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SUMMARY:Mouhcine Assouli (XLIM - Universitè de Limoges) - Initialization-
 driven neural generation and training for high-dimensional optimal control
  and first order mean field games
DTSTART:20260429T080000Z
DTEND:20260429T090000Z
DTSTAMP:20260502T105200Z
UID:indico-event-16459@indico.math.cnrs.fr
DESCRIPTION:We introduce a neural-network-based method for approximating t
 he value function of high-dimensional deterministic optimal control proble
 ms. The proposed approach exploits the relationship between Pontryagin's M
 aximum Principle (PMP) and the value function\, which is characterized as 
 the unique viscosity solution of the Hamilton–Jacobi–Bellman (HJB) equ
 ation. A neural network is first trained to obtain a coarse approximation 
 of the value function by minimizing the residual of the HJB equation. The 
 gradient of this approximation is then used to initialize the numerical so
 lution of the two-point boundary value problem arising from PMP\, enabling
  the generation of reliable optimal trajectories\, adjoint states\, and co
 sts. This dataset is then used to further train the neural network through
  a loss function enforcing the HJB equation..\nWe next address the computa
 tion of equilibria in first-order Mean Field Game (MFG) problems by integr
 ating the proposed methodology with the fictitious play algorithm. Such eq
 uilibria are characterized by a coupled system consisting of an HJB equati
 on and a continuity equation. To approximate the solution of the continuit
 y equation\, we introduce a second neural network that learns the flow map
  transporting the initial agent distribution along optimal trajectories. T
 his network is trained using data obtained by solving the ordinary differe
 ntial equations (ODEs) associated with the controlled dynamics. The iterat
 ive procedure is initialized via joint training of the value function and 
 the flow map\, based on a composite loss involving both HJB and ODE residu
 als. Several numerical experiments demonstrate the accuracy and robustness
  of the proposed method..\n\nhttps://indico.math.cnrs.fr/event/16459/
URL:https://indico.math.cnrs.fr/event/16459/
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