Verifiable Error Bounds for Physics-Informed Neural Network Solutions of Lyapunov and Hamilton-Jacobi-Bellman Equations

A new research paper by Jun Liu, published on arXiv, tackles a fundamental trust gap in applying AI to high-stakes engineering. The work develops a rigorous mathematical framework for calculating verifiable error bounds when using Physics-Informed Neural Networks (PINNs) to solve two critical classes of partial differential equations (PDEs): Lyapunov equations (for proving system stability) and Hamilton-Jacobi-Bellman (HJB) equations (for optimal control). Until now, a small error in the PINN's internal physics loss function didn't guarantee a small error in the actual solution, limiting trust in AI for safety-critical applications.

The core breakthrough is proving that a computable bound on the PDE residual directly translates to a relative error bound for the solution itself. For the HJB equation, this means the method can produce certified upper and lower bounds on the optimal value function and quantify the performance gap of any derived control policy. Furthermore, the paper shows that even one-sided residual bounds are sufficient to certify that the neural network's output itself constitutes a valid Lyapunov or control Lyapunov function—a key requirement for proving stability. This transforms PINNs from a promising but unverified tool into one with mathematically provable guarantees, enabling their use in designing and certifying controllers for autonomous systems, robotics, and power grids where failure is not an option.