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Solving probabilistic weather forecasts is challenging due to computational constraints and the nonlinear nature of Earth atmosphere. Our team proposes a proof-of-concept to address these challenges by solving the Liouville equation, i.e., 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 63 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.

The forecast probability in Lorenz 63 model (Left): Analytical Solution and (Right): AI-based Liouville equation. Both shows nearly identical probability density function suggesting the reliability of forecast provided by AI-based Liouville equation.