Verifiable Error Bounds for Physics-Informed Neural KKL Observers

Abstract

This paper proposes a computable state-estimation error bound for learning-based Kazantzis--Kravaris/Luenberger (KKL) observers. Recent work learns the KKL transformation map with a physics-informed neural network (PINN) and a corresponding left-inverse map with a conventional neural network. However, no computable state-estimation error bounds are currently available for this approach. We derive a state-estimation error bound that depends only on quantities that can be certified over a prescribed region using neural network verification. We further extend the result to bounded additive measurement noise and demonstrate the guarantees on nonlinear benchmark systems.

Figures (3)

• Figure 1: Overview of the learning-based KKL training and verification pipeline with L-BFGS fine-tuning.
• Figure 2: State-estimation error trajectories for the learned KKL observers on two nonlinear benchmarks: (a) reverse Duffing and (b) Van der Pol. Solid lines show $\|\hat{x}(t)-x(t)\|$ along representative simulated trajectories, and dashed horizontal lines indicate the certified ultimate bounds from \ref{['eq:prop_x_ultimate']} computed over the corresponding verification regions.
• Figure 3: Van der Pol state-estimation error trajectory $\|\hat{x}(t)-x(t)\|$ under additive measurement noise ($1.0\%$ of peak output; $\bar{v}=0.033$). The dashed line indicates the certified ultimate bound from \ref{['eq:prop_x_ultimate_noisy']}.

Theorems & Definitions (3)

• proof
• proof
• proof