Turning time series into algebraic equations: Symbolic machine learning for interpretable modeling of chaotic time series
Abstract
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 adapts a neural network-based equation learning architecture to the forecasting setting, enabling end-to-end differentiable discovery of compact and interpretable algebraic relations. The symbolic tree forecaster 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 one-step-ahead forecast setting using several accuracy metrics and compare against a broad suite of baselines spanning classical statistical models, tree ensembles, and state-of-the-art deep learning architectures. Numerical experiments cover a benchmark of 132 low-dimensional chaotic attractors and two real-world chaotic time series (weekly dengue incidence in San Juan and the Niño 3.4 sea-surface temperature index). Across datasets, symbolic forecasters achieve competitive one-step-ahead accuracy while providing transparent analytical equations that reveal salient aspects of the underlying dynamics.
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BibTeX
@article{panja2026turning,
title = {{Turning time series into algebraic equations: Symbolic machine learning for interpretable modeling of chaotic time series}},
author = {Madhurima Panja and Grace Younes and Chittaranjan Hens and Tanujit Chakraborty},
journal = {APL Machine Learning},
year = {2026},
month = nov,
doi = {10.1063/5.0358326},
eprint = {2603.07261},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05853621v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - Turning time series into algebraic equations: Symbolic machine learning for interpretable modeling of chaotic time series
AU - Madhurima Panja
AU - Grace Younes
AU - Chittaranjan Hens
AU - Tanujit Chakraborty
PY - 2026
DA - 2026/11/06
JO - APL Machine Learning
DO - 10.1063/5.0358326
UR - https://hal.science/hal-05853621v1
AB - 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 adapts a neural network-based equation learning architecture to the forecasting setting, enabling end-to-end differentiable discovery of compact and interpretable algebraic relations. The symbolic tree forecaster 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 one-step-ahead forecast setting using several accuracy metrics and compare against a broad suite of baselines spanning classical statistical models, tree ensembles, and state-of-the-art deep learning architectures. Numerical experiments cover a benchmark of 132 low-dimensional chaotic attractors and two real-world chaotic time series (weekly dengue incidence in San Juan and the Niño 3.4 sea-surface temperature index). Across datasets, symbolic forecasters achieve competitive one-step-ahead accuracy while providing transparent analytical equations that reveal salient aspects of the underlying dynamics.
SN - 2770-9019
ER -