This code implements an axample of multivariate regression problems in tensor modeling as Data- and Knowledge-driven Turbulence Modeling —— a universal, inherently interpretable ML-based framework (IMLF) with the kernel of a physics-inspired Parameterized Tensorial Neural Network (denoted as PTNN), which was first proposed by Chao Jiang (2020). Encouraging predictions are obtained, and in particular, physically realizable constraints are really statisfied for the first time using the Progressive Iteration-type Realizability (PIR) algorithm developed by C. Jiang. When building on this code and/or using aspects of IMLF and PTNN (including tensorial neural network, TNN), the following citations are suggested.

• Chao Jiang, Ricardo Vinuesa, Ruilin Chen, Junyi Mi, Shujin Laima, and Hui Li. "Data-driven Reliable Turbulence Modeling through A Physics-inspired Parameterized Tensorial Neural Network." Physics of Fluids (submitted).
• Chao Jiang, Junyi Mi, Ricardo Vinuesa, and Hui Li. "Machine Learning A General-purpose Turbulence Model with Built-in Domain-Knowledge." Computer Methods in Applied Mechanics and Engineering (under preparation).
• Chao Jiang and Hui Li. "Reliable and Explainable Machine-Learning Methods for Accelerating Closure Development." Journal of Fluid Mechanics (in progress).
• Chao Jiang and Hui Li. "On Realizability and Metrics of Data-driven Turbulence Models." Physical Review Fluids (in progress).
• Chao Jiang and Hui Li. "Searching a Unified Stress-strain Paradigm across the Flow- and Geometry-topology using Parameterized Tensorial Neural Network."
• Chao Jiang and Hui Li. "On Risks of Machine Learining in Turbulence Modeling: A Physical Perspective."

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 PTNN obtained at https://github.com/Jackachao0618/PTNN

• 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:

(coming soon ... ... )

Please free to contact me in case of questions: Chao Jiang [[email protected]]

More Information in the ResearchGate profile: https://www.researchgate.net/profile/Chao_Jiang44