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

Madhurima Panja; Grace Younes; Tanujit Chakraborty

cs.LGMar 7, 2026

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

Authors: Madhurima Panja, Grace Younes, Tanujit Chakraborty

Organizations: SAFIR, Sorbonne University Abu Dhabi, UAE · Sorbonne Center for Artificial Intelligence, Sorbonne University, Paris, France

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 (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.

Figures & tables

Figure 1: A dataset of 132 distinct low-dimensional chaotic systems, colored by largest Lyapunov exponent ( $\lambda_{\max}$ ).

Data	Time Span	Length	Granularity	Linearity	Stationarity	Normal	Seasonality	Chaotic
Sanjuan Dengue	1990–2013	1196	Weekly	✗ (0.026)	✓ (0.098)	✗ (0.001)	✓ (0.001)	✓ (0.001)
El Niño SST	1990–2021	1634	Weekly	✓ (0.777)	✓ (0.100)	✗ (0.001)	✓ (0.001)	✓ (0.001)

Table 1: Statistical characteristics of real-world datasets. In the table, a ✓ denotes the presence of a feature, while a ✗ denotes its absence. Values in parentheses represent the p-values from the corresponding statistical tests and the maximum Lyapunov exponent.

Figure 2: Real-world datasets along with their autocorrelation function (ACF) and partial autocorrelation function (PACF).

Figure 3: Symbolic Forecasting Architectures. The upper panel illustrates the architecture of the Symbolic Neural Forecaster (SyNF), where a differentiable neural training mechanism is used to learn interpretable forecasting equations from symbolic operators and lagged inputs. The lower panel presents the workflow of the Symbolic Tree Forecaster (SyTF), which employs an evolutionary search to construct and refine symbolic expression trees based on predictive performance and interpretability. The demonstration is conducted on a simulated dataset. The mathematical expression obtained from the SyNF model is $\hat{y}_{t}=0.1110+0.9421y_{t-1}+0.0001\left(y_{t-1}\right)^{2}$ (top image) whereas the SyTF architecture yields $\hat{y}_{t,\text{SyTF}}=0.9613y_{t-1}$ (bottom image) using a single historical lagged observation. The one-step-ahead rolling window forecasts are also presented in the upper and lower panels for the simulated example datasets.

Figure 4: Comparative performance of symbolic forecasting approaches and state-of-the-art frameworks for one-step ahead (rolling window) forecasting of chaotic datasets using five historical observations for each model. In the plot, panels show error distributions across four metrics: RMSE (top-left), MAE (top-right), MARRE (bottom-left), and SMAPE (bottom-right). Red dots denote median performance; error bars represent the interquartile range (IQR). For all panels, models are ranked by ascending median error (lower values indicate better performance).

Figure 5: Comparative performance of symbolic forecasting approaches and state-of-the-art frameworks for one-step ahead forecasting of chaotic datasets using ten historical observations for each model. In the plot, panels show error distributions across four metrics: RMSE (top-left), MAE (top-right), MARRE (bottom-left), and SMAPE (bottom-right). Red dots denote median performance; error bars represent the interquartile range (IQR). For all panels, models are ranked by ascending median error (lower values indicate better performance).

Figure 6: Comparative performance of symbolic forecasting approaches and state-of-the-art frameworks for one-step ahead forecasting of chaotic datasets using twenty-five historical observations for each model. In the plot, panels show error distributions across four metrics: RMSE (top-left), MAE (top-right), MARRE (bottom-left), and SMAPE (bottom-right). Red dots denote median performance; error bars represent the interquartile range (IQR). For all panels, models are ranked by ascending median error (lower values indicate better performance).

Figure 7: Visualization of the MCB test results for synthetic chaotic attractor forecasting tasks based on RMSE (left) and SMAPE (right) metrics. The vertical axis represents the average ranks of the forecasting methods, while the horizontal axis lists the corresponding techniques. For example, SyTF – 1.98 indicates that the SyTF framework obtained an average rank of 1.98 under the RMSE metric, with similar interpretations applicable to the remaining models.

Dataset	Sanjuan Dengue				El Niño SST
Models	SMAPE	RMSE	MAE	MARRE	SMAPE	RMSE	MAE	MARRE
NLinear	21.70	21.37	16.97	7.44	1.03	0.34	0.27	8.29
RandomForest	25.21	25.46	20.08	8.81	0.87	0.30	0.22	6.94
XGBoost	27.58	28.09	22.16	9.72	0.92	0.31	0.24	7.29
LightGBM	24.27	32.62	21.99	9.64	0.82	0.28	0.21	6.55
LSTM	50.21	72.43	49.98	21.92	5.23	1.62	1.40	42.41

Table 2: Comparative evaluation of forecasting performance for machine learning models and symbolic forecasting frameworks on real-world datasets across multiple performance indicators. In the table, the best and second-best performing models are highlighted.

Dataset	Equation from
Dataset	SyTF $\left(\hat{y}_{t,\text{SyTF}}=\right)$	SyNF $\left(\hat{y}_{t}=\right)$
Sanjuan	$y_{t-4}0.087\sin(y_{t-2}+$	$-1.1e^{-4}y_{t-1}^{2}+6.7e^{-5}y_{t-1}y_{t-2}+0.288y_{t-1}+3.7e^{-5}y_{t-1}^{2}+0.684y_{t-2}+0.938-$
Dengue	$\sin(0.280y_{t-3}-0.696))+y_{t-4}$	$0.203\sin(-3.6e^{-4}y_{t-1}+1.337y_{t-2}+1.439)+0.601\cos(1.568y_{t-1}-0.288y_{t-2}+0.063)$
El Niño	$0.985y_{t-1}+0.417$	$-8.2e^{-3}y_{t-1}^{2}+0.017y_{t-1}y_{t-2}+0.553y_{t-1}+0.019y_{t-2}^{2}-0.324y_{t-2}+0.031-$
SST	$0.985y_{t-1}+0.417$	$0.137\sin(1.438y_{t-1}+0.078y_{t-2}-0.107)+1.792\cos(0.536y_{t-1}+3.2e^{-5}y_{t-2}-0.463)$

Table 3: Symbolic equations learned by SyTF and SyNF for real-world datasets.

Figure 8: Visualization of the MCB test results for real-world forecasting tasks based on RMSE (left) and SMAPE (right) metrics. The vertical axis represents the average ranks of the forecasting methods, while the horizontal axis lists the corresponding techniques. For instance, SyNF-Div-Reg – 2.50 denotes that the SyNF-Div-Reg framework achieved an average rank of 2.50 according to the RMSE metric (lower metric indicates better performance), and similar interpretations apply to the other methods.

Figure 9: The plot visualizes the ground truth (red points) observations of (a) Sanjuan Dengue and (b) El Niño SST dataset during the test period, along with the point forecasts (blue line) and the conformal prediction interval of the SYNF-Div-Reg model (yellow shaded).

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Symbolic Neural ODEs: Learning interpretable models from time-series data

Nibodh Boddupalli, Jeff Moehlis

Dept. of Mechanical Engineering, University of California, Santa Barbara, CA 93106

Nonlinear System Identification System Identification Neural ODEs Sparse Identification of Nonlinear Dynamics Chaotic Time-Series Prediction

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We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.

Abstract

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