Dynamical errors in machine learning forecasts
This addresses the need for evaluating the physical fidelity of ML forecasts in scientific and engineering domains, though it is incremental in proposing new metrics rather than a paradigm shift.
The paper tackled the problem that standard error metrics like MAE and MSE do not assess the dynamical consistency of machine learning forecasts, which is critical in scientific applications. It found that larger forecast errors tend to occur in states with higher complexity and lower persistence, and proposed new metrics based on dynamical indices that revealed substantial distortions in dynamical properties, especially for long lead times.
In machine learning forecasting, standard error metrics such as mean absolute error (MAE) and mean squared error (MSE) quantify discrepancies between predictions and target values. However, these metrics do not directly evaluate the physical and/or dynamical consistency of forecasts, an increasingly critical concern in scientific and engineering applications. Indeed, a fundamental yet often overlooked question is whether machine learning forecasts preserve the dynamical behavior of the underlying system. Addressing this issue is essential for assessing the fidelity of machine learning models and identifying potential failure modes, particularly in applications where maintaining correct dynamical behavior is crucial. In this work, we investigate the relationship between standard forecasting error metrics, such as MAE and MSE, and the dynamical properties of the underlying system. To achieve this goal, we use two recently developed dynamical indices: the instantaneous dimension ($d$), and the inverse persistence ($θ$). Our results indicate that larger forecast errors -- e.g., higher MSE -- tend to occur in states with higher $d$ (higher complexity) and higher $θ$ (lower persistence). To further assess dynamical consistency, we propose error metrics based on the dynamical indices that measure the discrepancy of the forecasted $d$ and $θ$ versus their correct values. Leveraging these dynamical indices-based metrics, we analyze direct and recursive forecasting strategies for three canonical datasets -- Lorenz, Kuramoto-Sivashinsky equation, and Kolmogorov flow -- as well as a real-world weather forecasting task. Our findings reveal substantial distortions in dynamical properties in ML forecasts, especially for long forecast lead times or long recursive simulations, providing complementary information on ML forecast fidelity that can be used to improve ML models.