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Published at November 22

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

cs.NE

Released Date: November 22, 2024

Authors: Afrah Fareaa¹, Mustafa Serdar Celebi

Aff.: ¹Department of Computational Science and Engineering, Istanbul Technical University, Istanbul 34469, Turkey

Arxiv: http://arxiv.org/abs/2411.15111v1

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Basis	Formula	Pros	Cons	Suitability
B-splines	$B_{i,k}(x)$	Flexibility and smoothness Local support Efficient computation	Cannot handle large complex structures.	Efficient for small-scale problems with local refinement needs
Chebyshev Polynomials	$T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x)$	Numerical stability Efficient evaluation Global approximation	Global nature Difficulties in local refinement	High numerical stability and efficiency for global approximations
Radial Basis Functions (RBFs)	$\phi(r)=e^{-(\epsilon r)^{2}}$	Flexibility and adaptability Local and global support Mesh-free method	Computational cost Parameter sensitivity	Best for complex geometries and moving boundaries due to mesh-free nature and adaptability
Hermite Polynomials	$H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}$	Suitable for Gaussian problems Orthogonal	Less flexible for non-Gaussian features	Gaussian-related Navier-Stokes problems
Wavelet Transformation	$\psi_{j,k}(x)=2^{j/2}\psi(2^{j}x-k)$	Captures localized features Multi-resolution analysis	Complex implementation Selecting basis is crucial	Problems with localized features and discontinuities
Legendre Polynomials	$P_{n}(x)=\frac{1}{2^{n}n!}\frac{d^{n}}{dx^{n}}\left((x^{2}-1)^{n}\right)$	Orthogonal basis Numerical stability	Global support	Polynomial approximation and integration
Gaussian Process Basis Functions	$k(x,x^{\prime})=\exp\left(-\frac{\\|x-x^{\prime}\\|^{2}}{2\sigma^{2}}\right)$	Probabilistic interpretation Flexibility in Basis functions choice	Computationally expensive	Problems requiring uncertainty quantification