Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling
cs.LG, cs.AI, math.DS
Submitted: 2026-02-18
Updated: 2026-09-18
Code: https://github.com/Nixtla/statsforecast
License: http://creativecommons.org/licenses/by/4.0/
The gist: Time series (TS) modeling has come a long way from early statistical, mainly linear, approaches to the current trend in TS foundation models.
Terminology
Abstract
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.
Sources
- TiRex: Zero-Shot Forecasting Across Long and Short Horizons with Enhanced In-Context Learning
- Forecasting Sequential Data using Consistent Koopman Autoencoders
- DyNODE: Neural Ordinary Differential Equations for Dynamics Modeling in Continuous Control
- Chronos-2: From Univariate to Universal Forecasting
- Koopman-informed recurrent neural networks
- Neural Ordinary Differential Equations
- LLM-Integrated Bayesian State Space Models for Multimodal Time-Series Forecasting
- Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions
- Learning Partially Known Stochastic Dynamics with Empirical PAC Bayes
- Teacher Forcing as Generalized Bayes: Optimization Geometry Mismatch in Switching Surrogates for Chaotic Dynamics
- Parallel-in-Time Training of Recurrent Neural Networks for Dynamical Systems Reconstruction
- Reservoir computing with large valid prediction time for the Lorenz system
- Panda: A pretrained forecast model for chaotic dynamics
- Recurrent switching linear dynamical systems
- Mamba Integrated with Physics Principles Masters Long-term Chaotic System Forecasting
- Deep learning for universal linear embeddings of nonlinear dynamics
- An Operator Theoretic Approach for Analyzing Sequence Neural Networks
- Using Machine Learning to Replicate Chaotic Attractors and Calculate Lyapunov Exponents from Data
- Identifying nonlinear dynamical systems with multiple time scales and long-range dependencies
- When is a System Discoverable from Data? Discovery Requires Chaos
Related papers
- Polynomial-Augmented Neural Networks (PANNs) with Weak Orthogonality Constraints for Enhanced Function and PDE Approximation
- AIRL-S: Unifying Reinforcement Learning and Search-Based Test-Time Scaling via Adversarial Inverse Reinforcement Learning
- Transformers as Bayesian In-Context Experimenters: Smoothness-Adaptive Efficient ATE Estimation
- Convergence issues in Relational Concept Analysis based on AOC-posets
- Beliefs Beyond Posteriors: Local-Consistency Optimisation for Bayesian Neural Networks
- Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks