Probing forced responses and causality in data-driven climate emulators: Conceptual limitations and the role of reduced-order models

Physical Review A central challenge in climate science and applied mathematics is developing data-driven models of multiscale systems that capture both stationary statistics and responses to external perturbations. Current neural climate emulators aim to resolve the atmosphere-ocean system in all its complexity but often struggle to reproduce forced responses, limiting their use in causal studies such as Green's function experiments. To explore the origin of these limitations, we first examine a simplified dynamical system that retains key features of climate variability. We interpret the results through linear response theory, providing a rigorous framework to evaluate neural models beyond stationary statistics and probe causal mechanisms. We argue that the ability of emulators of multiscale systems to reproduce perturbed statistics depends critically on (1) the choice of an appropriate coarse-grained representation and (2) careful parameterizations of unresolved processes. These insights highlight reduced-order models, tailored to specific goals, processes, and scales, as valuable alternatives to general-purpose emulators. We next consider a real-world application, developing a neural model to investigate the joint variability of the surface temperature field and radiative fluxes. The model infers a multiplicative noise process directly from data, largely reproduces the system's probability distribution, and enables causal studies through forced responses. We discuss its limitations and outline directions for future work. These results expose key challenges in data-driven modeling of multiscale physical systems and underscore the value of coarse-grained, stochastic approaches, with response theory as a principled framework to guide model design and enhance causal understanding.