Observable Geometry for Derivative-Free Optimization of Dynamical Systems

Derivative-free optimization of dynamical systems avoids differentiating through a simulator, yet it often reduces structured outputs to scalar losses and discards their observation-wise responses. These responses retain optimization-relevant structure that scalarization hides. We exploit this structure through observable mapping (OM), which constructs local response-informed coordinates, and instantiate it in Observable-Mapping Ensemble Kalman Inversion (OM-EKI) as a derivative-free optimizer for structured-response dynamical-system problems. We call the resulting task-relevant response structure observable geometry and hypothesize that useful coordinates exhibit three properties: compactness, residual reachability, and local persistence. Across settings with available, costly, and unavailable exact gradients, OM-EKI achieves competitive held-out accuracy and runtime on a deterministic Neural ODE, reduces median time-to-quality by $7.72\times$ relative to Adam on a 19,587-parameter stiff UDE, and approaches surrogate-gradient validation performance on a 50,176-parameter hard-reset recurrence for Spiking Heidelberg Digits (SHD). Mechanism-focused controls further support this hypothesis. Together, these results show that structured forward responses can be used not only to evaluate derivative-free search in dynamical systems, but also to organize it.