintuitively, we find that low spectral bias alone does not guarantee better accuracy, as functions with broader NTK eigenvalue spectra may exhibit convergence instability. We demonstrate that activation function selection remains inherently problem-specific, with different bases showing distinct advantages for particular PDE characteristics. We believe these insights will help in the design of more robust neural PDE solvers.