Residual-Corrected Physics-Informed Neural Network KKL Observers with Input-to-State Stability Guarantees

arXiv:2611.13411v1 Announce Type: new Abstract: Physics-informed neural network (PINN) Kazantzis--Kravaris--Luenberger (KKL) observers require an approximate solution to a KKL partial differential equation (PDE) and an approximate inverse of this solution. Either approximation can leave a nonzero estimation bias even without measurement noise. In this paper, we remove both errors from the implemented observer by construction. We build an encoder containing an explicitly invertible neural map. Its closed-form inverse provides the internal state reconstruction used to evaluate the learned KKL residual and correct the latent dynamics. This construction makes the synchronization manifold exactly invariant for every value of the network weights. Based on contraction of the residual-corrected observer dynamics, we prove an input-to-state stability (ISS) result for the implemented error dynamics with respect to measurement noise. A direct consequence is that the estimation error converges exponentially to zero in the noise-free setting. We provide two numerical examples, the Van der Pol oscillator and the chaotic Lorenz system, in which the residual-corrected observers achieve exact noise-free convergence and improved error responses compared with released PINN--KKL baselines.