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

Researchers develop a hybrid neural network approach for solving Hamilton-Jacobi-Bellman equations in continuous-time reinforcement learning, combining physics-informed neural solvers with stabilized finite-difference methods. The work provides rigorous error analysis separating residual, policy, and model-identification errors, with experimental validation across multiple control benchmarks.

This research bridges a critical gap between classical numerical methods and modern deep learning approaches for optimal control problems. The hybrid solver architecture treats finite differences as shift operators on neural networks, enabling practitioners to leverage neural function approximation while maintaining the stability properties of traditional grid-based methods. This is technically significant because Hamilton-Jacobi-Bellman equations form the theoretical foundation of continuous-time optimal control, yet solving them at scale has remained computationally challenging.

The work addresses longstanding concerns in physics-informed neural networks by providing explicit error bounds that don't hide problematic factors like inverse-viscosity blow-up—a pathological amplification that plagued earlier approaches. The error decomposition separates multiple error sources (residuals, policy mismatches, learned dynamics errors), enabling practitioners to diagnose bottlenecks in their implementations. This transparency is rare in deep RL literature and supports principled debugging.

For the reinforcement learning community, this opens pathways for model-based RL in high-dimensional continuous control tasks. Experiments scaling to 64-dimensional LQR problems and complex systems like quadrotors suggest practical viability beyond toy problems. The approach potentially offers advantages over model-free methods by incorporating physics constraints directly, reducing sample complexity and improving stability—critical for safety-sensitive applications in robotics and autonomous systems.

Future work likely explores scaling to higher dimensions and more complex dynamics. Practitioners should monitor whether this method achieves practical advantages over established baselines in real-world control tasks where incorporating model uncertainty and computational efficiency matter.

• →Hybrid neural-finite-difference solver combines deep learning flexibility with classical numerical stability for HJB equations.
• →Rigorous error analysis provides explicit bounds without hidden inverse-viscosity blow-up common in physics-informed neural networks.
• →Experimental validation spans 64-dimensional LQR to quadrotor control, demonstrating practical scalability.
• →Approach enables model-based reinforcement learning with theoretical guarantees on error propagation through policy improvement.
• →Error decomposition enables practitioners to diagnose and address specific bottlenecks in implementation.