Learning Macroeconomic Dynamics from Data: Applications of Universal Differential Equations and SINDy

Abstract

This paper introduces scientific machine learning methods to macroeconomic model discovery, focusing on Sparse Identification of Nonlinear Dynamics (SINDy), Universal Differential Equations (UDEs), and a hybrid UDE-SINDy pipeline in which a neural network embedded within a differential equation learns unknown dynamics that are subsequently distilled into interpretable symbolic equations via sparse regression. The Goodwin growth cycle, a canonical predator-prey formulation of distributional conflict between labour and capital, serves as the testing ground throughout. We survey the relevant methodology, implement each approach in a unified computational framework, and benchmark them on problems calibrated to macroeconomic scales, where data are scarce, noisy, and nonstationary. On stationary synthetic Goodwin data, the hybrid pipeline recovers the bilinear interaction coefficients to four significant figures, with the vu term emerging as the sole active component of the sparse representation. On nonstationary synthetic data, consisting of a four-regime piecewise system with forced oscillations, abrupt parameter shifts, and transition damping, both direct SINDy and UDE-SINDy recover the predator-prey structure with correct structural identification. These experiments establish that the methods can operate under conditions characteristic of macroeconomic time series. When the validated pipeline is applied to 77 years of U.S. quarterly data (1948:Q1-2025:Q3) on employment, capacity utilisation, and the labour share, neither method recovers the vu interaction at any regularisation strength. Direct SINDy returns dense polynomial models; the UDE-SINDy pipeline finds near-zero neural network corrections and no dominant interaction term. The result is consistent across both methods and both empirical specifications, classical (v, u) and structuralist (TCU, u). The Lotka-Volterra functional form that defines the Goodwin model is not present in the data. Several caveats apply, including the restriction to secondorder polynomial libraries and the treatment of the system as a two-dimensional autonomous ODE. The methods developed here are not specific to the Goodwin model and can be applied to other macroeconomic dynamical systems for which candidate functional forms can be specified.

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DOI

10.2139/ssrn.6491442

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https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6491442

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