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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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