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. This is a joint work with Olivier Bokanowski and Richard Tsai.
Carlos Esteve-Yagüe is a faculty member (Ramón y Cajal researcher) in the Department of Mathematics at the University of Alicante (Spain). Following a Ph.D. at Université Paris 13, he held postdoctoral research positions at the Autonomous University of Madrid, the University of Deusto, and the University of Cambridge. His research lies at the intersection of applied mathematics, partial differential equations, control theory, and machine learning.