Séminaire Mathématiques Financières

Jeudi 15 octobre 2026

11h – 12h

Salle 3001

Carlos ESTEVE-YAGUE (Universidad de Alicante) “Solving Hamilton-Jacobi-Bellman Equations with Neural Networks: Monotone Discretizations and Dimension-Robust Convergence”

Summary:

In recent years, advancements in deep learning and new optimization algorithms have motivated the use of artificial neural networks to solve non-linear problems in high-dimensional setups. A relevant case of application is the approximation of the value function for optimal control problems, which can be characterized by the viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation. Along with the neural network architecture, one of the crucial steps in implementing any deep learning method is the choice of the loss functional used to train the network parameters, typically through gradient-based optimization. This talk takes up exactly that question for HJB equations: which loss functional design guarantees convergence to the viscosity solution, and at what rate? I will show that using a monotone discretization of the underlying differential operator ensures the associated loss functional has a unique critical point, which approximates the viscosity solution. Moreover, for sufficiently coarse discretizations, we prove a dimension-robust Polyak–Łojasiewicz inequality.

This implies linear convergence of the associated gradient flow at a rate that avoids the curse of dimensionality. Building on this theoretical foundation, we propose a multi-level training algorithm that exploits the faster convergence available on the coarser grids of the discretization.

Joint work: Olivier BOKANOWSKI, Richard TSAI