sayin/Data_Driven_Symbolic_Regression

Interpretable machine learning (symbolic regression) using Genetic programming/Gene expression programming and Sparse regression used to identify physical process, numerical schemes, and LES subgrid scale (eddy viscosity) turbulence models.

**In part1_eqation_discovery/stridge_eq_disc/ contains sparse Optimzation algorithm

Interpretable machine learning (symbolic regression) using Genetic programming/Gene expression programming and Sparse Optimization used to identify physical process, numerical schemes, and LES subgrid scale (eddy viscosity) turbulence models.

Abstarct:

In this study we put forth a modular approach for distilling hidden flow physics from discrete and sparse observations. To address functional expressiblity, a key limitation of the black-box machine learning methods, we have exploited the use of symbolic regression as a principle for identifying relations and operators that are related to the underlying processes. This approach combines evolutionary computation with feature engineering to provide a tool for discovering hidden parameterizations embedded in the trajectory of fluid flows in the Eulerian frame of reference. Our approach in this study mainly involves gene expression programming (GEP) and sequential threshold ridge regression (STRidge) algorithms. We demonstrate our results in three different applications: (i) equation discovery, (ii) truncation error analysis, and (iii) hidden physics discovery, for which we include both predicting unknown source terms from a set of sparse observations and discovering subgrid scale closure models. We illustrate that both GEP and STRidge algorithms are able to distill the Smagorinsky model from an array of tailored features in solving the Kraichnan turbulence problem. Our results demonstrate the huge potential of these techniques in complex physics problems, and reveal the importance of feature selection and feature engineering in model discovery approaches.

The repository consits of three parts:

1. Equation (PDE) Discovery using GEP and STRidge,
2. Trucation Error Analysis using GEP and STRidge,
3. Hidden Physics Discovery using GEP and LES sub grid scale Modelling using GEP and STRidge.

Genetic/Gene Expression Programming:

Chromosome	GEP Flow chart

Sparse Optimization:

Hidden source term identified from sparse data collected from 2D vortex merger.

2D vortex meger	GEP

Smagorinsky eddy viscosity turbulence model identified by GEP and Sparse regression along with ad-hoc (calibrated) parameter for 2D decaying trubulence.

GEP	Sparse regression	2D decaying trubulence with identified SGS model

Published Journals:

1. Vaddireddy, H., Rasheed, A., Staples, A. E., & San, O. (2020). Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensor observation data. Physics of Fluids, 32(1), 015113. (https://aip.scitation.org/doi/full/10.1063/1.5136351)

2. Vaddireddy, H., & San, O. (2019). Equation discovery using fast function extraction: A deterministic symbolic regression approach. Fluids, 4(2), 111. (https://www.mdpi.com/2311-5521/4/2/111)

ACKNOWLEDGMENTS:

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under Award No. DE-SC0019290. O.S. gratefully acknowledges their support.