Stabilized Neural Hamilton–Jacobi–Bellman Solvers: Error Analysis and Applications in Model-Based Reinforcement Learning
Recent advancements in physics-informed neural solvers have unveiled a novel approach to model-based reinforcement learning (RL) in continuous time. This technique is fundamentally rooted in the Hamilton–Jacobi–Bellman (HJB) equations, which govern optimal feedback synthesis. In practical scenarios, implementations often navigate a unique domain that does not conform strictly to conventional grid methods or continuous PDE physics-informed neural networks (PINNs).
The innovative framework introduced in the recent preprint (arXiv:2605.07116v1) characterizes the value function through a neural network, where finite-difference HJB policy-evaluation operators are computed via network queries at strategically shifted points. Residuals are minimized using random continuous collocation, effectively merging the benefits of stabilized finite-difference policy evaluation with the flexibility of non-grid-based value representation.
Error Theory Development
This research lays the groundwork for a comprehensive error theory tailored to this hybrid regime. By interpreting finite differences as shift operators functioning on neural networks, the authors establish a population $L^2$ stability estimate for a single policy-evaluation step that incorporates learned dynamics. This stability estimate is crucial as it delineates various error components, including:
- Residual error
- Initial and exterior-collar mismatch
- Policy mismatch
- Model-identification error
Moreover, it introduces a gradient amplification factor specifically for learned dynamics. Notably, the underlying linear evaluation stability is safeguarded against hidden inverse-viscosity blow-up, a common issue in traditional methods.
Finite-Sample Collocation and Multi-Step Propagation
The research further extends its findings to include a finite-sample collocation certificate alongside a conditional multi-step propagation result facilitated through greedy policy improvement. This aspect of the study is particularly significant as it provides a framework for understanding how errors propagate over multiple steps, which is critical for applications in RL.
Experimental Validation
To substantiate their theoretical contributions, the authors conducted a series of experiments across various benchmarks, including:
- Compact-control Linear Quadratic Regulator (LQR) up to 64 dimensions
- Allen–Cahn control
- Pendulum control
- Hopper control
- 3D quadrotor
These experiments provided a comparative analysis against established model-based and model-free RL baselines. The results effectively illustrated the anticipated trends in residual error, policy mismatch, and learned-model error, affirming the robustness of the proposed method.
Conclusion
The findings presented in this work offer significant insights into the field of model-based reinforcement learning, particularly in the context of continuous time and complex dynamics. By bridging the gap between classical methods and neural network representations, stabilized neural HJB solvers pave the way for more effective and efficient RL applications. As the field continues to evolve, this research serves as a vital stepping stone toward the development of more sophisticated and reliable RL algorithms.
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