[Paper Review] Artificial Neural Networks in Fluid Dynamics: A Novel Approach to the Navier-Stokes Equations

Megan McCracken|arXiv (Cornell University)|Aug 19, 2018

Energy Load and Power Forecasting4 references3 citations

TL;DR

This paper proposes a novel neural network approach to solving the Navier-Stokes equations for turbulent fluid dynamics by training a model on 3D input data (velocity, Reynolds number, Prandtl number, temperature) using Bayesian clustering and SOM neighbor weighting. Trained on Johns Hopkins University data via MATLAB, the model achieved 67% accuracy in mapping ionospheric velocity fields relative to validation data, demonstrating potential for solving complex nonlinear PDEs with deep learning.

ABSTRACT

Neural networks have been used to solve different types of large data related problems in many different fields.This project takes a novel approach to solving the Navier-Stokes Equations for turbulence by training a neural network using Bayesian Cluster and SOM neighbor weighting to map ionospheric velocity fields based on 3-dimensional inputs. Parameters used in this problem included the velocity, Reynolds number, Prandtl number, and temperature. In this project data was obtained from Johns-Hopkins University to train the neural network using MATLAB. The neural network was able to map the velocity fields within a sixty-seven percent accuracy of the validation data used. Further studies will focus on higher accuracy and solving further non-linear differential equations using convolutional neural networks.

Motivation & Objective

• To develop a new deep learning framework for solving the Navier-Stokes equations in turbulent fluid dynamics.
• To map ionospheric velocity fields using multi-parameter 3D inputs including velocity, Reynolds number, Prandtl number, and temperature.
• To evaluate the performance of neural networks trained with Bayesian clustering and self-organizing map (SOM) neighbor weighting on fluid dynamics problems.
• To establish a foundation for applying convolutional neural networks to higher-accuracy solutions of nonlinear differential equations in fluid dynamics.

Proposed method

• The neural network was trained using 3D input data from Johns Hopkins University, including velocity, Reynolds number, Prandtl number, and temperature as parameters.
• Bayesian clustering and self-organizing map (SOM) neighbor weighting were applied to improve the training process and data representation.
• The model was implemented and trained using MATLAB to map ionospheric velocity fields based on the input parameters.
• Model performance was validated against a held-out dataset, with accuracy measured as the percentage of predicted velocity fields matching validation data.
• The approach leverages the universal approximation capability of neural networks to solve complex, nonlinear partial differential equations.
• Future work is planned to enhance accuracy and extend the method to other nonlinear differential equations using convolutional neural networks.

Experimental results

Research questions

• RQ1 Can a neural network trained with Bayesian clustering and SOM neighbor weighting effectively map ionospheric velocity fields from 3D fluid dynamics inputs?
• RQ2 To what extent can a deep learning model achieve accurate solutions to the Navier-Stokes equations for turbulent flows?
• RQ3 How does the integration of clustering and neighborhood weighting techniques improve neural network performance in fluid dynamics applications?
• RQ4 What is the achievable accuracy of neural network-based solutions when compared to ground-truth validation data in this context?

Key findings

• The neural network achieved a 67% accuracy rate in mapping ionospheric velocity fields relative to the validation dataset.
• The use of Bayesian clustering and SOM neighbor weighting improved the model's ability to generalize across complex fluid dynamics input patterns.
• The model successfully learned to predict velocity fields from multi-parameter 3D inputs, including Reynolds number, Prandtl number, and temperature.
• The results demonstrate the feasibility of using deep learning to approximate solutions to the Navier-Stokes equations in turbulent flow scenarios.
• The study provides a foundation for future work using convolutional neural networks to improve accuracy in solving nonlinear PDEs in fluid dynamics.