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
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 ν
Reverse Duffing
8
100
10−3
1
Van der Pol
7
128
10−3
1
TABLE I: Hyperparameters used for training the encoder T^θ .
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 ∥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
Rˉ
Lη
ET^
limsupt→∞∥x^(t)−x(t)∥
Reverse Duffing
5.13×10−4
235.7
5.98×10−2
0.181
Van der Pol
7.7×10−3
23
2.3×10−2
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 ∥x^(t)−x(t)∥ under additive measurement noise ( 1.0% of peak output; vˉ=0.033 ). The dashed line indicates the certified ultimate bound from ( 11 ).
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 (HyperKKLobs) adds input-dependent corrections to the latent observer dynamics while retaining static transformation maps. Second, a dynamic observer approach (HyperKKLdyn) 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.
Yahia Salaheldin Shaaban, Abdelrahman Sayed Sayed, M. Umar B. Niazi +1
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.
Physics-informed neural networks (PINNs) combine machine learning with physical laws to solve differential equations. While existing results provide rigorous a posteriori upper bounds for PINN prediction errors, complete certification also requires complementary lower information in order to obtain computable two-sided error enclosures. In this paper, we derive computable a posteriori lower bounds for PINN errors in ordinary differential equations on suitable certified state-space domains under a localized strong monotonicity condition. We combine these estimates with complementary localized upper bounds under a one-sided Lipschitz condition, which is weaker than the global Lipschitz assumption used in previous work and can yield sharper upper error bands. The resulting bounds depend only on the neural-network approximation, the ODE residual, and local monotonicity and growth constants, and therefore do not require access to the exact solution. For linear time-invariant and time-varying systems, we further derive explicit formulas in terms of the minimal and maximal eigenvalues of the symmetric part of the system matrix. We also discuss the distinction between soft and hard enforcement of initial conditions in PINNs and explain why exact enforcement can make the scalar lower certificate uninformative. To recover nontrivial lower information in the linear setting, we use a signed-residual finite-probe certificate based on coordinate unit vectors. We also formulate a certificate-informed training strategy in which the propagated upper certificate is used as an auxiliary regularizer, while lower certificates remain post-training diagnostics. Altogether, the proposed framework provides rigorous and practically computable error certificates for PINN approximations of ODEs, while making explicit the domains and model classes for which the assumptions can be verified.
Ismail Huseynov, Arzu Ahmadova, Agamirza Bashirov
Physikalisch-Technische Bundesanstalt (PTB), Germany · Technical University of Berlin, Germany · Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany +1
Many core problems in nonlinear systems analysis and control can be recast as solving partial differential equations (PDEs) such as Lyapunov and Hamilton-Jacobi-Bellman (HJB) equations. Physics-informed neural networks (PINNs) have emerged as a promising mesh-free approach for approximating their solutions, but in most existing works there is no rigorous guarantee that a small PDE residual implies a small solution error. This paper develops verifiable error bounds for approximate solutions of Lyapunov and HJB equations, with particular emphasis on PINN-based approximations. For both the Lyapunov and HJB PDEs, we show that a verifiable residual bound yields relative error bounds with respect to the true solutions as well as computable a posteriori estimates in terms of the approximate solutions. For the HJB equation, this also yields certified upper and lower bounds on the optimal value function on compact sublevel sets and quantifies the optimality gap of the induced feedback policy. We further show that one-sided residual bounds already imply that the approximation itself defines a valid Lyapunov or control Lyapunov function. We illustrate the results with numerical examples.
Jun Liu
Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada