Understanding Deep Neural Networks via Linear Separability of Hidden Layers

In this paper, we measure the linear separability of hidden layer outputs to study the characteristics of deep neural networks. In particular, we first propose Minkowski difference based linear separability measures (MD-LSMs) to evaluate the linear separability degree of two points sets. Then, we demonstrate that there is a synchronicity between the linear separability degree of hidden layer outputs and the network training performance, i.e., if the updated weights can enhance the linear separability degree of hidden layer outputs, the updated network will achieve a better training performance, and vice versa. Moreover, we study the effect of activation function and network size (including width and depth) on the linear separability of hidden layers. Finally, we conduct the numerical experiments to validate our findings on some popular deep networks including multilayer perceptron (MLP), convolutional neural network (CNN), deep belief network (DBN), ResNet, VGGNet, AlexNet, vision transformer (ViT) and GoogLeNet.

Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Method

Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs), and have been widely used in a variety of PDE problems. However, there still remain some challenges in the application of PINNs: 1) the mechanism of PINNs is unsuitable (at least cannot be directly applied) to exploiting a small size of (usually very few) extra informative samples to refine the networks; and 2) the efficiency of training PINNs often becomes low for some complicated PDEs. In this paper, we propose the generative adversarial physics-informed neural network (GA-PINN), which integrates the generative adversarial (GA) mechanism with the structure of PINNs, to improve the performance of PINNs by exploiting only a small size of exact solutions to the PDEs. Inspired from the weighting strategy of the Adaboost method, we then introduce a point-weighting (PW) method to improve the training efficiency of PINNs, where the weight of each sample point is adaptively updated at each training iteration. The numerical experiments show that GA-PINNs outperform PINNs in many well-known PDEs and the PW method also improves the efficiency of training PINNs and GA-PINNs.