Physics-integrated machine learning: embedding a neural network in the Navier-Stokes equations. Part I

TL;DR

This paper introduces a physics-integrated machine learning framework that embeds neural networks within Navier-Stokes equations, enabling training without labeled data and offering advantages over classical ML approaches in fluid dynamics modeling.

Contribution

The paper presents a novel integration of neural networks with PDE-based Navier-Stokes solutions, allowing training on field data without explicit target outputs, and demonstrating its effectiveness in a fluid flow case study.

Findings

Physics-integrated ML achieves similar accuracy to classical ML.

It can recover target outputs without labeled training data.

Offers advantages in data extraction and model closure without scale assumptions.

Abstract

In this paper the physics- (or PDE-) integrated machine learning (ML) framework is investigated. The Navier-Stokes (NS) equations are solved using Tensorflow library for Python via Chorin's projection method. The methodology for the solution is provided, which is compared with a classical solution implemented in Fortran. This solution is integrated with a neural network (NN). Such integration allows one to train a NN embedded in the NS equations without having the target (labeled training) data for the direct outputs from the NN; instead, the NN is trained on the field data (quantities of interest), which are the solutions for the NS equations. To demonstrate the performance of the framework, a case study is formulated: the 2D lid-driven cavity with non-constant velocity-dependent dynamic viscosity is considered. A NN is trained to predict the dynamic viscosity from the velocity fields. The performance of the physics-integrated ML is compared with classical ML framework, when a NN is directly trained on the available data (fields of the dynamic viscosity). Both frameworks showed similar accuracy; however, despite its complexity and computational cost, the physics-integrated ML offers principal advantages, namely: (i) the target outputs (labeled training data) for a NN might be unknown and can be recovered using 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. The advantage (i) is demonstrated in this paper, while the advantages (ii) and (iii) are the subjects for future work. Such integration of PDEs with ML opens a door for a tighter data-knowledge connection, which may potentially influence the further development of the physics-based modelling with ML for data-driven thermal fluid models.