Should physical models be replaced by machine learning? A physics-corrected probabilistic forecasting framework

Recent advances in machine learning have led to claims that purely data-driven models can outperform traditional physical models for forecasting complex dynamical systems. This tension is especially evident in numerical weather prediction, where both physically derived models and deep learning methods are actively employed. This raises a fundamental question: Does the available data contain sufficient information for an empirical model to outperform a simulation model? If so, what are the broader implications? Should physical models be replaced entirely? The question is addressed by first constructing a controlled system–model environment based on the classic Lorenz63 system. A physics-corrected methodology is introduced, in which a data-driven model is trained to probabilistically correct the systematic errors in the output of an imperfect physical model, allowing the two components to complement each other. Rather than imposing governing-equation constraints directly within the learning process, the proposed methodology treats the output of an imperfect physical model as a fallible forecast component to be corrected using observational data. Moreover, it differs from conventional residual-correction approaches by formulating the correction itself probabilistically: instead of learning a single pointwise adjustment, a Mixture Density Network is used to learn the conditional distribution of the physical-model forecast error. Physical-model ensembles are likewise represented probabilistically through kernel dressing, allowing all forecasting approaches to be evaluated within a common probabilistic framework. Extensive experiments are conducted under a range of conditions, including varying observational noise levels, degrees of physical model imperfection, and forecast lead times. Evaluation based on probabilistic forecasting metrics shows that physics-corrected models outperform the standalone physical and purely data-driven models across most experimental settings, with the magnitude of improvement varying across different settings. The results underscore the broader value of integrating physical and data-driven perspectives, suggesting a general modeling paradigm for building robust, interpretable, and uncertainty-aware forecasting systems across a wide range of complex dynamical processes.