Turning mechanistic models into forecasters by using machine learning

2026-02-04

AI Summary

This work addresses the limitation of conventional data-driven approaches, which typically assume constant model parameters and thus struggle to capture time-evolving mechanisms in complex dynamical systems. The study proposes a novel framework that explicitly incorporates time-varying parameters into data-driven discovery of differential equations for the first time. By constructing a library of candidate functions and employing sparse regression to identify governing equations, the method further leverages machine learning models—such as CNN-LSTM and gradient boosting machines (GBM)—to forecast the temporal evolution of these parameters, yielding a forward-looking dynamic predictor. Evaluated on multiple real-world datasets, the approach achieves learning errors below 3% and forecasting errors under 6% within a one-month horizon, significantly outperforming purely data-driven baselines and establishing an interpretable, high-accuracy paradigm for modeling time-varying dynamical systems.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Search and Optimization: Learning to Search
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

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
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

The equations of complex dynamical systems may not be identified by expert knowledge, especially if the underlying mechanisms are unknown. Data-driven discovery methods address this challenge by inferring governing equations from time-series data using a library of functions constructed from the measured variables. However, these methods typically assume time-invariant coefficients, which limits their ability to capture evolving system dynamics. To overcome this limitation, we allow some of the parameters to vary over time, learn their temporal evolution directly from data, and infer a system of equations that incorporates both constant and time-varying parameters. We then transform this framework into a forecasting model by predicting the time-varying parameters and substituting these predictions into the learned equations. The model is validated using datasets for Susceptible-Infected-Recovered, Consumer--Resource, greenhouse gas concentration, and Cyanobacteria cell count. By dynamically adapting to temporal shifts, our proposed model achieved a mean absolute error below 3\% for learning a time series and below 6\% for forecasting up to a month ahead. We additionally compare forecasting performance against CNN-LSTM and Gradient Boosting Machine (GBM), and show that our model outperforms these methods across most datasets. Our findings demonstrate that integrating time-varying parameters into data-driven discovery of differential equations improves both modeling accuracy and forecasting performance.

Problem

• time-varying parameters
• data-driven discovery
• dynamical systems
• forecasting
• governing equations

Innovation

• time-varying parameters
• data-driven discovery
• differential equations
• forecasting model
• dynamic systems