AI summaryⓘ
The authors study how to build models that predict complex time-dependent systems and how to measure the uncertainty in those predictions properly. They find a problem they call the dynamic-probabilistic consistency gap, where usual methods make the model's uncertainty estimates unreliable or disconnected from the system's true behavior. To fix this, they propose a new training approach called KAFFEE, which uses a type of Kalman filter to better track both the predictions and their uncertainties. When tested on chaotic systems, KAFFEE improved the accuracy of modeled dynamics and uncertainty estimates compared to standard methods. The authors also show that their approach helps maintain good performance when adapting models to new chaotic systems.
Dynamical systems reconstructionUncertainty quantificationDynamic-probabilistic consistencyExtended Kalman filterJacobian matrixChaotic systemsLorenz-96 modelBayesian filteringTime-series modelingProbabilistic objectives
Authors
Andre Herz, Matthijs Pals, Daniel Durstewitz, Georgia Koppe
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.