A Distributional Evaluation Metric for Chaotic Physical Systems

Long-horizon predictions of chaotic physical systems eventually decorrelate from reference trajectories, making pointwise errors unreliable for distinguishing simulation failures from plausible alternative evolutions. We address this evaluation gap by comparing distributions of predicted and reference states using a limited number of independent trajectories. We introduce PDE-KD, a distributional evaluation metric for neural physics simulators based on a kernel discrepancy between predicted and reference distributions of states embedded in a learned feature space. To evaluate feature spaces for this application, we assess their statistical discriminative power under controlled perturbations of the spatial field and the initial conditions. PDE-KD is sensitive to disturbances across a broad range of spatial scales on several physical systems; it is sample efficient, requiring few trajectories to reliably detect discrepancies, and it reliably distinguishes conditional distributions. On trained neural simulators, it orders the models of each family consistently with measures of rollout quality, whereas pixel-space errors favor over-smoothed predictions. We thus present PDE-KD as a general, sample-efficient distributional evaluation metric that complements pointwise errors and targeted physical diagnostics for model evaluations.