SK-PINN: Accelerated physics-informed deep learning by smoothing kernel gradients

TL;DR

This paper introduces SK-PINNs, a novel framework that accelerates physics-informed neural network training by using smoothing kernel discretization, significantly reducing computation time especially for large and complex problems.

Contribution

The paper proposes SK-PINNs, integrating smoothing kernel discretization into PINNs to enhance training efficiency and scalability for complex differential equations.

Findings

Training speed surpasses vanilla PINNs by up to tens of times.

Convergence rates are consistent with vanilla PINNs according to NTK analysis.

Effective in solving complex problems in fluid dynamics and solid mechanics.

Abstract

The automatic differentiation (AD) in the vanilla physics-informed neural networks (PINNs) is the computational bottleneck for the high-efficiency analysis. The concept of derivative discretization in smoothed particle hydrodynamics (SPH) can provide an accelerated training method for PINNs. In this paper, smoothing kernel physics-informed neural networks (SK-PINNs) are established, which solve differential equations using smoothing kernel discretization. It is a robust framework capable of solving problems in the computational mechanics of complex domains. When the number of collocation points gradually increases, the training speed of SK-PINNs significantly surpasses that of vanilla PINNs. In cases involving large collocation point sets or higher-order problems, SK-PINN training can be up to tens of times faster than vanilla PINN. Additionally, analysis using neural tangent kernel (NTK) theory shows that the convergence rates of SK-PINNs are consistent with those of vanilla PINNs. The superior performance of SK-PINNs is demonstrated through various examples, including regular and complex domains, as well as forward and inverse problems in fluid dynamics and solid mechanics.