Toward an Analytical Solution of the Liouville Equation via Data-Driven Methods: Applications to Ensemble Forecasting

Solving probabilistic weather forecasts is challenging due to computational constraints and the nonlinear nature of Earth atmosphere. This study proposes a proof-of-concept to address these challenges by solving the Liouville equation, that is, the analytical solution for probabilistic forecasts, with data-driven method. Using the sparse identification of nonlinear dynamics (SINDy) algorithm, our research demonstrates that data-driven models can achieve accuracy levels in probabilistic forecasts comparable to analytical solutions. Through various experiments, including Bernoulli differential equations, the Lorenz 84 model, and subseasonal forecasts of tropical intraseasonal variability, we show that the data-driven Liouville equations yield simple functional forms or smoothness across physical space when predictability is present. These findings suggest the potential of these advancements in tackling higher-dimensional weather forecasting problems. Additionally, we discuss potential applications and future challenges.