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

TL;DR

This paper proposes HyperKKL, a novel learning approach for designing Kazantzis-Kravaris/Luenberger observers for non-autonomous nonlinear systems 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.

Abstract

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.

Figures (6)

• Figure 1: Dynamic HyperKKL Architecture (Phase 2). The base encoder $\hat{\mathcal{T}}_{\theta^{\text{base}}}$ and decoder $\hat{\mathcal{T}}^*_{\phi^{\text{base}}}$, pre-trained on autonomous dynamics in Phase 1, are frozen. A shared LSTM encoder processes the input window $u_{[t-w,\,t]}$ and produces a hidden state $h_t$, which is passed to two separate MLP decoder heads that predict chunked weight perturbations $\Delta\theta_{\text{enc}}$ and $\Delta\phi_{\text{dec}}$. The perturbations are added to the frozen base weights, yielding input-conditioned maps $\hat{\mathcal{T}}(x;\,\theta^{\text{base}} + \Delta\theta)$ and $\hat{\mathcal{T}}^*(z;\,\phi^{\text{base}} + \Delta\phi)$.
• Figure 2: Duffing (Square Input): The Dynamic HyperKKL adapts to discontinuous square waves without the transient spikes seen in static methods.
• Figure 3: Duffing System: State estimation time-series.
• Figure 4: Van der Pol System: State estimation time-series.
• Figure 5: Rossler System (Chaotic): State estimation time-series.