The DOI matches its Crossref record; 31 of the 31 references it cites were found in a registry or an index.
Learning partial differential equations via data discovery and sparse optimization
Open accessAbstract
We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.
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Cites
Boyd, S., & Vandenberghe, L. (2004) · doi:10.1017/cbo9780511804441 · Cited by 56,601
Regression shrinkage and selection via the lasso
Tibshirani, R. (1996) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1996.tb02080.x · Cited by 51,482
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
Beck, A., & Teboulle, M. (2009) · SIAM Journal on Imaging Sciences · doi:10.1137/080716542 · Cited by 11,952
De-noising by soft-thresholding
Donoho, D. L. (1995) · IEEE Transactions on Information Theory · doi:10.1109/18.382009 · Cited by 9,619
Stable signal recovery from incomplete and inaccurate measurements
Candès, E. J., Romberg, J. K., & Tao, T. (2006) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.20124 · Cited by 7,036
Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries
Elad, M., & Aharon, M. (2006) · IEEE Transactions on Image Processing · doi:10.1109/tip.2006.881969 · Cited by 5,321
Discovering governing equations from data by sparse identification of nonlinear dynamical systems
Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1517384113 · Cited by 4,796
A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging
Chambolle, A., & Pock, T. (2011) · Journal of Mathematical Imaging and Vision · doi:10.1007/s10851-010-0251-1 · Cited by 4,336
The Split Bregman Method for L1-Regularized Problems
Goldstein, T., & Osher, S. (2009) · SIAM Journal on Imaging Sciences · doi:10.1137/080725891 · Cited by 4,282
Compressive Sensing [Lecture Notes]
Baraniuk, R. (2007) · IEEE Signal Processing Magazine · doi:10.1109/msp.2007.4286571 · Cited by 4,177
Cited by
Machine Learning for Fluid Mechanics
Brunton, S. L., Noack, B. R., & Koumoutsakos, P. (2020) · Annual Review of Fluid Mechanics · doi:10.1146/annurev-fluid-010719-060214 · Cited by 3,063
Data-driven discovery of coordinates and governing equations
Champion, K., Lusch, B., Kutz, J. N., & Brunton, S. L. (2019) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1906995116 · Cited by 909
Universal Differential Equations for Scientific Machine Learning
Rackauckas, C., Ma, Y., Martensen, J., Warner, C., Zubov, K., Supekar, R., Skinner, D., Ramadhan, A., & Edelman, A. (2020) · doi:10.21203/rs.3.rs-55125/v1 · Cited by 643
Chaos as an intermittently forced linear system
Brunton, S. L., Brunton, B. W., Proctor, J. L., Kaiser, E., & Kutz, J. N. (2017) · Nature Communications · doi:10.1038/s41467-017-00030-8 · Cited by 608
PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network
Long, Z., Lu, Y., & Dong, B. (2019) · Journal of Computational Physics · doi:10.1016/j.jcp.2019.108925 · Cited by 549
Metadata
- Authors
- Schaeffer, H.
- Venue
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Publisher
- The Royal Society
- Published
- 2017-01
- Volume
- 473
- Issue
- 2197
- Pages
- 20160446
- ISSN
- 1364-5021 (ISSN-L)
- 1471-2946
- OpenAlex
- W2573798107
- Cited by
- 432
- References
- 31
- FWCI
- 14.05
- Type
- journal article
- Language
- en
- Open access
- green
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Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
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