Gradient Information and Regularization for Gene Expression Programming to Develop Data-Driven Physics Closure Models

Flow Turbulence and Combustion, vol. 114, pp. 145–175

Abstract

Abstract Learning accurate numerical constants when developing algebraic models is a known challenge for evolutionary algorithms, such as Gene Expression Programming (GEP). This paper introduces the concept of adaptive symbols to the GEP framework by Weatheritt and Sandberg (J Comput Phys 325:22–37, 2016a) to develop advanced physics closure models. Adaptive symbols utilize gradient information to learn locally optimal numerical constants during model training, for which we investigate two types of nonlinear optimization algorithms. The second contribution of this work is implementing two regularization techniques to incentivize the development of implementable and interpretable closure models. We apply $$L_2$$ L 2 regularization to ensure small magnitude numerical constants and devise a novel complexity metric that supports the development of low complexity models via custom symbol complexities and multi-objective optimization. This extended framework is employed to four use cases, namely rediscovering Sutherland’s viscosity law, developing laminar flame speed combustion models and training two types of fluid dynamics turbulence models. The model prediction accuracy and the convergence speed of training are improved significantly across all of the more and less complex use cases, respectively. The two regularization methods are essential for developing implementable closure models and we demonstrate that the developed turbulence models substantially improve simulations over state-of-the-art models.

Authors 8

1. Fabian Waschkowski corresponding

   The University of Melbourne

   Affiliation as printed

   Department of Mechanical Engineering, University of Melbourne, Melbourne, Australia

   Department of Mechanical Engineering, University of Melbourne, Melbourne, VIC, 3010, Australia

2. Haochen Li

   Peking University

   Affiliation as printed

   Center for Applied Physics and Technology, Peking University, Beijing, China

   Center for Applied Physics and Technology, Peking University, Beijing, 100871, China

3. Abhishek J. Deshmukh Aachen Institute for Combustion Technology

   RWTH Aachen University

   Affiliation as printed

   Institute for Combustion Technology, RWTH Aachen University, Aachen, Germany

   Institute for Combustion Technology, RWTH Aachen University, 52062, Aachen, Germany

4. Temistocle Grenga Aachen Institute for Combustion Technology

   RWTH Aachen University

   Affiliation as printed

   Institute for Combustion Technology, RWTH Aachen University, Aachen, Germany

   Institute for Combustion Technology, RWTH Aachen University, 52062, Aachen, Germany

5. Yaomin Zhao

   Peking University

   Affiliation as printed

   Center for Applied Physics and Technology, Peking University, Beijing, China

   Center for Applied Physics and Technology, Peking University, Beijing, 100871, China

6. Heinz G. Pitsch Aachen Institute for Combustion Technology

   RWTH Aachen University

   Affiliation as printed

   Institute for Combustion Technology, RWTH Aachen University, Aachen, Germany

   Institute for Combustion Technology, RWTH Aachen University, 52062, Aachen, Germany

7. JOSEPH C. KLEWICKI

   The University of Melbourne

   Affiliation as printed

   Department of Mechanical Engineering, University of Melbourne, Melbourne, Australia

   Department of Mechanical Engineering, University of Melbourne, Melbourne, VIC, 3010, Australia

8. Richard D. Sandberg

   The University of Melbourne

   Affiliation as printed

   Department of Mechanical Engineering, University of Melbourne, Melbourne, Australia

   Department of Mechanical Engineering, University of Melbourne, Melbourne, VIC, 3010, Australia