The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling

2026-05-29

AI Summary

This work addresses the dynamic–probabilistic consistency (DPC) gap in surrogate modeling of chaotic systems, where finite-horizon probabilistic objectives often misalign predictive uncertainty with true system dynamics. To bridge this gap, the authors propose KAFFEE, a novel framework that introduces a differentiable extended Kalman filter into surrogate model training. KAFFEE aligns uncertainty quantification with system dynamics by learning local Jacobian matrices to propagate covariance and constructing an innovation-sequence likelihood based on local prediction residuals. Theoretical analysis identifies three root causes of the DPC gap: kernel collapse, noise masking, and blind uncertainty. Experiments demonstrate that KAFFEE effectively mitigates the DPC gap in the stochastic hyperchaotic Lorenz-96 system, more accurately reconstructs dynamical invariants, and enables context-aware Bayesian filtering across 13 chaotic systems while preserving zero-shot dynamical generalization capability.

Abstract

Dynamical systems reconstruction (DSR) aims to learn surrogate models that capture the dynamics underlying time-series data. Reliably deploying these surrogates requires uncertainty estimates consistent with the learned dynamics. We expose a dynamic-probabilistic consistency (DPC) gap: the pursuit of finite-horizon probabilistic objectives can degrade dynamics or decouple predictive uncertainty from the local tangent dynamics it ought to reflect. We isolate three mechanisms behind this gap: core collapse, noise masking, and blind uncertainty. Specifically, we show that open-loop Gaussian rollout objectives can penalize Jacobian-generated covariance growth in chaotic systems, encouraging optimization shortcuts that weaken physical expansion or decouple uncertainty from it. To mitigate this gap, we propose KAFFEE (Kalman-Aware Framework For Ergodic Emulation), a differentiable extended Kalman filter-based training framework that evaluates likelihood on local predictive residuals (innovations) while transporting covariance through learned local Jacobians. On stochastic hyperchaotic Lorenz-96, KAFFEE reduces the identified failure modes, improves reconstruction of dynamical invariants relative to open-loop objectives, and maintains competitive predictive scores. We further show that the DPC gap appears when probabilistically adapting a DSR foundation model across 13 chaotic systems, where KAFFEE enables in-context Bayesian filtering while largely preserving zero-shot dynamics.

Problem

• Dynamic-Probabilistic Consistency
• Chaotic Surrogate Modeling
• Dynamical Systems Reconstruction
• Predictive Uncertainty
• Jacobian Dynamics

Innovation

• dynamic-probabilistic consistency
• chaotic surrogate modeling
• KAFFEE
• extended Kalman filter
• Jacobian-aware uncertainty