| Abstract: |
| We propose a data-driven framework for approximating value functions of Hamilton--Jacobi--Bellman (HJB) equations to the time-dependent setting, targeting finite-horizon optimal control problems governed by nonlinear control-affine dynamics. The approach exploits the structural link between the Pontryagin Maximum Principle (PMP) and dynamic programming to generate augmented datasets containing values, gradients, Hessians, and temporal derivatives of the value function. A backward temporal decomposition reduces the problem to a sequence of short-horizon subproblems, on each of which the enriched data is used within a sparse polynomial regression framework based on hyperbolic cross index sets and $\ell^2$ regularization. The inclusion of high-order spatial and temporal information enables accurate approximation of the value function and the associated feedback law across the full time horizon. We assess the methodology on nonlinear control problems of moderate dimension. The numerical results demonstrate that derivative-enriched regression combined with temporal decomposition yields stable and accurate approximations in Sobolev norms. |
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