Finite-Horizon Hamilton--Jacobi--Bellman Equations with State-Dependent Diffusion in Spectral Barron Spaces

What the math gives to ML

The transferable asset is the paper's parametrix treatment of state-dependent diffusion: instead of approximating a variable-coefficient parabolic operator directly, freeze the diffusion locally, apply an exactly computable Gaussian propagator, and represent the coefficient mismatch as a correction. This suggests a preconditioned neural PDE solver in which the network learns only the residual left after analytic diffusion smoothing, rather than the full HJB solution. The same construction can be implemented as a short-horizon Picard or policy iteration with frozen Gaussian convolutions and automatic-differentiation correction terms. The approach is most promising for high-dimensional stochastic-control surrogates, where grid methods fail and ordinary PINN residual optimization is poorly conditioned.