Abstract
Differential equations based on physical principals are used to represent complex dynamic systems in all fields of science and engineering. When known, these equations have been shown to well represent real-world dynamics. However, since the true dynamics of complex systems are generally unknown, learning the governing equations can improve our understanding of the mechanisms driving the systems. Here, we develop a Bayesian approach to data-driven discovery of nonlinear spatio-temporal dynamic equations. Our approach can accommodate measurement error and missing data, both of which are common in real-world data, and accounts for parameter uncertainty. The proposed framework is illustrated using three simulated systems with varying amounts of measurement uncertainty and missing data and applied to a real-world system to infer the temporal evolution of the vorticity of the streamfunction.
Funding Statement
This research was partially supported by the U.S. National Science Foundation (NSF) grant SES-1853096, the U.S. Geological Survey Midwest Climate Adaptation Science Center (CASC) grant No. G20AC00096, and by the Director, Office of Science, Office of Biological and Environmental Research of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 and by the Regional and Global Model Analysis Program area within the Earth and Environmental Systems Modeling Program.
Acknowledgments
The authors would like to acknowledge Dr. Ralph Milliff for comments on an early draft and for helpful discussions concerning the results from the barotropic vorticity example.
This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California.
Authors declare they have no competing interests.
The data is publically available at https://cds.climate.copernicus.eu/cdsapp#!/dataset/10.24381/cds.bd0915c6?tab=overview and code can be found on GitHub at https://github.com/jsnowynorth/BayesianDiscovery.jl.
Citation
Joshua S. North. Christopher K. Wikle. Erin M. Schliep. "A Bayesian Approach for Spatio-Temporal Data-Driven Dynamic Equation Discovery." Bayesian Anal. Advance Publication 1 - 30, 2023. https://doi.org/10.1214/23-BA1406
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