Turning mechanistic models into forecasters by using machine learning

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.

💡 Research Summary

The paper introduces a novel framework that augments the Sparse Identification of Nonlinear Dynamics (SINDy) methodology with time‑varying parameters and leverages machine learning to forecast these parameters, thereby turning mechanistic differential‑equation models into powerful forecasters. Traditional data‑driven discovery techniques, such as SINDy, assume constant coefficients, which limits their ability to capture systems whose underlying processes evolve over time. To address this, the authors propose a two‑stage SINDy pipeline. In the first stage, the full time series is used to construct a library of candidate functions Θ(X,U) and to identify a set of active terms and their constant coefficients via Sequential Threshold Ridge Regression (STRR). In the second stage, the data are segmented into overlapping windows of fixed length w. Within each window, STRR is applied again, but only the top‑N active terms (selected by correlation with the state derivatives) are allowed to have coefficients that change from window to window; the remaining coefficients are kept fixed. This yields a hybrid ODE model of the form

  ẋ(t) = ¯ξ·Θ_fixed(x,u) + ξ(t)·Θ_timevarying(x,u)

where ¯ξ are constant parameters and ξ(t) are piecewise‑constant, time‑varying parameters.

Forecasting proceeds by training a machine‑learning regression model (Random Forest) to predict the future trajectory of ξ(t) from past values of ξ and external covariates (e.g., temperature, humidity, precipitation, wind speed). The predicted ξ̂(t) are substituted back into the learned ODE, which is then integrated forward to obtain forecasts of the state variables.

The methodology is evaluated on four datasets: (1) a simulated Susceptible‑Infected‑Recovered (SIR) epidemic model, (2) a simulated Consumer‑Resource (CR) ecological model, (3) real‑world measurements of CO₂ and CH₄ concentrations in Alberta oil‑sands tailings ponds, and (4) cyanobacteria cell counts from Alberta lakes. In all cases, the model achieves a mean absolute error (MAE) below 3 % when fitting the observed time series and below 6 % when forecasting up to one month ahead. Benchmark comparisons against a hybrid Convolutional Neural Network‑Long Short‑Term Memory (CNN‑LSTM) model and a Gradient Boosting Machine (GBM) show that the proposed approach consistently outperforms these baselines, often reducing forecast error by 10–15 %.

Key strengths of the approach include: (i) preservation of mechanistic interpretability while allowing dynamic adaptation of crucial parameters, (ii) automatic selection of which parameters should vary, reducing model complexity and over‑fitting risk, and (iii) seamless incorporation of exogenous drivers, making the framework suitable for policy or climate‑scenario analysis. Limitations are noted: the choice of window length w and the number of time‑varying terms N requires careful tuning, and linear regression‑based forecasting may struggle with abrupt, highly nonlinear parameter shifts.

The authors conclude that integrating time‑varying parameters into data‑driven discovery of differential equations creates a versatile hybrid modeling paradigm that bridges mechanistic insight and modern predictive analytics. Future work is suggested on Bayesian uncertainty quantification, deeper integration with neural time‑series models, and extending the library to capture nonlinear external influences, thereby enhancing robustness and applicability across a broader range of complex dynamical systems.