The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling
This work addresses a critical problem for researchers and engineers deploying surrogate models for chaotic systems, ensuring that uncertainty estimates are consistent with the underlying dynamics, rather than being an incremental improvement.
The paper identifies a "dynamic-probabilistic consistency gap" in surrogate modeling of chaotic systems, where standard probabilistic objectives can degrade learned dynamics or decouple uncertainty from local dynamics. They propose KAFFEE, a Kalman filter-based training framework, which reduces failure modes and improves reconstruction of dynamical invariants on stochastic hyperchaotic Lorenz-96 while maintaining competitive predictive scores.
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.