A Dimension-Augmented Physics-Informed Neural Network (DaPINN) with High Level Accuracy and Efficiency

TL;DR

This paper introduces DaPINN, a novel neural network that significantly enhances the accuracy and efficiency of physics-informed neural networks by augmenting input dimensions and incorporating regularization, verified through various augmentation techniques.

Contribution

The paper proposes DaPINN, a dimension-augmented PINN that improves accuracy and efficiency by adding sample features and regularization, outperforming traditional PINNs in experiments.

Findings

DaPINN achieves 1-2 orders of magnitude lower error than PINN.

DaPINN reduces dependence on sample points for accuracy.

Effective augmentation techniques include power series, Fourier series, and replica augmentation.

Abstract

Physics-informed neural networks (PINNs) have been widely applied in different fields due to their effectiveness in solving partial differential equations (PDEs). However, the accuracy and efficiency of PINNs need to be considerably improved for scientific and commercial use. To address this issue, we systematically propose a novel dimension-augmented physics-informed neural network (DaPINN), which simultaneously and significantly improves the accuracy and efficiency of the PINN. In the DaPINN model, we introduce inductive bias in the neural network to enhance network generalizability by adding a special regularization term to the loss function. Furthermore, we manipulate the network input dimension by inserting additional sample features and incorporating the expanded dimensionality in the loss function. Moreover, we verify the effectiveness of power series augmentation, Fourier series augmentation and replica augmentation, in both forward and backward problems. In most experiments, the error of DaPINN is 1$\sim$2 orders of magnitude lower than that of PINN. The results show that the DaPINN outperforms the original PINN in terms of both accuracy and efficiency with a reduced dependence on the number of sample points. We also discuss the complexity of the DaPINN and its compatibility with other methods.