Charlotte Ricarda Doll

Daniel Durstewitz, Christoph Jürgen Hemmer, Florian Hess et al.

Time series (TS) modeling has come a long way from early statistical, mainly linear, approaches to the current trend in TS foundation models. With a lot of hype and industrial demand in this field, it is not always clear how much progress there really is. To advance TS forecasting and analysis to the next level, here we argue that the field needs a dynamical systems (DS) perspective. TS of observations from natural or engineered systems almost always originate from some underlying DS, and arguably access to its governing equations would yield theoretically optimal forecasts. This is the promise of DS reconstruction (DSR), a class of ML/AI approaches that aim to infer surrogate models of the underlying DS from data. But models based on DS principles offer other profound advantages: Beyond short-term forecasts, they enable to predict the long-term statistics of an observed system, which in many practical scenarios may be the more relevant quantities. DS theory furthermore provides domain-independent theoretical insight into mechanisms underlying TS generation, and thereby will inform us, e.g., about upper bounds on performance of any TS model, generalization into unseen regimes as in tipping points, or potential control strategies. After reviewing some of the central concepts, methods, measures, and models in DS theory and DSR, we will discuss how insights from this field can advance TS modeling in crucial ways, enabling better forecasting with much lower computational and memory footprints. We conclude with a number of specific suggestions for translating insights from DSR into TS modeling.

Georg Trede, Charlotte Ricarda Doll, Elias Weber et al.

Predicting the behavior of dynamical systems (DS) beyond the dynamical and parameter regimes observed in training is a pivotal and essentially unresolved problem in scientific ML. It is central to any good scientific theory, which we expect to be able to make predictions about regimes not covered by currently available data. Recent hierarchical and hyper-network guided approaches for DS reconstruction (DSR) enable training on many DS simultaneously, and revealed that extracted latent features are often related to crucial control parameters of the underlying DS that varied across the training corpus. However, true out-of-domain forecasting abilities of these models, e.g., across tipping points, remain limited, and fine-tuning, or even full model retraining, on time series from the new dynamical regime is usually required. Here, we mathematically analyze the root of these limitations in previous model formulations and identify three core shortcomings rooted in a mismatch between structural assumptions of the reconstruction model and typical properties of physical systems. We propose a combination of remedies for these shortcomings, most importantly feature splitting, and furthermore derive a closed-form bound on the reliable extrapolation range. We demonstrate empirically that our techniques allow for accurate zero-shot prediction into new dynamical regimes, outside the observed training regime, as, e.g., encountered across tipping points.