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Platt JA, Penny SG, Smith TA, Chen TC, Abarbanel HDI. Constraining chaos: Enforcing dynamical invariants in the training of reservoir computers. Chaos 2023; 33:103107. [PMID: 37788385 DOI: 10.1063/5.0156999] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2023] [Accepted: 08/14/2023] [Indexed: 10/05/2023]
Abstract
Drawing on ergodic theory, we introduce a novel training method for machine learning based forecasting methods for chaotic dynamical systems. The training enforces dynamical invariants-such as the Lyapunov exponent spectrum and the fractal dimension-in the systems of interest, enabling longer and more stable forecasts when operating with limited data. The technique is demonstrated in detail using reservoir computing, a specific kind of recurrent neural network. Results are given for the Lorenz 1996 chaotic dynamical system and a spectral quasi-geostrophic model of the atmosphere, both typical test cases for numerical weather prediction.
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Affiliation(s)
- Jason A Platt
- Department of Physics, University of California San Diego, San Diego, California 92093, USA
| | - Stephen G Penny
- Sofar Ocean, 28 Pier Annex, San Francisco, California 94105, USA
- Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, Colorado 80309, USA
| | - Timothy A Smith
- Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, Colorado 80309, USA
- Physical Sciences Laboratory, National Oceanic and Atmospheric Administration, Boulder, Colorado 80305, USA
| | - Tse-Chun Chen
- Pacific Northwest National Laboratory, 902 Battelle Blvd, Richland, Washington 99354, USA
| | - Henry D I Abarbanel
- Department of Physics, University of California San Diego, San Diego, California 92093, USA
- Marine Physical Laboratory, Scripps Institution of Oceanography, 9500 Gilman Drive, La Jolla, California 92093, USA
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Platt JA, Penny SG, Smith TA, Chen TC, Abarbanel HDI. A systematic exploration of reservoir computing for forecasting complex spatiotemporal dynamics. Neural Netw 2022; 153:530-552. [PMID: 35839598 DOI: 10.1016/j.neunet.2022.06.025] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2022] [Revised: 06/14/2022] [Accepted: 06/20/2022] [Indexed: 11/16/2022]
Abstract
A reservoir computer (RC) is a type of recurrent neural network architecture with demonstrated success in the prediction of spatiotemporally chaotic dynamical systems. A further advantage of RC is that it reproduces intrinsic dynamical quantities essential for its incorporation into numerical forecasting routines such as the ensemble Kalman filter-used in numerical weather prediction to compensate for sparse and noisy data. We explore here the architecture and design choices for a "best in class" RC for a number of characteristic dynamical systems. Our analysis points to the importance of large scale parameter optimization. We also note in particular the importance of including input bias in the RC design, which has a significant impact on the forecast skill of the trained RC model. In our tests, the use of a nonlinear readout operator does not affect the forecast time or the stability of the forecast. The effects of the reservoir dimension, spinup time, amount of training data, normalization, noise, and the RC time step are also investigated. Finally, we detail how our investigation leads to optimal design choices for a parallel RC scheme applied to the 40 dimensional spatiotemporally chaotic Lorenz 1996 dynamics.
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Affiliation(s)
- Jason A Platt
- Department of Physics, University of California San Diego, United States of America.
| | - Stephen G Penny
- Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, CO 80309, United States of America; Physical Sciences Laboratory, National Oceanic and Atmospheric Administration, Boulder, CO 80305, United States of America
| | - Timothy A Smith
- Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, CO 80309, United States of America; Physical Sciences Laboratory, National Oceanic and Atmospheric Administration, Boulder, CO 80305, United States of America
| | - Tse-Chun Chen
- Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, CO 80309, United States of America; Physical Sciences Laboratory, National Oceanic and Atmospheric Administration, Boulder, CO 80305, United States of America
| | - Henry D I Abarbanel
- Department of Physics, University of California San Diego, United States of America; Marine Physical Laboratory, Scripps Institution of Oceanography, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
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Platt JA, Wong A, Clark R, Penny SG, Abarbanel HDI. Robust forecasting using predictive generalized synchronization in reservoir computing. Chaos 2021; 31:123118. [PMID: 34972341 DOI: 10.1063/5.0066013] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2021] [Accepted: 11/15/2021] [Indexed: 06/14/2023]
Abstract
Reservoir computers (RCs) are a class of recurrent neural networks (RNNs) that can be used for forecasting the future of observed time series data. As with all RNNs, selecting the hyperparameters in the network to yield excellent forecasting presents a challenge when training on new inputs. We analyze a method based on predictive generalized synchronization (PGS) that gives direction in designing and evaluating the architecture and hyperparameters of an RC. To determine the occurrences of PGS, we rely on the auxiliary method to provide a computationally efficient pre-training test that guides hyperparameter selection. We provide a metric for evaluating the RC using the reproduction of the input system's Lyapunov exponents that demonstrates robustness in prediction.
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Affiliation(s)
- Jason A Platt
- Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
| | - Adrian Wong
- Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
| | - Randall Clark
- Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
| | - Stephen G Penny
- Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, Colorado 80305-3328, USA
| | - Henry D I Abarbanel
- Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
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Penny SG, Akella S, Balmaseda MA, Browne P, Carton JA, Chevallier M, Counillon F, Domingues C, Frolov S, Heimbach P, Hogan P, Hoteit I, Iovino D, Laloyaux P, Martin MJ, Masina S, Moore AM, de Rosnay P, Schepers D, Sloyan BM, Storto A, Subramanian A, Nam S, Vitart F, Yang C, Fujii Y, Zuo H, O’Kane T, Sandery P, Moore T, Chapman CC. Observational Needs for Improving Ocean and Coupled Reanalysis, S2S Prediction, and Decadal Prediction. Front Mar Sci 2019; 6:391. [PMID: 31534949 PMCID: PMC6750049 DOI: 10.3389/fmars.2019.00391] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Developments in observing system technologies and ocean data assimilation (DA) are symbiotic. New observation types lead to new DA methods and new DA methods, such as coupled DA, can change the value of existing observations or indicate where new observations can have greater utility for monitoring and prediction. Practitioners of DA are encouraged to make better use of observations that are already available, for example, taking advantage of strongly coupled DA so that ocean observations can be used to improve atmospheric analyses and vice versa. Ocean reanalyses are useful for the analysis of climate as well as the initialization of operational long-range prediction models. There are many remaining challenges for ocean reanalyses due to biases and abrupt changes in the ocean-observing system throughout its history, the presence of biases and drifts in models, and the simplifying assumptions made in DA solution methods. From a governance point of view, more support is needed to bring the ocean-observing and DA communities together. For prediction applications, there is wide agreement that protocols are needed for rapid communication of ocean-observing data on numerical weather prediction (NWP) timescales. There is potential for new observation types to enhance the observing system by supporting prediction on multiple timescales, ranging from the typical timescale of NWP, covering hours to weeks, out to multiple decades. Better communication between DA and observation communities is encouraged in order to allow operational prediction centers the ability to provide guidance for the design of a sustained and adaptive observing network.
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Affiliation(s)
- Stephen G. Penny
- Department of Atmospheric and Oceanic Science, University of Maryland, College Park, MD, United States
| | - Santha Akella
- National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, MD, United States
| | | | - Philip Browne
- European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
| | - James A. Carton
- Department of Atmospheric and Oceanic Science, University of Maryland, College Park, MD, United States
| | | | | | - Catia Domingues
- Antarctic Climate and Ecosystems Cooperative Research Centre, Hobart, TAS, Australia
| | - Sergey Frolov
- Naval Research Laboratory, Monterey, CA, United States
| | | | - Patrick Hogan
- Naval Research Laboratory, Stennis Space Center, MS, United States
| | - Ibrahim Hoteit
- King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
| | | | - Patrick Laloyaux
- European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
| | | | - Simona Masina
- Euro-Mediterranean Center on Climate Change, Lecce, Italy
| | - Andrew M. Moore
- University of California, Santa Cruz, Santa Cruz, CA, United States
| | - Patricia de Rosnay
- European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
| | - Dinand Schepers
- European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
| | - Bernadette M. Sloyan
- Commonwealth Scientific and Industrial Research Organisation, Canberra, ACT, Australia
| | - Andrea Storto
- NATO Centre for Maritime Research and Experimentation, La Spezia, Italy
| | - Aneesh Subramanian
- Department of Atmospheric and Oceanic Science, University of Colorado, Boulder, Boulder, CO, United States
| | | | - Frederic Vitart
- European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
| | - Chunxue Yang
- Istituto di Scienze Marine, Consiglio Nazionale delle Ricerche, Rome, Italy
| | - Yosuke Fujii
- JMA Meteorological Research Institute, Tsukuba, Japan
| | - Hao Zuo
- European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
| | - Terry O’Kane
- Commonwealth Scientific and Industrial Research Organisation, Canberra, ACT, Australia
| | - Paul Sandery
- Commonwealth Scientific and Industrial Research Organisation, Canberra, ACT, Australia
| | - Thomas Moore
- Commonwealth Scientific and Industrial Research Organisation, Canberra, ACT, Australia
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Abstract
The state-of-the-art data assimilation methods used today in operational weather prediction centers around the world can be classified as generalized one-way coupled impulsive synchronization. This classification permits the investigation of hybrid data assimilation methods, which combine dynamic error estimates of the system state with long time-averaged (climatological) error estimates, from a synchronization perspective. Illustrative results show how dynamically informed formulations of the coupling matrix (via an Ensemble Kalman Filter, EnKF) can lead to synchronization when observing networks are sparse and how hybrid methods can lead to synchronization when those dynamic formulations are inadequate (due to small ensemble sizes). A large-scale application with a global ocean general circulation model is also presented. Results indicate that the hybrid methods also have useful applications in generalized synchronization, in particular, for correcting systematic model errors.
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Affiliation(s)
- Stephen G Penny
- Department of Atmospheric and Oceanic Science, University of Maryland, College Park, Maryland 20742, USA; National Centers for Environmental Prediction (NCEP), College Park, Maryland 20740, USA; and RIKEN Advanced Institute for Computational Science, Kobe, Hyogo 650-0047, Japan
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BozorgMagham AE, Motesharrei S, Penny SG, Kalnay E. Causality Analysis: Identifying the Leading Element in a Coupled Dynamical System. PLoS One 2015; 10:e0131226. [PMID: 26125157 PMCID: PMC4488350 DOI: 10.1371/journal.pone.0131226] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2014] [Accepted: 05/30/2015] [Indexed: 11/18/2022] Open
Abstract
Physical systems with time-varying internal couplings are abundant in nature. While the full governing equations of these systems are typically unknown due to insufficient understanding of their internal mechanisms, there is often interest in determining the leading element. Here, the leading element is defined as the sub-system with the largest coupling coefficient averaged over a selected time span. Previously, the Convergent Cross Mapping (CCM) method has been employed to determine causality and dominant component in weakly coupled systems with constant coupling coefficients. In this study, CCM is applied to a pair of coupled Lorenz systems with time-varying coupling coefficients, exhibiting switching between dominant sub-systems in different periods. Four sets of numerical experiments are carried out. The first three cases consist of different coupling coefficient schemes: I) Periodic-constant, II) Normal, and III) Mixed Normal/Non-normal. In case IV, numerical experiment of cases II and III are repeated with imposed temporal uncertainties as well as additive normal noise. Our results show that, through detecting directional interactions, CCM identifies the leading sub-system in all cases except when the average coupling coefficients are approximately equal, i.e., when the dominant sub-system is not well defined.
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Affiliation(s)
- Amir E. BozorgMagham
- Department of Atmospheric and Oceanic Science (AOSC), University of Maryland, College Park, MD, 20742, USA
- * E-mail:
| | - Safa Motesharrei
- National Socio–Environmental Synthesis Center (SESYNC), Annapolis, Maryland 21401, USA
- Department of Physics, University of Maryland, College Park, Maryland, 20742, USA
- Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
| | - Stephen G. Penny
- Department of Atmospheric and Oceanic Science (AOSC), University of Maryland, College Park, MD, 20742, USA
- National Centers for Environmental Prediction (NCEP), College Park, MD 20740, USA
| | - Eugenia Kalnay
- Department of Atmospheric and Oceanic Science (AOSC), University of Maryland, College Park, MD, 20742, USA
- Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
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