Turning Time Series into Algebraic Equations: Symbolic Machine Learning for Interpretable Modeling of Chaotic Time Series

Chaotic time series are notoriously difficult to forecast. Small uncertainties in initial conditions amplify rapidly, while strong nonlinearities and regime dependent variability constrain predictability. Although modern deep learning often delivers strong short horizon accuracy, its black box nature limits scientific insight and practical trust in settings where understanding the underlying dynamics matters. To address this gap, we propose two complementary symbolic forecasters that learn explicit, interpretable algebraic equations from chaotic time series data. Symbolic Neural Forecaster (SyNF) adapts a neural network based equation learning architecture to the forecasting setting, enabling fully differentiable discovery of compact and interpretable algebraic relations. The Symbolic Tree Forecaster (SyTF) builds on evolutionary symbolic regression to search directly over equation structures under a principled accuracy complexity trade off. We evaluate both approaches in a rolling window nowcasting setting with one step ahead forecasting using several accuracy metrics and compare against a broad suite of baselines spanning classical statistical models, tree ensembles, and modern deep learning architectures. Numerical experiments cover a benchmark of 132 low dimensional chaotic attractors and two real world chaotic time series, namely weekly dengue incidence in San Juan and the Nino 3.4 sea surface temperature index. Across datasets, symbolic forecasters achieve competitive one step ahead accuracy while providing transparent equations that reveal salient aspects of the underlying dynamics.

💡 Research Summary

This paper presents a novel approach to forecasting chaotic time series by transforming the problem into one of learning interpretable algebraic equations. The core challenge addressed is the inherent difficulty in predicting chaotic systems, where small initial errors amplify rapidly, coupled with the lack of interpretability in high-accuracy black-box deep learning models.

The authors propose two complementary symbolic forecasting methods: the Symbolic Neural Forecaster (SyNF) and the Symbolic Tree Forecaster (SyTF). SyNF adapts a neural network-based equation learning architecture, specifically an Equation Learner (EQL) network, to the forecasting setting. It represents potential mathematical expressions through a neural network whose weights correspond to coefficients and operator selections, enabling end-to-end differentiable learning of both the equation’s structure and parameters. SyTF, on the other hand, builds upon evolutionary symbolic regression using the PySR library. It searches the space of expression trees via genetic programming, explicitly optimizing for a Pareto frontier that balances prediction accuracy (e.g., Mean Squared Error) against model complexity (e.g., tree size).

The methodology is rigorously evaluated on an extensive benchmark. The synthetic dataset comprises 132 distinct low-dimensional chaotic attractors (e.g., Lorenz, Rössler, Chua’s circuit), providing a diverse testbed for generalization. Two real-world chaotic time series are also included: weekly dengue fever incidence in San Juan, Puerto Rico, and the Niño 3.4 Sea Surface Temperature index. The forecasting task is framed as a rolling-window, one-step-ahead prediction (nowcasting). A broad suite of baseline models is used for comparison, encompassing classical statistical methods (ARIMA), tree ensembles (Random Forest, XGBoost), and modern deep learning architectures (LSTM, Transformer, N-BEATS).

The experimental results demonstrate that both SyNF and SyTF achieve competitive, and often superior, one-step-ahead forecasting accuracy compared to the baselines across the synthetic and real-world datasets. Crucially, they deliver this performance while providing fully transparent, compact algebraic equations as their output models. These equations are not merely predictive tools but can offer scientific insight by revealing dominant nonlinear terms, feedback loops, or linear approximations around equilibria within the underlying dynamics.

The paper concludes by discussing limitations and future directions. Current work focuses on one-step prediction and lower-dimensional systems; extensions to multi-step forecasting, higher-dimensional data, and the incorporation of exogenous variables are noted as important next steps. The research signifies a paradigm shift, advocating for forecasting models that do not sacrifice interpretability for accuracy, thereby bridging the gap between data-driven prediction and scientific discovery. It establishes symbolic regression as a serious and effective contender in the chaotic time series forecasting landscape.