Provably Stable Neural Dynamics via Koopman Operator Certificates

Aryan Dadwal

Keywords: Koopman operator, stability certificates, neural dynamics, dynamical systems, physics-informed machine learning

TL;DR: We introduce a deep Koopman architecture that guarantees Schur stability by construction, preventing catastrophic divergence in long-horizon physics simulations.

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 $S \succ 0$ is symmetric positive definite and $A$ is skew-symmetric. This yields a native stability certificate: $V(z) = \|z\|^2$ is a strict latent Lyapunov function. We prove that this latent certificate transfers to practical Input-to-State Stability in the original state space under bi-Lipschitz conditions on the encoder--decoder pair, and establish a limitation theorem characterizing when strictly stable reduced representations cannot exactly capture a given dynamical system. Experiments on four benchmarks---the Duffing oscillator, an unstable saturating node, the Lorenz attractor, and the 1D viscous Burgers' PDE---demonstrate that the certified model maintains bounded predictions where unconstrained baselines diverge catastrophically.

Submission Number: 151