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 Rössler systems.

💡 Research Summary

The paper introduces HyperKKL, a learning‑based framework that extends Kazantzis‑Kravaris/Luenberger (KKL) observers to non‑autonomous nonlinear systems. Classical KKL observers rely on an immersion map T that satisfies a PDE linking the nonlinear dynamics to a stable linear latent space. For autonomous systems this PDE involves only spatial derivatives, but in the presence of an exogenous input u(t) a time‑derivative term ∂T/∂t appears, making static immersion maps insufficient. Existing learning‑based KKL methods have focused on autonomous dynamics and therefore cannot handle input‑driven systems without costly retraining or online gradient updates.

HyperKKL tackles this limitation by conditioning the observer’s parameters on the external input using a hypernetwork. The authors propose two complementary architectures:

1. Static HyperKKL – retains the original input‑independent immersion T(x) and augments the linear observer dynamics with an input‑injection term φ̂(z,u;ξ). An LSTM processes a sliding window of the input signal to produce a context embedding that is concatenated with the latent state z and fed to a small MLP representing φ̂. The injection network is trained to output zero when u≡0, guaranteeing exact recovery of the autonomous observer. This approach is computationally cheap and works well when the input acts as a bounded perturbation that does not fundamentally reshape the attractor.

2. Dynamic HyperKKL – implements a truly time‑varying immersion T(x,t) by letting a hypernetwork Hψ generate input‑dependent perturbations to the weights of both the encoder (lifting map T̂) and decoder (inverse map T̂*). The context c is the recent input history u