Verifiable Error Bounds for Physics-Informed Neural KKL Observers

Hannah Berin-Costain; Harry Wang; Kirsten Morris; Jun Liu

eess.SYMar 20, 2026

Verifiable Error Bounds for Physics-Informed Neural KKL Observers

Authors: Hannah Berin-Costain, Harry Wang, Kirsten Morris, Jun Liu

Organizations: Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada · University of Toronto, Toronto, ON M5S 1A1, Canada

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 & tables

Fig. 1: Overview of the learning-based KKL training and verification pipeline with L-BFGS fine-tuning.

System	Hidden layers	Layer size	Learning rate	Physics weight $\nu$
Reverse Duffing	8	100	$10^{-3}$	1
Van der Pol	7	128	$10^{-3}$	1

TABLE I: Hyperparameters used for training the encoder

\hat{\mathcal{T}}_{\theta}

Panel of Figure 3 — Fig. 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 ( 10 ) computed over the corresponding verification regions.

System	$\bar{\mathcal{R}}$	$\mathcal{L}_{\eta}$	$\mathcal{E}_{\hat{\mathcal{T}}}$	$\limsup_{t\to\infty}$ $\\|\hat{x}(t)-x(t)\\|$
Reverse Duffing	$5.13\times 10^{-4}$	$235.7$	$5.98\times 10^{-2}$	$0.181$
Van der Pol	$7.7\times 10^{-3}$	$23$	$2.3\times 10^{-2}$	$0.112$

TABLE II: Certified quantities and resulting estimation error bound for reverse Duffing and Van der Pol.

Fig. 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 ( 11 ).

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Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, UAE. · Univ Gustave Eiffel, COSYS-ESTAS, F-59657 Villeneuve d’Ascq, France. · Department of Decision and Control Systems and Digital Futures, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden.
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Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

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HyperKKL: Learning KKL Observers for Non-Autonomous Nonlinear Systems via Hypernetwork-Based Input Conditioning

Yahia Salaheldin Shaaban, Abdelrahman Sayed Sayed, M. Umar B. Niazi, Karl Henrik Johansson

Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, UAE. · Univ Gustave Eiffel, COSYS-ESTAS, F-59657 Villeneuve d’Ascq, France. · Department of Decision and Control Systems and Digital Futures, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden.

Dynamical Systems

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Kazantzis-Kravaris/Luenberger (KKL) observers are a class of state observers for nonlinear systems that rely on an injective map to transform the nonlinear dynamics into a stable quasi-linear latent space, from where the state estimate is obtained in the original coordinates via a left inverse of the transformation map. Current learning-based methods for these maps are designed exclusively for autonomous systems and do not generalize well to controlled or non-autonomous systems. In this paper, we propose two learning-based designs of neural KKL observers for non-autonomous systems whose dynamics are influenced by exogenous inputs. To this end, a hypernetwork-based framework (

HyperKKL

) is proposed with two input-conditioning strategies. First, an augmented observer approach (

HyperKKL_{obs}

) adds input-dependent corrections to the latent observer dynamics while retaining static transformation maps. Second, a dynamic observer approach (

HyperKKL_{dyn}

) employs a hypernetwork to generate encoder and decoder weights that are input-dependent, yielding time-varying transformation maps. We derive a theoretical worst-case bound on the state estimation error. Numerical evaluations on four nonlinear benchmark systems show that input conditioning yields consistent improvements in estimation accuracy over static autonomous maps, with an average symmetric mean absolute percentage error (SMAPE) reduction of 29% across all non-zero input regimes.

Abstract

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