The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling

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.

Teacher Forcing as Generalized Bayes: Optimization Geometry Mismatch in Switching Surrogates for Chaotic Dynamics

Identity teacher forcing (ITF) enables stable training of deterministic recurrent surrogates for chaotic dynamical systems and has been highly effective for dynamical systems reconstruction (DSR) with recurrent neural networks (RNNs), including interpretable almost-linear RNNs (AL-RNNs). However, as an intervention-based prediction loss (and thus a generalized Bayes update), teacher forcing need not match the free-running model's marginal likelihood geometry. We compare the objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs, estimating ambiguity-aware observed information via Louis' identity. In the switching setting studied here, conditioning on a single forced regime path (as ITF does) inflates curvature, while marginal likelihood curvature is reduced by a missing-information correction when multiple switching explanations remain plausible. In Lorenz-63 experiments, windowed evidence fine-tuning improves held-out evidence but can degrade dynamical quantities of interest (QoIs) relative to ITF-pretrained models.