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Abstract This work proposes a general framework for analyzing noise-driven transitions in spatially extended non-equilibrium systems and explaining the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic parameterization formulas to reduce the complexity of the original equations while preserving the essential dynamical effects of unresolved scales. The approach is flexible and operates for both Gaussian noise and non-Gaussian noise with jumps. Our stochastic parameterization formulas offer two key advantages. First, they can approximate stochastic invariant manifolds when these manifolds exist. Second, even when such manifolds break down, our formulas can be adapted through a simple optimization of its constitutive parameters. This allows us to handle scenarios with weak time-scale separation where the system has undergone multiple transitions, resulting in large-amplitude solutions not captured by invariant manifolds or other time-scale separation methods. The optimized stochastic parameterizations capture then how small-scale noise impacts larger scales through the system’s nonlinear interactions. This effect is achieved by the very fabric of our parameterizations incorporating non-Markovian (memory-dependent) coefficients into the reduced equation. These coefficients account for the noise’s past influence, not just its current value, using a finite memory length that is selected for optimal performance. The specific memory function, which determines how this past influence is weighted, depends on both the strength of the noise and how it interacts with the system’s nonlinearities. Remarkably, training our theory-guided reduced models on a single noise path effectively learns the optimal memory length for out-of-sample predictions. This approach retains indeed good accuracy in predicting noise-induced transitions, including rare events, when tested against a large ensemble of different noise paths. This success stems from our hybrid approach, which combines analytical understanding with data-driven learning. This combination avoids a key limitation of purely data-driven methods: their struggle to generalize to unseen scenarios, also known as the ‘extrapolation problem.’ 

Detection and attribution (DA) studies are cornerstones of climate science, providing crucial evidence for policy decisions. Their goal is to link observed climate change patterns to anthropogenic and natural drivers via the optimal fingerprinting method (OFM). We show that response theory for nonequilibrium systems offers the physical and dynamical basis for OFM, including the concept of causality used for attribution. Our framework clarifies the method’s assumptions, advantages, and potential weaknesses. We use our theory to perform DA for prototypical climate change experiments performed on an energy balance model and on a low-resolution coupled climate model. We also explain the underpinnings of degenerate fingerprinting, which offers early warning indicators for tipping points. Finally, we extend the OFM to the nonlinear response regime. Our analysis shows that OFM has broad applicability across diverse stochastic systems influenced by time-dependent forcings, with potential relevance to ecosystems, quantitative social sciences, and finance, among others. Published by the American Physical Society2024