Learning Controls and Interactions in Mean-Field Games

Abstract

Neural networks provide an efficient parametrization class for high-dimensional functions. In physics-informed approaches such as PINNs, the solution is represented by a neural network and trained by penalizing the residual of the goveming partial differential equation. These methods, however, treat the PDE primarily as a constraint and do not fully exploit the underlying model's structure.

We introduce a framework that incorporates model information directly into the training objective. Instead of learning only the solution, two neural networks parametrize the control and the mean-field interaction term, and are optimized via sampling. The method is illustrated on a price-formation model and a crowd-dynamics model. Numerical experiments show that the neural network parametrization remains stable and accurate, including in the presence of common noise. Moreover, the optimization structure allows for a posteriori performance guarantees via estimates derived from the model formulation.

Speaker Biography

Dr Julian received both B.S. in Mathematics and B.S. in Engineering from Escuela Colombiana de Ingenieria, Colombia. After receiving his M.Sc. in Mathematical Sciences from Universidad Nacional Autónoma de México, México, he joined Professor Diogo Gomes as a Ph.D. student in applied mathematics at King Abdullah University of Science and Technology, Saudi Arabia. Upon completing his Ph.D., he joined Ecole Polytechnique as a Postdoctoral Researcher. His research interests are primarily in optimal control, optimal transport, calculus of variations, partial differential equations, and mean-field games, with a particular focus on applications to engineering and finance.