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1. Machine Learning for Robust Identification of Complex Nonlinear Dynamical Systems: Applications to Earth Systems Modeling

   Yadav, Nishant ; Ravela, Sai ; Ganguly, Auroop R. ( August 2020 , ArXivorg)

   null (Ed.)

   Systems exhibiting nonlinear dynamics, including but not limited to chaos, are ubiquitous across Earth Sciences such as Meteorology, Hydrology, Climate and Ecology, as well as Biology such as neural and cardiac processes. However, System Identification remains a challenge. In climate and earth systems models, while governing equations follow from first principles and understanding of key processes has steadily improved, the largest uncertainties are often caused by parameterizations such as cloud physics, which in turn have witnessed limited improvements over the last several decades. Climate scientists have pointed to Machine Learning enhanced parameter estimation as a possible solution, with proof-of-concept methodological adaptations being examined on idealized systems. While climate science has been highlighted as a "Big Data" challenge owing to the volume and complexity of archived model-simulations and observations from remote and in-situ sensors, the parameter estimation process is often relatively a "small data" problem. A crucial question for data scientists in this context is the relevance of state-of-the-art data-driven approaches including those based on deep neural networks or kernel-based processes. Here we consider a chaotic system - two-level Lorenz-96 - used as a benchmark model in the climate science literature, adopt a methodology based on Gaussian Processes for parameter estimation and compare the gains in predictive understanding with a suite of Deep Learning and strawman Linear Regression methods. Our results show that adaptations of kernel-based Gaussian Processes can outperform other approaches under small data constraints along with uncertainty quantification; and needs to be considered as a viable approach in climate science and earth system modeling. 

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2. Stochastic‐Deep Learning Parameterization of Ocean Momentum Forcing

   https://doi.org/10.1029/2021MS002534  

   Guillaumin, Arthur P. ; Zanna, Laure ( September 2021 , Journal of Advances in Modeling Earth Systems)

   Coupled climate simulations that span several hundred years cannot be run at a high‐enough spatial resolution to resolve mesoscale ocean dynamics. Recently, several studies have considered Deep Learning to parameterize subgrid forcing within macroscale ocean equations using data from ocean‐only simulations with idealized geometry. We present a stochastic Deep Learning parameterization that is trained on data generated by CM2.6, a high‐resolution state‐of‐the‐art coupled climate model. We train a Convolutional Neural Network for the subgrid momentum forcing using macroscale surface velocities from a few selected subdomains with different dynamical regimes. At each location of the coarse grid, rather than predicting a single number for the subgrid momentum forcing, we predict both the mean and standard deviation of a Gaussian probability distribution. This approach requires training our neural network to minimize a negative log‐likelihood loss function rather than the Mean Square Error, which has been the standard in applications of Deep Learning to the problem of parameterizations. Each estimate of the conditional mean subgrid forcing is thus associated with an uncertainty estimate–the standard deviation—which will form the basis for a stochastic subgrid parameterization. Offline tests show that our parameterization generalizes well to the global oceans and a climate with increasedlevels without further training. We then implement our learned stochastic parameterization in an eddy‐permitting idealized shallow water model. The implementation is stable and improves some statistics of the flow. Our work demonstrates the potential of combining Deep Learning tools with a probabilistic approach in parameterizing unresolved ocean dynamics.

    

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3. Domain-driven models yield better predictions at lower cost than reservoir computers in Lorenz systems

   https://doi.org/10.1098/rsta.2020.0246  

   Pyle, Ryan ; Jovanovic, Nikola ; Subramanian, Devika ; Palem, Krishna V. ; Patel, Ankit B. ( April 2021 , Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   null (Ed.)

   Recent advances in computing algorithms and hardware have rekindled interest in developing high-accuracy, low-cost surrogate models for simulating physical systems. The idea is to replace expensive numerical integration of complex coupled partial differential equations at fine time scales performed on supercomputers, with machine-learned surrogates that efficiently and accurately forecast future system states using data sampled from the underlying system. One particularly popular technique being explored within the weather and climate modelling community is the echo state network (ESN), an attractive alternative to other well-known deep learning architectures. Using the classical Lorenz 63 system, and the three tier multi-scale Lorenz 96 system (Thornes T, Duben P, Palmer T. 2017 Q. J. R. Meteorol. Soc. 143 , 897–908. ( doi:10.1002/qj.2974 )) as benchmarks, we realize that previously studied state-of-the-art ESNs operate in two distinct regimes, corresponding to low and high spectral radius (LSR/HSR) for the sparse, randomly generated, reservoir recurrence matrix. Using knowledge of the mathematical structure of the Lorenz systems along with systematic ablation and hyperparameter sensitivity analyses, we show that state-of-the-art LSR-ESNs reduce to a polynomial regression model which we call Domain-Driven Regularized Regression (D2R2). Interestingly, D2R2 is a generalization of the well-known SINDy algorithm (Brunton SL, Proctor JL, Kutz JN. 2016 Proc. Natl Acad. Sci. USA 113 , 3932–3937. ( doi:10.1073/pnas.1517384113 )). We also show experimentally that LSR-ESNs (Chattopadhyay A, Hassanzadeh P, Subramanian D. 2019 ( http://arxiv.org/abs/1906.08829 )) outperform HSR ESNs (Pathak J, Hunt B, Girvan M, Lu Z, Ott E. 2018 Phys. Rev. Lett. 120 , 024102. ( doi:10.1103/PhysRevLett.120.024102 )) while D2R2 dominates both approaches. A significant goal in constructing surrogates is to cope with barriers to scaling in weather prediction and simulation of dynamical systems that are imposed by time and energy consumption in supercomputers. Inexact computing has emerged as a novel approach to helping with scaling. In this paper, we evaluate the performance of three models (LSR-ESN, HSR-ESN and D2R2) by varying the precision or word size of the computation as our inexactness-controlling parameter. For precisions of 64, 32 and 16 bits, we show that, surprisingly, the least expensive D2R2 method yields the most robust results and the greatest savings compared to ESNs. Specifically, D2R2 achieves 68 × in computational savings, with an additional 2 × if precision reductions are also employed, outperforming ESN variants by a large margin. This article is part of the theme issue ‘Machine learning for weather and climate modelling’. 

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4. A framework for machine learning of model error in dynamical systems

   https://doi.org/10.1090/cams/10  

   Levine, Matthew ; Stuart, Andrew ( October 2022 , Communications of the American Mathematical Society)

   The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems from noisily and partially observed data. We compare pure data-driven learning with hybrid models which incorporate imperfect domain knowledge, referring to the discrepancy between an assumed truth model and the imperfect mechanistic model as model error. Our formulation is agnostic to the chosen machine learning model, is presented in both continuous- and discrete-time settings, and is compatible both with model errors that exhibit substantial memory and errors that are memoryless. First, we study memoryless linear (w.r.t. parametric-dependence) model error from a learning theory perspective, defining excess risk and generalization error. For ergodic continuous-time systems, we prove that both excess risk and generalization error are bounded above by terms that diminish with the square-root of T T , the time-interval over which training data is specified. Secondly, we study scenarios that benefit from modeling with memory, proving universal approximation theorems for two classes of continuous-time recurrent neural networks (RNNs): both can learn memory-dependent model error, assuming that it is governed by a finite-dimensional hidden variable and that, together, the observed and hidden variables form a continuous-time Markovian system. In addition, we connect one class of RNNs to reservoir computing, thereby relating learning of memory-dependent error to recent work on supervised learning between Banach spaces using random features. Numerical results are presented (Lorenz ’63, Lorenz ’96 Multiscale systems) to compare purely data-driven and hybrid approaches, finding hybrid methods less datahungry and more parametrically efficient. We also find that, while a continuous-time framing allows for robustness to irregular sampling and desirable domain- interpretability, a discrete-time framing can provide similar or better predictive performance, especially when data are undersampled and the vector field defining the true dynamics cannot be identified. Finally, we demonstrate numerically how data assimilation can be leveraged to learn hidden dynamics from noisy, partially-observed data, and illustrate challenges in representing memory by this approach, and in the training of such models. 

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5. Use of Neural Networks for Stable, Accurate and Physically Consistent Parameterization of Subgrid Atmospheric Processes With Good Performance at Reduced Precision

   https://doi.org/10.1029/2020GL091363  

   Yuval, Janni ; O'Gorman, Paul A. ; Hill, Chris N. ( March 2021 , Geophysical Research Letters)

   A promising approach to improve climate‐model simulations is to replace traditional subgrid parameterizations based on simplified physical models by machine learning algorithms that are data‐driven. However, neural networks (NNs) often lead to instabilities and climate drift when coupled to an atmospheric model. Here, we learn an NN parameterization from a high‐resolution atmospheric simulation in an idealized domain by accurately calculating subgrid terms through coarse graining. The NN parameterization has a structure that ensures physical constraints are respected, such as by predicting subgrid fluxes instead of tendencies. The NN parameterization leads to stable simulations that replicate the climate of the high‐resolution simulation with similar accuracy to a successful random‐forest parameterization while needing far less memory. We find that the simulations are stable for different horizontal resolutions and a variety of NN architectures, and that an NN with substantially reduced numerical precision could decrease computational costs without affecting the quality of simulations.

    

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