{
  "id": "2009.11213",
  "version": "v1",
  "published": "2020-09-23T15:24:53.000Z",
  "updated": "2020-09-23T15:24:53.000Z",
  "title": "Physics-integrated machine learning: embedding a neural network in the Navier-Stokes equations. Part II",
  "authors": [
    "Arsen S. Iskhakov",
    "Nam T. Dinh"
  ],
  "categories": [
    "physics.flu-dyn",
    "physics.comp-ph"
  ],
  "abstract": "The work is a continuation of a paper by Iskhakov A.S. and Dinh N.T. \"Physics-integrated machine learning: embedding a neural network in the Navier-Stokes equations\". Part I // arXiv:2008.10509 (2020) [1]. The proposed in [1] physics-integrated (or PDE-integrated (partial differential equation)) machine learning (ML) framework is furtherly investigated. The Navier-Stokes equations are solved using the Tensorflow ML library for Python programming language via the Chorin's projection method. The Tensorflow solution is integrated with a deep feedforward neural network (DFNN). Such integration allows one to train a DFNN embedded in the Navier-Stokes equations without having the target (labeled training) data for the direct outputs from the DFNN; instead, the DFNN is trained on the field variables (quantities of interest), which are solutions for the Navier-Stokes equations (velocity and pressure fields). To demonstrate performance of the framework, two additional case studies are formulated: 2D turbulent lid-driven cavities with predicted by a DFNN (a) turbulent viscosity and (b) derivatives of the Reynolds stresses. Despite its complexity and computational cost, the proposed physics-integrated ML shows a potential to develop a \"PDE-integrated\" closure relations for turbulent models and offers principal advantages, namely: (i) the target outputs (labeled training data) for a DFNN might be unknown and can be recovered using the knowledge base (PDEs); (ii) it is not necessary to extract and preprocess information (training targets) from big data, instead it can be extracted by PDEs; (iii) there is no need to employ a physics- or scale-separation assumptions to build a closure model for PDEs. The advantage (i) is demonstrated in the Part I paper [1], while the advantage (ii) is the subject of the current paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-23T15:24:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "navier-stokes equations",
      "physics-integrated machine learning",
      "2d turbulent lid-driven cavities",
      "tensorflow ml library",
      "offers principal advantages"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}