Neural-network approaches for high-dimensional optimal control problems

Decanato - Facoltà di scienze informatiche

Data: 23 Febbraio 2022 / 16:30 - 17:30

USI Campus EST, room D1.14, Sector D // online on MS Teams

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Speaker: Prof. Lars Ruthotto, Emory University, United States

Abstract: We consider neural network approaches to solving high-dimensional optimal control problems with deterministic and randomly perturbed dynamics. The training process simultaneously approximates the value function of the control problem and identifies the part of the state space likely to be visited by optimal trajectories. The latter is important to avoid the curse of dimensionality associated with solving these problems globally (that is, for all states). We, therefore, consider our approach an approximate semi-global method.
Our network design and the training problem leverage insights from optimal control theory. We approximate the value function of the control problem using a neural network and use the Pontryagin maximum principle to express the optimal control (and therefore the sampling) in terms of the value function. Our training loss consists of a weighted sum of the objective functional of the control problem and penalty terms that enforce the Hamilton Jacobi Bellman equations along the sampled trajectories. Importantly, training is unsupervised in that it does not require solutions of the control problem.
Our approach reduces to the method of characteristics when the dynamics is deterministic. Hence, it can thus be seen as a generalization of recent approaches for solving high-dimensional deterministic control problems. In our numerical experiments, we compare our method to existing solvers for a more general class of semilinear PDEs. Using a two-dimensional toy problem, we demonstrate the importance of the stochastic PMP to inform the sampling. For a 100-dimensional benchmark problem, we demonstrate that approach improves accuracy and time-to-solution.

Biography: Prof. Ruthotto is an applied mathematician developing computational methods for machine learning and inverse problems. He's an Associate Professor in the Department of Mathematics and the Department of Computer Science at Emory University and a member of Emory’s Scientific Computing Group. He leads the Emory REU/RET site for Computational Mathematics for Data Science. Prior to joining Emory, he was a postdoc at the University of British Columbia and helds PhD positions at the University of Lübeck and the University of Münster.

Host: Prof. Igor Pivkin