This code implements an axample of multivariate regression problems in turbulence modeling —— a universal, inherently interpretable machine leanring framework (IMLF) with the kernel of a physics-inspired residual neural network (denoted as PiResNet), which is both data- and knowledge-driven. Encouraging predictions are obtained, and in particular, physically realizable constraints are really statisfied for the first time using the Progressive Iteration Realizability (PIR) algorithm developed by C. Jiang. When building on this code and/or using aspects of IMLF and PiResNet, the following citations are suggested:

• Chao Jiang, Ricardo Vinuesa, Ruilin Chen, Junyi Mi, Shujin Laima, and Hui Li. "An Interpretable Framework of Data-driven Turbulence Modeling using Deep Neural Networks." Physics of Fluids (submitted).

Also see an earlier version:

• Chao Jiang, Junyi Mi, Shujin Laima, and Hui Li. "Data-driven Turbulence Modeling Using A Physics-informed Deep Residual Neural Network." 14th World Congress in Computational Mechanics (WCCM) & ECCOMAS Congress 2020, 19-July, 2020: Paris, France.

Code of PiResNet obtained at https://github.com/Jackachao0618/PiResNet

• Chao Jiang, Junyi Mi, Shujin Laima, and Hui Li. "A Nonlocal Algebraic Tensor Model for Reynolds-stress Closures." 71st Annual Meeting of the APS Division of Fluid Mechanics, Bulletin of the American Physical Society, 18-November, 2018: Atlanta, USA.
• Chao Jiang, Junyi Mi, Shujin Laima, and Hui Li. "A Novel Algebraic Stress Model with Machine-Learning-Assisted Parameterization." Energies 13 (2020): 258.

This code is built on the open-source library TensorFlow, which requires the following python packages:

python 3.6.10
pytorch 1.6.0
matplotlib 3.2.2
numpy 1.19.1
scipy 1.5.2

Please free to contact me in case of questions.