We consider the numerical solution of mean field games and optimal control problems whose state space dimension is in the tens or hundreds. In this setting, most existing numerical solvers are affected by the curse of dimensionality (CoD). To mitigate the CoD, we present a machine learning framework that combines the approximation power of neural networks with the scalability of Lagrangian PDE solvers. Specifically, we parameterize the value function with a neural network and train its weights using the objective function with additional penalties that enforce the Hamilton Jacobi Bellman equations. A key benefit of this approach is that no training data is needed, e.g., no numerical solutions to the problem need to be computed before training.
We illustrate our approach and its efficacy using numerical experiments. To show the framework's generality, we consider applications such as optimal transport, deep generative modeling, mean field games for crowd motion, and multi-agent optimal control.
Lars Ruthotto is an applied mathematician developing computational methods for machine learning and inverse problems. He is an Associate Professor in the Department of Mathematics and the Department of Computer Science at Emory University and a member of the Scientific Computing Group. Prior to joining Emory, he was a postdoc at the University of British Columbia and he held PhD positions at the University of Lübeck and the University of Münster.
Lars received an NSF CAREER award and is also supported by grants from the US Israeli Binational Science Foundation, the US Department of Energy’s Advanced Scientific Computing Research program, and the Air Force Office of Scientific Research.