Topological Out-of-Domain Generalization in Dynamical Systems Reconstruction

Recent foundation models (FMs) for zero-shot reconstruction of dynamical systems (DS) achieve strong out-of-domain generalization but provide little insight into the mechanisms that underlie their forecasts. Such an understanding could help to strip down overladen FM architectures to their bare essence and expose the minimal requirements for in-context learning in the DS domain. Toward this goal, here we iteratively reduce a recent powerful SOTA model for DS reconstruction, DynaMix (Hemmer & Durstewitz, 2025), to a minimal interpretable two-parameter form, which we call DynaBase. DynaBase produces forecasts through a linear blend of the current latent state and the nearest in-context neighbor and its temporal successor. Surprisingly, despite its extreme simplicity, DynaBase produces highly competitive zero-shot DS reconstructions across chaotic and cyclic systems, with a negligible parameter load, many orders of magnitude below that of other FMs. Even more, this extreme simplicity permits direct model optimization on DS reconstruction measures, as well as closed-form one-step analytical solutions on prediction MSE. Theoretical and empirical analysis of DynaBase further leads to a 1-parameter family of maps, with the context-parroting algorithm of (Zhang & Gilpin, 2026) recovered at one end, and chaotic (divergent but bounded) behavior at the other. We further show how different training strategies lead to models either optimal for short-term prediction or for DS reconstruction. Thus, DynaBase not only exposes the minimal mechanisms required for producing zero-shot DS reconstruction, but also reconciles within an accessible mathematical frame divergent observations in the literature.

Successful deep neural networks discover salient features of data. We show when and why they fail to learn out-of-distribution (OOD)-relevant representations from an in-distribution (ID) training window. This requires decoupling feature learning from data-generating-process (DGP) identifiability. From a single training window, OOD extrapolation is non-identifiable: infinitely many DGPs are $\varepsilon$-observationally equivalent on the training data but diverge arbitrarily outside it, and no in-distribution criterion alone reliably breaks the tie. A structural commitment, the feature map, label map, and model class $(\varphi, ψ, \mathcal{M})$, dictates the assumed DGP and governs OOD generalization while leaving ID performance essentially unchanged. When architecture, pretraining, augmentation, input formats, or domain knowledge implicitly inject the missing commitment, the model succeeds. When it cannot infer OOD-relevant structure from ID evidence, it fails. Changing only the representation can make the same architecture, at the same in-distribution loss, differ by ${\sim}520\times$ out of distribution. When the commitment is correct and identifiable, OOD error vanishes. For example, Fourier coordinates turn periodic extrapolation into interpolation on $\mathbb{S}^1$. The same mechanism predicts outcomes in three natural-science settings (mass-action chemistry; Kepler's-third-law exoplanet prediction, $n=2{,}362$; and cross-species coding-DNA detection) and in a 264-run positional-encoding study across Transformer, Mamba, and S4D. Finally, a controlled study shows: correct features are necessary but not sufficient. The model class must express the target, and the transformed training data must cover the relevant representation space.

Many systems resist analytical modeling, making data-driven inference of dynamics important. Yet data-driven methods can fail to converge or generalize, leaving open a central question: When can system behavior be learned reliably from data, and when is such learning impossible? We answer this question using adversarial dynamical systems to identify the boundary between accessible and inaccessible regimes. In Koopman operator learning, a leading framework for representing nonlinear dynamics through linear spectral objects, we design optimal data-driven spectral algorithms with convergence and certification guarantees under conditions arising broadly in physical systems. This yields a convergence theory for Koopman-operator approximations and resolves a longstanding open problem in Koopman spectral analysis. Conversely, by constructing adversarial systems, we prove matching impossibility results: without these conditions, no single-sequence limiting procedure can guarantee learning, regardless of data quality. These results sharply characterize when data-driven spectral learning can succeed and when it must fail. We validate the framework on oscillators, chaotic fluid flows and Arctic sea ice concentration forecasting. In the latter, we uncover hidden modes of Arctic sea ice decline, deliver long-range forecasts with geographic error bounds, and outperform state-of-the-art dynamical and deep learning models at substantially lower computational cost, enabling real-time deployment on standard CPUs.