Physics-Informed with Power-Enhanced Residual Network: Residual Network

Table of Links

Abstract & Introduction

Neural Networks

PINN for Solving Inverse Burgers’ Equation

Residual Network

Numerical Results

Results, Acknowledgments & References

4 Residual Network

In this study, we propose a power-enhanced variant of the ResNet that skips every other layer, denoted as the “Power-Enhanced SkipResNet.” The modification involves altering the recursive definition in (9) as follows:

Figure 2: Three neural network architectures: (a) plain neural network (Plain NN), (b) residual network (ResNet), (c) power-enhanced SkipResNet, and (d) Unraveled SQRSkipResNet (plot (c) with p = 2) where ⊙ denotes element-wise multiplication.

For the purpose of comparison among Plain NN, ResNet, and SQR-SkipResNet (Figs. 2(a)-(c), respectively), we evaluate the output of the third hidden layer concerning the input y0 = X . The results for the plain neural network are as follows:

Figure 2(d) visually represents the “expression tree” for the case with p = 2, providing an insightful illustration of the data flow from input to output. The graph demonstrates the existence of multiple paths that the data can traverse. Each of these paths represents a distinct configuration, determining which residual modules are entered and which ones are skipped.

Our extensive numerical experiments support our approach, indicating that a power of 2 is effective for networks with fewer than 30 hidden layers. However, for deeper networks, a larger power can contribute to network stability. Nonetheless, deploying such deep networks does not substantially enhance accuracy and notably increases CPU time. In tasks like interpolation and solving PDEs, a power of 2 generally suffices, and going beyond may not justify the added complexity in terms of accuracy and efficiency