PadéKoop: Learning Rational Observables for Long-Horizon Dynamics

Finite-dimensional Koopman models depend critically on the observables used to represent nonlinear dynamics. Generic neural encoders can fit in-distribution trajectories, but provide no structural preference for inverse-power, fractional, or saturation dynamics and can extrapolate poorly beyond the training region. We introduce **PadéKoop**, a Koopman framework that learns observables as ratios of independent numerator and strictly positive denominator networks. This Padé-style parameterization represents denominator-driven structure while keeping every observable well-defined; the resulting features augment the physical state and evolve through a linear Koopman operator. To separate representation error from latent amplification, PadéKoop combines rational observables with optional spectral control of the Koopman matrix. We derive well-posedness, rational-approximation, and rollout-error properties that clarify these complementary roles. Across four equation-generated physical and biochemical benchmarks, PadéKoop achieves the lowest mean horizon-100 out-of-distribution error among controlled MLP- and KAN-observable Koopman baselines. A parameter-matched enzyme-kinetics ablation shows 69% lower error than an MLP and 54% lower error than a constant-denominator control. Spectral-control experiments further show that hard projection can suppress catastrophic latent amplification, but does not consistently improve predictive accuracy. Together, these results support rational observables as an effective inductive bias for denominator-driven dynamics and spectral control as a complementary robustness mechanism, rather than a guarantee of universally accurate long-horizon prediction.