Stabilized neural Hamilton--Jacobi--Bellman solvers: Error analysis and applications in model-based reinforcement learning

• URL: http://arxiv.org/abs/2605.07116v1
• Date: Fri, 08 May 2026 01:48:02 GMT
• Title: Stabilized neural Hamilton--Jacobi--Bellman solvers: Error analysis and applications in model-based reinforcement learning
• Abstract summary: Theory of Hamilton--Jacobi--Bellman (HJB) equations can be applied to model-based reinforcement learning.<n>We develop a hybrid regime that preserves stabilized finite-difference policy-evaluation structure while avoiding grid-based value unknowns.<n>Experiments on compact-control LQR upto 64 dimensions, Allen--Cahn control, pendulum, Hopper, and 3D quadrotor benchmarks compare against representative model-based and model-free RL baselines.
• Score: 3.8280665730760366
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Physics-informed neural solvers offer a promising route to model-based reinforcement learning in continuous time, where optimal feedback synthesis is governed by Hamilton--Jacobi--Bellman (HJB) equations. Practical implementations often occupy a regime that is neither a classical grid method nor a continuous-PDE PINN: the value function is represented by a neural network, finite-difference HJB policy-evaluation operators are evaluated by network queries at shifted points, and residuals are minimized by random continuous collocation. This regime preserves the stabilized finite-difference policy-evaluation structure while avoiding grid-based value unknowns. We develop an error theory for this hybrid regime. Interpreting finite differences as shift operators acting on neural networks, we prove a population $L^2$ stability estimate for one policy-evaluation step with learned dynamics. The bound separates residual error, initial and exterior-collar mismatch, policy mismatch, and model-identification error, with an explicit gradient amplification factor for learned dynamics, while the underlying linear evaluation stability remains free of hidden inverse-viscosity blow-up. We further give a finite-sample collocation certificate and a conditional multi-step propagation result through greedy policy improvement. Experiments on compact-control LQR upto 64 dimensions, Allen--Cahn control, pendulum, Hopper, and 3D quadrotor benchmarks compare against representative model-based and model-free RL baselines, demonstrating the predicted residual, policy-mismatch, and learned-model error trends.

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