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Title: Strong Stationarity for Optimal Control Problems with Non-smooth Integral Equation Constraints: Application to a Continuous DNN
Abstract

Motivated by the residual type neural networks (ResNet), this paper studies optimal control problems constrained by a non-smooth integral equation associated to a fractional differential equation. Such non-smooth equations, for instance, arise in the continuous representation of fractional deep neural networks (DNNs). Here the underlying non-differentiable function is the ReLU or max function. The control enters in a nonlinear and multiplicative manner and we additionally impose control constraints. Because of the presence of the non-differentiable mapping, the application of standard adjoint calculus is excluded. We derive strong stationary conditions by relying on the limited differentiability properties of the non-smooth map. While traditional approaches smoothen the non-differentiable function, no such smoothness is retained in our final strong stationarity system. Thus, this work also closes a gap which currently exists in continuous neural networks with ReLU type activation function.

 
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Award ID(s):
2110263
PAR ID:
10530061
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer
Date Published:
Journal Name:
Applied Mathematics & Optimization
Volume:
88
Issue:
3
ISSN:
0095-4616
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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