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Only a subset of degrees of freedom are typically accessible or measurable in real-world systems. As a consequence, the proper setting for empirical modeling is that of partially-observed systems. Notably, data-driven models consistently outperform physics-based models for systems with few observable degrees of freedom; e.g. hydrological systems. Here, we provide an operator-theoretic explanation for this empirical success. To predict a partially-observed system’s future behavior with physics-based models, the missing degrees of freedom must be explicitly accounted for using data assimilation and model parametrization. Data-driven models, in contrast, employ delay-coordinate embeddings and their evolution under the Koopman operator to implicitly model the effects of the missing degrees of freedom. We describe in detail the statistical physics of partial observations underlying data-driven models using novel maximum entropy and maximum caliber measures. The resulting nonequilibrium Wiener projections applied to the Mori–Zwanzig formalism reveal how data-driven models may converge to the true dynamics of the observable degrees of freedom. Additionally, this framework shows how data-driven models infer the effects of unobserved degrees of freedom implicitly, in much the same way that physics models infer the effects explicitly. This provides a unified implicit-explicit modeling framework for predicting partially-observed systems, with hybrid physics-informed machine learning methods combining both implicit and explicit aspects.

Several Earth system components are at a high risk of undergoing rapid, irreversible qualitative changes or “tipping” with increasing climate warming. It is therefore necessary to investigate the feasibility of arresting or even reversing the crossing of tipping thresholds. Here, we study feedback control of an idealized energy balance model (EBM) for Earth’s climate, which exhibits a “small icecap” instability responsible for a rapid transition to an ice-free climate under increasing greenhouse gas forcing. We develop an optimal control strategy for the EBM under different forcing scenarios to reverse sea-ice loss while minimizing costs. Control is achievable for this system, but the cost nearly quadruples once the system tips. While thermal inertia may delay tipping, leading to an overshoot of the critical forcing threshold, this leeway comes with a steep rise in requisite control once tipping occurs. Additionally, we find that the optimal control is localized in the polar region.

With shifting environmental trends, many Earth system elements may be poised to undergo critical transitions or ‘tipping’. Reliable anticipation of these tipping elements is vital to inform policy decisions. Many of the current methods for tipping point detection are based on loss of resilience or ‘critical slowdown’ of the system as it approaches a tipping point. However, these methods are prone to false alarms; the detected slowdown may be an artifact of nonstationary noise unrelated to tipping behavior. Here, we explore the efficacy of early warning signs based on a nonequilibrium thermodynamics framework. The model-free detection method relies on the increased intrinsic time-irreversibility due to detailed balance breaking, preceding the onset of tipping or instabilities. We demonstrate that these EWSs are effective for tipping point detection and robust against false alarms due to nonstationary noise, using idealized models for two key elements of the Earth system that are prone to tipping: the Atlantic Meridional Overturning Circulation and Arctic sea-ice loss. Increased intrinsic time-irreversibility provides a robust early warning for Earth system tipping points, outperforming traditional methods in the presence of nonstationary noise, according to a study based on a nonequilibrium thermodynamics framework.

Causal discovery aims to identify cause-effect mechanisms for better scientific understanding, explainable decision-making, and more accurate modeling. Standard statistical frameworks, such as Granger causality, lack the ability to quantify causal relationships in nonlinear dynamics due to the presence of complex feedback mechanisms, timescale mixing, and nonstationarity. Thus, applying these methods to study causal dynamics in real-world systems, such as the Earth, is a major challenge. Addressing this shortcoming, we leverage deep learning and a Koopman operator-theoretic formalism to present a class of causal discovery algorithms. Kausal uses deep Koopman operator methods to approximate nonlinear dynamics in a linearized vector space in which traditional causal inference methods such as Granger causality can be more easily applied. Our idealized experiments demonstrate Kausal ’s superior ability in discovering and characterizing causal signals compared to existing deep learning and non-deep learning state-of-the-art approaches. Finally, the successful identification of major El Niño and La Niña events in observations showcases Kausal ’s skill to handle real-world applications. The authors develop Kausal, a deep learning framework that combines causal discovery with Koopman operator theory to study complex, nonlinear systems. They show that it accurately identifies causal signals and successfully detects major El Niño and La Niña events, outperforming existing methods.

Abstract Only a subset of degrees of freedom are typically accessible or measurable in real-world systems. As a consequence, the proper setting for empirical modeling is that of partially-observed systems. Notably, data-driven models consistently outperform physics-based models for systems with few observable degrees of freedom; e.g. hydrological systems. Here, we provide an operator-theoretic explanation for this empirical success. To predict a partially-observed system’s future behavior with physics-based models, the missing degrees of freedom must be explicitly accounted for using data assimilation and model parametrization. Data-driven models, in contrast, employ delay-coordinate embeddings and their evolution under the Koopman operator to implicitly model the effects of the missing degrees of freedom. We describe in detail the statistical physics of partial observations underlying data-driven models using novel maximum entropy and maximum caliber measures. The resulting nonequilibrium Wiener projections applied to the Mori–Zwanzig formalism reveal how data-driven models may converge to the true dynamics of the observable degrees of freedom. Additionally, this framework shows how data-driven models infer the effects of unobserved degrees of freedom implicitly, in much the same way that physics models infer the effects explicitly. This provides a unified implicit-explicit modeling framework for predicting partially-observed systems, with hybrid physics-informed machine learning methods combining both implicit and explicit aspects.

We generalize the exact predictive regularity of symmetry groups to give an algebraic theory of patterns, building from a core principle of future equivalence. For topological patterns in fully-discrete one-dimensional systems, future equivalence uniquely specifies a minimal semiautomaton. We demonstrate how the latter and its semigroup algebra generalizes translation symmetry to partial and hidden symmetries. This generalization is not as straightforward as previously considered. Here, though, we clarify the underlying challenges. A stochastic form of future equivalence, known as predictive equivalence, captures distinct statistical patterns supported on topological patterns. Finally, we show how local versions of future equivalence can be used to capture patterns in spacetime. As common when moving to higher dimensions, there is not a unique local approach, and we detail two local representations that capture different aspects of spacetime patterns. A previously developed local spacetime variant of future equivalence captures patterns as generalized symmetries in higher dimensions, but we show that this representation is not a faithful generator of its spacetime patterns. This motivates us to introduce a local representation that is a faithful generator, but we demonstrate that it no longer captures generalized spacetime symmetries. Taken altogether, building on future equivalence, the theory defines and quantifies patterns present in a wide range of classical field theories.

Coherent structures form spontaneously in far-from-equilibrium spatiotemporal systems and are found at all spatial scales in natural phenomena from laboratory hydrodynamic flows and chemical reactions to ocean and atmosphere dynamics. Phenomenologically, they appear as key components that organize macroscopic dynamical behaviors. Unlike their equilibrium and near-equilibrium counterparts, there is no general theory to predict what patterns and structures may emerge in far-from-equilibrium systems. Each system behaves differently; details and history matter. The complex behaviors that emerge cannot be explicitly described mathematically, nor can they be directly deduced from the governing equations (e.g. what is the mathematical expression for a hurricane, and how can you derive it from the equations of a general circulation climate model?). It is thus appealing to bring the instance-based data-driven models of machine learning to bear on the problem. Supervised learning models have been the most successful, but they require ground-truth training labels which do not exist for far-from-equilibrium structures. Unsupervised models that leverage physical principles of self-organization are required.The work developed in this thesis utilizes a notion of intrinsic computation to construct a physics-based machine learning model, the local causal states, to extract emergent pattern and structure in complex spatiotemporal systems. As a behavior-driven theory, it does so without requiring the governing equations or ground-truth training labels. After motivating the need for history-dependent, instance-based modeling for studying far-from-equilibrium phenomena, parallels between models of computation and complex dynamical system will be developed to argue for the use of machine learning models based on intrinsic computation. The mathematical foundations in symbolic dynamics and shift spaces is then given for the local causal states. Spacetime invariant sets are shown to be equivalent to spacetime symmetries in the local causal states. These behaviors, known as domains, capture pattern as generalized symmetries. Using this, the local causal states are used to give a formal definition of coherent structures as spatially-localized, temporally-persistent deviations from generalized spacetime symmetries. The utility of the local causal states in capturing pattern and structure is demonstrated using cellular automata models and complex fluid flows (using both simulations and observations). The fluid flow results require high-performance computing; we will briefly describe our distributed implementation in Python and how we were able to process almost 90TB of data from the CAM5.1 climate model in under 7 minutes end-to-end on 1024 Intel Haswell nodes of the Cori supercomputer.

We generalize the exact predictive regularity of symmetry groups to give an algebraic theory of patterns, building from a core principle of future equivalence. For topological patterns in fully-discrete one-dimensional systems, future equivalence uniquely specifies a minimal semiautomaton. We demonstrate how the latter and its semigroup algebra generalizes translation symmetry to partial and hidden symmetries. This generalization is not as straightforward as previously considered. Here, though, we clarify the underlying challenges. A stochastic form of future equivalence, known as predictive equivalence, captures distinct statistical patterns supported on topological patterns. Finally, we show how local versions of future equivalence can be used to capture patterns in spacetime. As common when moving to higher dimensions, there is not a unique local approach, and we detail two local representations that capture different aspects of spacetime patterns. A previously-developed local spacetime variant of future equivalence captures patterns as generalized symmetries in higher dimensions, but we show this representation is not a faithful generator of its spacetime patterns. This motivates us to introduce a local representation that is a faithful generator, but we demonstrate that it no longer captures generalized spacetime symmetries. Taken altogether, building on future equivalence, the theory defines and quantifies patterns present in a wide range of classical field theories.

Only a subset of degrees of freedom are typically accessible or measurable in real-world systems. As a consequence, the proper setting for empirical modeling is that of partially-observed systems. Notably, data-driven models consistently outperform physics-based models for systems with few observable degrees of freedom; e.g., hydrological systems. Here, we provide an operator-theoretic explanation for this empirical success. To predict a partially-observed system's future behavior with physics-based models, the missing degrees of freedom must be explicitly accounted for using data assimilation and model parametrization. Data-driven models, in contrast, employ delay-coordinate embeddings and their evolution under the Koopman operator to implicitly model the effects of the missing degrees of freedom. We describe in detail the statistical physics of partial observations underlying data-driven models using novel Maximum Entropy and Maximum Caliber measures. The resulting nonequilibrium Wiener projections applied to the Mori-Zwanzig formalism reveal how data-driven models may converge to the true dynamics of the observable degrees of freedom. Additionally, this framework shows how data-driven models infer the effects of unobserved degrees of freedom implicitly, in much the same way that physics models infer the effects explicitly. This provides a unified implicit-explicit modeling framework for predicting partially-observed systems, with hybrid physics-informed machine learning methods combining implicit and explicit aspects.

We present a theory of causality in dynamical systems using Koopman operators. Our theory is grounded on a rigorous definition of causal mechanism in dynamical systems given in terms of flow maps. In the Koopman framework, we prove that causal mechanisms manifest as particular flows of observables between function subspaces. While the flow map definition is a clear generalization of the standard definition of causal mechanism given in the structural causal model framework, the flow maps are complicated objects that are not tractable to work with in practice. By contrast, the equivalent Koopman definition lends itself to a straightforward data-driven algorithm that can quantify multivariate causal relations in high-dimensional nonlinear dynamical systems. The coupled Rossler system provides examples and demonstrations throughout our exposition. We also demonstrate the utility of our data-driven Koopman causality measure by identifying causal flow in the Lorenz 96 system. We show that the causal flow identified by our data-driven algorithm agrees with the information flow identified through a perturbation propagation experiment. Our work provides new theoretical insights into causality for nonlinear dynamical systems, as well as a new toolkit for data-driven causal analysis.