Sensitivity Fidelity Reveals Physical Dependencies Beyond Forecast Accuracy in Deep-Learning Earth-System Models

Deep-learning models increasingly rival numerical simulation in Earth-system forecasting, yet forecast error alone cannot determine whether their learned input-output dependencies are physically consistent. Here we introduce sensitivity fidelity as an additional criterion: alongside prediction error, model responses to input perturbations should be examined against a trusted reference or established physical knowledge. In an idealized Rayleigh-B'enard convection benchmark, four surrogate architectures achieved low forecast errors yet encoded markedly different sensitivities. Quantitative comparison with a matched adjoint distinguished the models far more sharply than forecast accuracy and showed that predictive skill and sensitivity fidelity need not coincide, while finite-perturbation experiments further linked these differences to the propagation of initial-condition uncertainty. We then applied this strategy to El Ni~no-Southern Oscillation prediction, where no adjoint reference exists and learned sensitivities showed event-wise robustness, validity under finite perturbations, and consistency with established precursors. These results identify gradient-based evaluation as a discriminating complement to forecast-error metrics, providing a practical test of whether data-driven Earth-system models learn physically credible dependencies.