
Deep neural network approximation for highdimensional parabolic HamiltonJacobiBellman equations
The approximation of solutions to second order HamiltonβJacobiβBellman (...
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Largetime asymptotics in deep learning
It is by now wellknown that practical deep supervised learning may roug...
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Deep learning method for solving stochastic optimal control problem via stochastic maximum principle
In this paper, we aim to solve the stochastic optimal control problem vi...
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DepthAdaptive Neural Networks from the Optimal Control viewpoint
In recent years, deep learning has been connected with optimal control a...
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Martingale Functional Control variates via Deep Learning
We propose blackboxtype control variate for Monte Carlo simulations by...
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Newton vs the machine: solving the chaotic threebody problem using deep neural networks
Since its formulation by Sir Isaac Newton, the problem of solving the eq...
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Solving for best linear approximates
Our goal is to finally settle a persistent problem in Diophantine Approx...
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Control On the Manifolds Of Mappings As a Setting For Deep Learning
We use a controltheoretic setting to model the process of training (deep learning) of Artificial Neural Networks (ANN), which are aimed at solving classification problems. A successful classifier is the network whose inputoutput map approximates well the classifying map defined on a finite or an infinite training set. A fruitful idea is substitution of a multilayer ANN by a continuoustime control system, which can be seen as a neural network with infinite number of layers. Under certain conditions it can achieve high rate of approximation with presumably not so high computational cost. The problem of best approximation for this model results in optimal control problem of Bolza type for ensembles of points. The two issues to be studied are: i) possibility of a satisfactory approximation of complex classification profiles; ii) finding the values of parameters (controls) which provide the best approximation. In controltheoretic terminology it corresponds respectively to the verification of an ensemble controllability property and to the solution of an ensemble optimal control problem. In the present contribution we concentrate on the first type of problems; our main results include examples of control systems, which are approximately controllable in the groups of diffeomorphisms of β^n, π^n, π^2.
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