Predators, Prey, and Sloppy Parameters: Fine-Tuning Amortized Inference for Generalized Lotka–Volterra

Accurate parameters do not guarantee accurate trajectories in nonlinear ordinary differential equation inverse problems. We establish this distinction through a controlled generalized Lotka-Volterra benchmark with $S\in \{3,6,10\}$ species, comparing amortized point estimators, neural posterior estimation, and test-time optimization on identical observations. Amortized regression achieves low parameter error but often produces inaccurate or invalid simulations. Fisher information and posterior diagnostics connect this mismatch to highly unequal sensitivity across parameter directions. We introduce optimizer-in-the-loop (OIL) tuning: a three-term objective trains a parameter initializer through finite trajectory-matching updates, and test-time trajectory matching refines its prediction. OIL improves trajectory recovery and simulation reliability while preserving competitive parameter accuracy. Matched-budget comparisons expose where the learned initialization helps and where ordinary warm starts remain competitive. Experiments with additive noise, FitzHugh-Nagumo and Lorenz-63 dynamics, and a data-conditioned Hudson Bay hare-lynx case study extend the comparison beyond clean gLV trajectories. Together, these results establish the need to evaluate parameter recovery, trajectory accuracy, numerical validity, and uncertainty as distinct properties of an inverse method.