Finite-sample guarantees for point-guided training

Establish finite-sample guarantees for point-guided Lyapunov training of neural network observers, including conditions under which finitely many sampled error states ensure the desired training outcome.

Background

The paper’s Stage I procedure trains a neural network observer by enforcing a strict Lyapunov-decrease condition at sampled points in a prescribed compact error-state domain. The theoretical analysis proves that successful samples induce local certified neighborhoods and provides a probabilistic coverage bound as the number of successful samples increases.

However, the paper does not establish finite-sample guarantees for the point-guided training procedure itself. In particular, the existing coverage result assumes successful samples and therefore does not characterize when the optimization process will produce those samples or achieve the required sample-wise Lyapunov margin.

References

The analysis characterizes Stage~I geometry. Finite-time optimization and sample-complexity guarantees for training success remain open.

— Learning Provable Neural Network Observer for Uncertain Dynamical Systems  (2609.30819 - Wang et al., 25 Sep 2026) in Section 3.2, “Convergence and Stability Analysis”; Appendix, Section “Limitations and Future Work” (Section sec-lim)

Finite-sample guarantees for point-guided training remain open, as do the required sample size and conditions for successful gradient-based certification.

— Learning Provable Neural Network Observer for Uncertain Dynamical Systems  (2609.30819 - Wang et al., 25 Sep 2026) in Appendix, Section “Limitations and Future Work” (Section sec-lim)