HyperKKL: Enabling Non-Autonomous State Estimation through Dynamic Weight Conditioning

This paper proposes HyperKKL, a novel learning approach for designing Kazantzis-Kravaris/Luenberger (KKL) observers for non-autonomous nonlinear systems. While KKL observers offer a rigorous theoretical framework by immersing nonlinear dynamics into a stable linear latent space, its practical realization relies on solving Partial Differential Equations (PDE) that are analytically intractable. Current existing learning-based approximations of the KKL observer are mostly designed for autonomous systems, failing to generalize to driven dynamics without expensive retraining or online gradient updates. HyperKKL addresses this by employing a hypernetwork architecture that encodes the exogenous input signal to instantaneously generate the parameters of the KKL observer, effectively learning a family of immersion maps parameterized by the external drive. We rigorously evaluate this approach against a curriculum learning strategy that attempts to generalize from autonomous regimes via training heuristics alone. The novel approach is illustrated on four numerical simulations in benchmark examples including the Duffing, Van der Pol, Lorenz, and R\"ossler systems.