Provably Stable Neural Dynamics via Koopman Operator Certificates

Aryan Dadwal

Keywords: Koopman Operator, Neural Dynamics, Stability Certificates, Dynamical Systems, Forward Models, Surrogate Simulators

TL;DR: We learn an approximate reduced Koopman representation guaranteed to be Schur stable via a continuous-time generator $G=-S+A$, achieving 0% divergence on chaotic physics rollouts like the Lorenz attractor and Burgers' PDE.

Abstract: Learning neural forward models of dynamical systems that remain stable over long rollout horizons is a fundamental challenge in scientific computing and physics-informed machine learning. We introduce Koopman-Stable Neural Dynamics, a deep Koopman architecture that learns a finite-dimensional, stable latent approximation of Koopman-style dynamics. The latent transition operator is parameterized as the matrix exponential of a continuous-time generator $G=-S+A$ where $S>0$ is symmetric positive definite and A is skew-symmetric. This yields a native stability certificate: $V(z)=||z||^{2}$ is a strict Lyapunov function requiring no post-hoc verification. We prove that this certificate transfers to practical Input-to-State Stability in the original state space, and establish a limitation theorem characterizing when strictly stable reduced representations cannot exactly represent a given system. Experiments on four benchmarks-the Duffing oscillator, an unstable saturating node, the Lorenz attractor, and the ID viscous Burgers' PDE-demonstrate that the certified model maintains bounded predictions where unconstrained baselines diverge catastrophically.

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Submission Number: 179