A framework of deep neural networks via the solution operator of partial differential equations

There is a close connection between deep neural networks (DNN) and partial differential equations (PDEs). Many DNN architectures can be modeled by PDEs and have been proposed in the literature. However, their neural network design space is restricted due to the specific form of PDEs, which prevents the design of more effective neural network structures. In this paper, we attempt to derive a general form of PDEs for the design of ResNet-like DNN. To achieve this goal, we first formulate DNN as an adjustment operator applied on the base classifier. Then based on several reasonable assumption, we show the adjustment operator for ResNet-like DNN is the solution operator of PDEs. To show the effectiveness for general form of PDEs, we show that several effective networks can be interpreted by our general form of PDEs and design a training method motivated by PDEs theory to train DNN models for better robustness and less chance of overfitting. Theoretically, we prove that the robustness of DNN trained with our method is certifiable and our training method reduces the generalization gap for DNN. Furthermore, we demonstrate that DNN trained with our method can achieve better generalization and is more resistant to adversarial perturbations than baseline model.