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

2026-03-07

AI Summary

This work addresses the challenge of predicting chaotic time series, which are notoriously difficult due to their sensitivity to initial conditions, strong nonlinearity, and dependence on underlying dynamical mechanisms, while also lacking interpretability in most existing deep learning approaches. To this end, we propose two symbolic predictors—Symbolic Neural Forecaster (SyNF) and Symbolic Tree Forecaster (SyTF)—which, for the first time, integrate differentiable neural equation discovery with evolutionary symbolic regression. Our methods simultaneously achieve high short-term prediction accuracy and automatically uncover concise, explicit algebraic dynamical equations that govern the system. Experiments on 132 chaotic attractors and two real-world datasets demonstrate that the proposed approaches match the predictive performance of state-of-the-art models while yielding interpretable equations that reveal the intrinsic dynamics of the underlying systems.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• chaotic time series
• interpretable modeling
• symbolic regression
• forecasting
• black-box models

Innovation

• symbolic regression
• interpretable modeling
• chaotic time series forecasting
• equation discovery
• differentiable architecture