Physics-informed kernel function neural networks for solving partial differential equations.

Area of Science:

• Computational Mathematics
• Machine Learning
• Numerical Analysis

Background:

• Partial differential equations (PDEs) are fundamental in modeling complex physical phenomena.
• Traditional numerical methods for PDEs can be computationally intensive.
• Physics-informed neural networks (PINNs) integrate physical laws into neural networks but rely on loss function constraints.

Purpose of the Study:

• To introduce physics-informed kernel function neural networks (PIKFNNs) as an advancement over standard PINNs.
• To demonstrate the efficacy of embedding PDE information directly into activation functions.
• To provide a novel neural network architecture for solving linear and specific nonlinear PDEs.

Main Methods:

• Developed PIKFNNs using a shallow neural network with one hidden layer.
• Utilized physics-informed kernel functions (PIKFs) as customized activation functions.
• PIKFs incorporate PDE information, such as fundamental solutions or Green's functions.

Main Results:

• PIKFNNs successfully solve various linear and specific nonlinear PDEs.
• The approach embeds PDE information within activation functions, differing from PINNs' loss function constraints.
• Validated the feasibility and accuracy of PIKFNNs through benchmark examples.

Conclusions:

• PIKFNNs represent a novel and effective neural network architecture for solving PDEs.
• Embedding physics information into activation functions offers a promising alternative to traditional PINN methods.
• The proposed PIKFNNs demonstrate high accuracy and feasibility for a range of PDE problems.