Using gene expression programming to discover macroscopic governing equations hidden in the data of molecular simulations

Abstract

The unprecedented amount of data and the advancement of machine learning methods are driving the rapid development of data-driven modeling in the community of fluid mechanics. In this work, a data-driven strategy is developed by the combination of the direct simulation Monte Carlo (DSMC) method and the gene expression programming (GEP) method. DSMC is a molecular simulation method without any assumed macroscopic governing equations a priori and is employed to generate data of flow fields, while the enhanced GEP method is leveraged to discover governing equations. We first validate our idea using two benchmarks, such as the Burgers equation and Sine-Gordon equation. Then we apply the strategy to discover governing equations hidden in the complex fluid dynamics. Our results demonstrate that in the continuum regime, the discovered equations are consistent with the traditional ones with linear constitutive relations, while in the non-continuum regime such as shock wave, the discovered equation comprises of high-order constitutive relations, which are similar to those in the Burnett equation but with modified coefficients. Compared to the Navier-Stokes-Fourier equations and the Burnett equation, the prediction of the viscous stress and heat flux in the shock wave via the presented data-driven model has the best match to the DSMC data. It is promising to extend the proposed data-driven strategy to more complex problems and discover hidden governing equations which may be unknown so far.

Benchmarking of Machine Learning Ocean Subgrid Parameterizations in an Idealized Model

“…To address these challenges, in Section 6 we introduce a novel algorithm for equation discovery based on genetic programming, an alternative form of symbolic regression that is stochastic but can more efficiently explore higher‐order operations (Koza, 1994; Schmidt & Lipson, 2009; Xing et al., 2022). We adapt this algorithm to search over spatial differential operators, and combine it with linear regression and residual‐fitting to more efficiently and accurately fit constants.…”

Dimensional homogeneity constrained gene expression programming for discovering governing equations

“…2020; Xing et al. 2022), which learn the forms of functions and their corresponding coefficients concurrently. The preselected elements for GEP include only mathematical operators, physical constants and physical variables.…”

Benchmarking of machine learning ocean subgrid parameterizations in an idealized model

“…To address these challenges, in Section 6 we introduce a novel algorithm for equation discovery based on genetic programming, an alternative form of symbolic regression that is stochastic but can more efficiently explore higher-order operations (Koza, 1994;Schmidt & Lipson, 2009;Xing et al, 2022). We adapt this algorithm to search over spatial differential operators, and combine it with linear regression and residual-fitting to more efficiently and accurately fit constants.…”