An optimal time variable learning framework for Deep Neural Networks

Harbir Antil (Center for Mathematics and Artificial Intelligence (CMAI) and Department of Mathematical Sciences, George Mason University, Fairfax, Virginia, U.S.A.)

Hugo Díaz (Department of Mathematical Sciences, University of Delaware, Newark, Del., U.S.A.)

Evelyn Herberg (Center for Mathematics and Artificial Intelligence (CMAI) and Department of Mathematical Sciences, George Mason University, Fairfax, Virginia, U.S.A.)

Abstract

Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. The novelty, in this paper, lies in letting the discretization parameter (time step-size) vary from layer to layer, which needs to be learned, in an optimization framework. The proposed framework can be applied to any of the existing networks such as ResNet, DenseNet or Fractional‑DNN. This framework is shown to help overcome the vanishing and exploding gradient issues. Stability of some of the existing continuous DNNs such as Fractional‑DNN is also studied. The proposed approach is applied to an ill-posed 3D‑Maxwell’s equation.

Keywords

deep learning, Deep Neural Network, fractional time derivatives, fractional neural network, residual neural network, optimal network architecture, exploding gradients, vanishing gradients

2010 Mathematics Subject Classification

34A08, 49J15, 68T05, 82C32

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This work is partially supported by NSF grants DMS-2110263, DMS-1913004, and DMS-2111315; by the Air Force Office of Scientific Research (AFOSR) under Award No. FA9550-22-1-0248; and by the Department of the Navy, Naval Postgraduate School, under Award No. N00244-20-1-0005.

Received 21 July 2022

Accepted 14 August 2023

Published 14 November 2023