Equation Discovery Using Fast Function Extraction: a Deterministic Symbolic Regression Approach

Vaddireddy, Harsha; San, Omer

doi:10.3390/fluids4020111

Title: Equation Discovery Using Fast Function Extraction: a Deterministic Symbolic Regression Approach

Abstract

Advances in machine learning (ML) coupled with increased computational power have enabled identification of patterns in data extracted from complex systems. ML algorithms are actively being sought in recovering physical models or mathematical equations from data. This is a highly valuable technique where models cannot be built using physical reasoning alone. In this paper, we investigate the application of fast function extraction (FFX), a fast, scalable, deterministic symbolic regression algorithm to recover partial differential equations (PDEs). FFX identifies active bases among a huge set of candidate basis functions and their corresponding coefficients from recorded snapshot data. This approach uses a sparsity-promoting technique from compressive sensing and sparse optimization called pathwise regularized learning to perform feature selection and parameter estimation. Furthermore, it recovers several models of varying complexity (number of basis terms). FFX finally filters out many identified models using non-dominated sorting and forms a Pareto front consisting of optimal models with respect to minimizing complexity and test accuracy. Numerical experiments are carried out to recover several ubiquitous PDEs such as wave and heat equations among linear PDEs and Burgers, Korteweg–de Vries (KdV), and Kawahara equations among higher-order nonlinear PDEs. Additional simulations are conducted on the same PDEs under noisy conditionsmore »« less

Authors:

Vaddireddy, Harsha ^[1];

^[1]

Oklahoma State Univ., Stillwater, OK (United States)

Publication Date:: 2019-06-15

Research Org.:: Oklahoma State Univ., Stillwater, OK (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1593572

Grant/Contract Number:: SC0019290

Resource Type:: Accepted Manuscript

Journal Name:: Fluids

Additional Journal Information:: Journal Volume: 4; Journal Issue: 2; Journal ID: ISSN 2311-5521

Publisher:: MDPI

Country of Publication:: United States

Language:: English

Subject:: 42 ENGINEERING; deterministic symbolic regression; fast function extraction; compressive sensing; pathwise regularized learning; non-dominated sorting

Citation Formats


                    Vaddireddy, Harsha, and San, Omer. Equation Discovery Using Fast Function Extraction: a Deterministic Symbolic Regression Approach.  United States: N. p., 2019. 
Web.  doi:10.3390/fluids4020111.

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                    Vaddireddy, Harsha, & San, Omer. Equation Discovery Using Fast Function Extraction: a Deterministic Symbolic Regression Approach.  United States.  https://doi.org/10.3390/fluids4020111

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                    Vaddireddy, Harsha, and San, Omer. Sat .  
"Equation Discovery Using Fast Function Extraction: a Deterministic Symbolic Regression Approach".  United States.  https://doi.org/10.3390/fluids4020111.  https://www.osti.gov/servlets/purl/1593572.

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@article{osti_1593572,

  title        = {Equation Discovery Using Fast Function Extraction: a Deterministic Symbolic Regression Approach},

  author       = {Vaddireddy, Harsha and San, Omer},

  abstractNote = {Advances in machine learning (ML) coupled with increased computational power have enabled identification of patterns in data extracted from complex systems. ML algorithms are actively being sought in recovering physical models or mathematical equations from data. This is a highly valuable technique where models cannot be built using physical reasoning alone. In this paper, we investigate the application of fast function extraction (FFX), a fast, scalable, deterministic symbolic regression algorithm to recover partial differential equations (PDEs). FFX identifies active bases among a huge set of candidate basis functions and their corresponding coefficients from recorded snapshot data. This approach uses a sparsity-promoting technique from compressive sensing and sparse optimization called pathwise regularized learning to perform feature selection and parameter estimation. Furthermore, it recovers several models of varying complexity (number of basis terms). FFX finally filters out many identified models using non-dominated sorting and forms a Pareto front consisting of optimal models with respect to minimizing complexity and test accuracy. Numerical experiments are carried out to recover several ubiquitous PDEs such as wave and heat equations among linear PDEs and Burgers, Korteweg–de Vries (KdV), and Kawahara equations among higher-order nonlinear PDEs. Additional simulations are conducted on the same PDEs under noisy conditions to test the robustness of the proposed approach.},

  doi          = {10.3390/fluids4020111},

  journal      = {Fluids},

  number       = 2,

  volume       = 4,

  place        = {United States},

  year         = {2019},

  month        = {6}

}

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Works referenced in this record:

Stable signal recovery from incomplete and inaccurate measurements
journal, January 2006

Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence
Communications on Pure and Applied Mathematics, Vol. 59, Issue 8, p. 1207-1223
DOI: 10.1002/cpa.20124

Application of an evolutionary algorithm to LES modelling of turbulent transport in premixed flames
journal, December 2018

Schoepplein, Matthias; Weatheritt, Jack; Sandberg, Richard
Journal of Computational Physics, Vol. 374
DOI: 10.1016/j.jcp.2018.08.016

Deep learning
journal, May 2015

LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey
Nature, Vol. 521, Issue 7553
DOI: 10.1038/nature14539

Distilling Free-Form Natural Laws from Experimental Data
journal, April 2009

Schmidt, Michael; Lipson, Hod
Science, Vol. 324, Issue 5923
DOI: 10.1126/science.1165893

Force identification of dynamic systems using genetic programming
journal, January 2005

Yang, Y. W.; Wang, C.; Soh, C. K.
International Journal for Numerical Methods in Engineering, Vol. 63, Issue 9
DOI: 10.1002/nme.1323

Closed-Loop Turbulence Control: Progress and Challenges
journal, August 2015

Brunton, Steven L.; Noack, Bernd R.
Applied Mechanics Reviews, Vol. 67, Issue 5
DOI: 10.1115/1.4031175

Model selection for dynamical systems via sparse regression and information criteria
journal, August 2017

Mangan, N. M.; Kutz, J. N.; Brunton, S. L.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 473, Issue 2204
DOI: 10.1098/rspa.2017.0009

Oscillatory Solitary Waves in Dispersive Media
journal, July 1972

Kawahara, Takuji
Journal of the Physical Society of Japan, Vol. 33, Issue 1
DOI: 10.1143/JPSJ.33.260

The application of robust Bayesian analysis to hypothesis testing and Occam's Razor
journal, February 1992

Berger, James O.; Jefferys, William H.
Journal of the Italian Statistical Society, Vol. 1, Issue 1
DOI: 10.1007/BF02589047

Extracting Sparse High-Dimensional Dynamics from Limited Data
journal, January 2018

Schaeffer, Hayden; Tran, Giang; Ward, Rachel
SIAM Journal on Applied Mathematics, Vol. 78, Issue 6
DOI: 10.1137/18M116798X

The perceptron: A probabilistic model for information storage and organization in the brain.
journal, January 1958

Rosenblatt, F.
Psychological Review, Vol. 65, Issue 6
DOI: 10.1037/h0042519

An Introduction To Compressive Sampling
journal, March 2008

Candes, E. J.; Wakin, M. B.
IEEE Signal Processing Magazine, Vol. 25, Issue 2
DOI: 10.1109/MSP.2007.914731

Discovering governing equations from data by sparse identification of nonlinear dynamical systems
journal, March 2016

Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan
Proceedings of the National Academy of Sciences, Vol. 113, Issue 15
DOI: 10.1073/pnas.1517384113

Compressed sensing
journal, April 2006

Donoho, D. L.
IEEE Transactions on Information Theory, Vol. 52, Issue 4
DOI: 10.1109/TIT.2006.871582

A novel evolutionary algorithm applied to algebraic modifications of the RANS stress–strain relationship
journal, November 2016

Weatheritt, Jack; Sandberg, Richard
Journal of Computational Physics, Vol. 325
DOI: 10.1016/j.jcp.2016.08.015

Deep Reinforcement Learning framework for Autonomous Driving
journal, January 2017

Sallab, AhmadEL; Abdou, Mohammed; Perot, Etienne
Electronic Imaging, Vol. 2017, Issue 19
DOI: 10.2352/ISSN.2470-1173.2017.19.AVM-023

Regularization and variable selection via the elastic net
journal, April 2005

Zou, Hui; Hastie, Trevor
Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 67, Issue 2
DOI: 10.1111/j.1467-9868.2005.00503.x

The big challenges of big data
journal, June 2013

Marx, Vivien
Nature, Vol. 498, Issue 7453
DOI: 10.1038/498255a

Data-driven discovery of partial differential equations
journal, April 2017

Rudy, Samuel H.; Brunton, Steven L.; Proctor, Joshua L.
Science Advances, Vol. 3, Issue 4
DOI: 10.1126/sciadv.1602614

Prediction of dynamical systems by symbolic regression
journal, July 2016

Quade, Markus; Abel, Markus; Shafi, Kamran
Physical Review E, Vol. 94, Issue 1
DOI: 10.1103/PhysRevE.94.012214

New exact travelling wave solutions for the Kawahara and modified Kawahara equations
journal, January 2004

Sirendaoreji,
Chaos, Solitons & Fractals, Vol. 19, Issue 1
DOI: 10.1016/S0960-0779(03)00102-4

Parse-matrix evolution for symbolic regression
journal, September 2012

Luo, Changtong; Zhang, Shao-Liang
Engineering Applications of Artificial Intelligence, Vol. 25, Issue 6
DOI: 10.1016/j.engappai.2012.05.015

Extreme Learning Machines as Encoders for Sparse Reconstruction
journal, November 2018

Al Mamun, S.; Lu, Chen; Jayaraman, Balaji
Fluids, Vol. 3, Issue 4
DOI: 10.3390/fluids3040088

Exact Recovery of Chaotic Systems from Highly Corrupted Data
journal, January 2017

Tran, Giang; Ward, Rachel
Multiscale Modeling & Simulation, Vol. 15, Issue 3
DOI: 10.1137/16M1086637

Regularization Paths for Generalized Linear Models via Coordinate Descent
journal, January 2010

Friedman, Jerome; Hastie, Trevor; Tibshirani, Robert
Journal of Statistical Software, Vol. 33, Issue 1
DOI: 10.18637/jss.v033.i01

Sparse dynamics for partial differential equations
journal, March 2013

Schaeffer, H.; Caflisch, R.; Hauck, C. D.
Proceedings of the National Academy of Sciences, Vol. 110, Issue 17
DOI: 10.1073/pnas.1302752110

Nonlinear Interaction between Short and Long Capillary-Gravity Waves
journal, November 1975

Kawahara, Takuji; Sugimoto, Nobumasa; Kakutani, Tsunehiko
Journal of the Physical Society of Japan, Vol. 39, Issue 5
DOI: 10.1143/JPSJ.39.1379

Multi-objective genetic programming for nonlinear system identification
journal, January 1998

Rodríguez-Vázquez, K.; Fleming, P. J.
Electronics Letters, Vol. 34, Issue 9
DOI: 10.1049/el:19980632

Machine learning: Trends, perspectives, and prospects
journal, July 2015

Jordan, M. I.; Mitchell, T. M.
Science, Vol. 349, Issue 6245
DOI: 10.1126/science.aaa8415

Occam's Razor
journal, April 1987

Blumer, Anselm; Ehrenfeucht, Andrzej; Haussler, David
Information Processing Letters, Vol. 24, Issue 6
DOI: 10.1016/0020-0190(87)90114-1

Generalized Linear Models
journal, January 1972

Nelder, J. A.; Wedderburn, R. W. M.
Journal of the Royal Statistical Society. Series A (General), Vol. 135, Issue 3
DOI: 10.2307/2344614

Closed-loop separation control using machine learning
journal, April 2015

Gautier, N.; Aider, J. -L.; Duriez, T.
Journal of Fluid Mechanics, Vol. 770
DOI: 10.1017/jfm.2015.95

Learning partial differential equations via data discovery and sparse optimization
journal, January 2017

Schaeffer, Hayden
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 473, Issue 2197
DOI: 10.1098/rspa.2016.0446

A fast and elitist multiobjective genetic algorithm: NSGA-II
journal, April 2002

Deb, K.; Pratap, A.; Agarwal, S.
IEEE Transactions on Evolutionary Computation, Vol. 6, Issue 2
DOI: 10.1109/4235.996017

Regression Shrinkage and Selection Via the Lasso
journal, January 1996

Tibshirani, Robert
Journal of the Royal Statistical Society: Series B (Methodological), Vol. 58, Issue 1
DOI: 10.1111/j.2517-6161.1996.tb02080.x

Existence of perturbed solitary wave solutions to a model equation for water waves
journal, September 1988

Hunter, John K.; Scheurle, Jurgen
Physica D: Nonlinear Phenomena, Vol. 32, Issue 2
DOI: 10.1016/0167-2789(88)90054-1

The Roles of Plastic Surgeons in Advancing Artificial Intelligence in Plastic Surgery
journal, April 2021

Luan, Fang; Gao, Xuejun; Zhao, Shanbaga
Aesthetic Plastic Surgery
DOI: 10.1007/s00266-021-02302-7

Deep Representational Similarity Learning for Analyzing Neural Signatures in Task-based fMRI Dataset
journal, October 2020

Yousefnezhad, Muhammad; Sawalha, Jeffrey; Selvitella, Alessandro
Neuroinformatics, Vol. 19, Issue 3
DOI: 10.1007/s12021-020-09494-4

Machine learning identifies a strong association between warming and reduced primary productivity in an oligotrophic ocean gyre
journal, February 2020

D’Alelio, Domenico; Rampone, Salvatore; Cusano, Luigi Maria
Scientific Reports, Vol. 10, Issue 1
DOI: 10.1038/s41598-020-59989-y

Occam's Razor
journal, May 1940

Rogers, Clement F.
Theology, Vol. 40, Issue 239
DOI: 10.1177/0040571x4004023904

Circulating Proteomic Signature of Early Death in Heart Failure Patients with Reduced Ejection Fraction
journal, January 2019

Cuvelliez, Marie; Vandewalle, Vincent; Brunin, Maxime
SSRN Electronic Journal
DOI: 10.2139/ssrn.3396013

Generalized Linear Models
journal, December 2000

McCulloch, Charles E.
Journal of the American Statistical Association, Vol. 95, Issue 452
DOI: 10.1080/01621459.2000.10474340

Deep Learning
text, January 2018

Ganatra, Sanket
Zenodo
DOI: 10.5281/zenodo.1189359

Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
text, January 2015

Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan
arXiv
DOI: 10.48550/arxiv.1509.03580

Prediction of Dynamical Systems by Symbolic Regression
text, January 2016

Quade, Markus; Abel, Markus; Shafi, Kamran
arXiv
DOI: 10.48550/arxiv.1602.04648

Data-driven discovery of partial differential equations
preprint, January 2016

Rudy, Samuel H.; Brunton, Steven L.; Proctor, Joshua L.
arXiv
DOI: 10.48550/arxiv.1609.06401

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Title: Equation Discovery Using Fast Function Extraction: a Deterministic Symbolic Regression Approach

Abstract

Citation Formats

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Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensor observation data journal, January 2020

Stable signal recovery from incomplete and inaccurate measurements
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Application of an evolutionary algorithm to LES modelling of turbulent transport in premixed flames
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Distilling Free-Form Natural Laws from Experimental Data
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Force identification of dynamic systems using genetic programming
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Oscillatory Solitary Waves in Dispersive Media
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The application of robust Bayesian analysis to hypothesis testing and Occam's Razor
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journal, January 1958

An Introduction To Compressive Sampling
journal, March 2008

Discovering governing equations from data by sparse identification of nonlinear dynamical systems
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A novel evolutionary algorithm applied to algebraic modifications of the RANS stress–strain relationship
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Deep Reinforcement Learning framework for Autonomous Driving
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