Nonlinear Dynamics in Semiconductor Lasers 2023 - Abstract

Kebiri, Omar

Deep learning for solving initial path optimization of mean-field systems with memory

We consider the problem of finding the optimal initial investment strategy for a system modelled by a linear McKean-Vlasov (mean-field) stochastic dif-ferential equation with delay, driven by a Brownian motion and a pure jump Poisson random measure. The problem is to find the optimal initial values for the system in the period [−δ, 0], where δ > 0 is a delay constant, before the system starts at t = 0. Because of the delay in the dynamics, the system will after startup be influenced by these initial investment values. It is known that linear stochastic delay differential equations are equivalent to stochastic Volterra integral equations. By using this equivalence we can find implicit expression for the optimal investment. Moreover, we propose a deep neural network-based algorithm to solve stochastic control problem with delay. Specifically, we employ multi-layer feed- forward neural network for control modeling in the interval [−δ, 0]), and the backpropagation for training the feedforward neural network,where we com- pute the gradient of the loss function using stochastic gradient descent (SGD) with respect to the weights of the network. This method can also apply for delay differential equations for mode-locked semiconductor lasers .

Joint work with B. Oksendal, N. Agram and M. Grid