Abstract
<title>Abstract</title> <p>Deterministic AI weather and ocean models are far more computationally efficient and achieve lower mean-square error than state-of-the-art operational forecasts, yet they are widely criticized for smoothing small scales, underestimating extremes, and violating physical constraints. We offer a unified Markov-Process explanation: MSE-trained deterministic models learn the conditional mean E[x_{t+Δt} | x_t] rather than the full transition law P(x_{t+Δt} | x_t), which links low RMSE to over-smoothing and underestimation of extremes. Moreover, this conditional mean represents a linear combination of potential physical states, and thus violates the basic property of nonlinear dynamics that linear combinations of solutions are in general not solutions of the governing equations, which accounts for the fundamental origin of physical-conservation violations in deterministic models. We first establish the corresponding mathematical analysis, and then support the theory with a finite-state Markov experiment and a stochastically perturbed Kelvin-Helmholtz fluid experiment using matched U-Net backbones. Both experiments show that deterministic models can attain lower next-step RMSE, but at the cost of almost completely discarding next-step distributional information relative to models that recover the full transition law. Although our analysis focuses on deterministic models trained with an RMSE loss as a representative case, the conclusions apply to deterministic forecasting models more generally. From an information-entropy perspective, deterministic forecasts are low-entropy at each step and overconfident relative to atmospheric and oceanic variability, whereas diffusion models sample from conditional transition distributions and better match intrinsic complexity. Accordingly, distributional generative methods may be better suited to future AI weather prediction, reconstruction, and bias correction.</p>