Physics constraints and response validation in discrete-time reduced-order modeling: from idealized turbulent systems to climate dynamics

A central challenge across science and engineering is to build data-driven reduced-order models of turbulent dynamical systems that reproduce stationary statistics, predict responses to external perturbations, and remain practical for real-world applications. To this end, we introduce an abstract discrete-time formulation of turbulent dynamical systems with exact energy-conserving nonlinearities. Parameterizing this structure with neural networks yields stable, physics-constrained reduced-order models. We then use the fluctuation-dissipation theorem (FDT) to validate the emulators' forced responses from unperturbed data alone, testing models beyond stationary statistics. The FDT also identifies candidate direct causal links, which are used as regularization terms only when they pass validation. We first test the framework on two idealized models of geophysical turbulence: the proposed emulators reproduce stationary statistics and accurately predict responses to weak and strong forcings, despite being trained solely on unperturbed data. We then move beyond idealized systems to model tropical climate dynamics from reanalysis data, where data scarcity, partial observability, and sensitivity to choices of stochastic parameterizations become central challenges. The resulting physics-constrained model reproduces key statistics of the El Niño-Southern Oscillation (ENSO) and qualitatively captures its cumulative responses to perturbations; augmenting it with a non-Markovian stochastic closure substantially improves quantitative agreement with the FDT benchmark. This response-validated model is then used to characterize long-term causal drivers of ENSO variability. The proposed methodology establishes a modular framework for stable reduced-order models capable of probing causal mechanisms in realistic, partially observed turbulent systems.