There has been rapid progress recently on the application of deep learning to solve fluid mechanics problems. From the initial Artificial Neural Networks (ANN) black box approach of Tompson et al. to the more recent Physics-Informed Neural Network (PINN) of Raissi et al., the accuracy of the predictions has increased tremendously by first estimating the errors of predicted outputs of the ANN from the actual physical equations which the outputs are expected to satisfy and then including these errors with those of the ANN during the minimisation of the mean square error. While PINN results in a better overall prediction than ANN during the course of these projects several issues associated with the operation of the PINN have shown up during the assessment of the impact of network hyper-parameters (such as the number of neurons, no. of hidden layers, size of training data and the random nature of selecting data for training the ANN, the weightage of the error from PINN) on the quality of prediction. When using neural network, there is no theoretical basis to know the size of neural network architecture and the amount of data needed. Hence there is no guarantee that the algorithm will not hit a local minima during the error minimisation stages. The impact of the weightage of the error from the PINN also plays a role in the prediction. More research needs to be done to assess the role this weightage plays in the prediction.

At the same time, there are also other potential alternative networks, one of which is the Extreme Learning Machine (ELM). The basic ELM was proposed by Huang et al. for a single hidden layer feed forward networks (SLFNs) and later it was extended to generalized SLFNs. The essence of ELM is that the weights of the hidden layer of SLFNs need not to be learnt. It is much faster than the traditional gradient based optimization methods alleviating the learning time problem. The idea is then to combine ELM with the physics informed portion of PINN, known as physics informed extreme learning machine (PIELM) which is addressed in the paper by Dwivedi and Srinivasan. Another idea worth exploring is the distributed version of PIELM called DPIELM, also developed by Dwivedi et al. This is motivated by ideas from the finite volume methods in which the whole computational domain is partitioned into multiple cells and governing equations are solved at each cell. The solutions of these individual cells are stitched together with additional convective and diffusive fluxes conditions at the cell interfaces.

Hence, in this paper, we aim to investigate the prediction performance of various deep learning methods on different internal and external 2D aerodynamic flow problems. The deep learning methods include:

1\) Physics-Informed Neural Network (PINN),

2\) physics informed extreme learning machine (PIELM) and

3\) distributed version of PIELM called DPIELM.

We will experiment with using different number of neurons, number of hidden layers and percentage of training data to assess the effects of these three factors on the prediction performance. We may also employ the use of proper orthogonal decomposition (POD) to reduce the dimensionality of the problem. The numerical flow solver OpenFOAM and our in-house immersed boundary method (IBM) numerical solver will be used to generate training and test data for the simulations. Work is currently in progress on the 2D nozzle flow prediction using TensorFlow. Results from our past interns working on 1D nozzle flow with PINN have shown good flow predictions with less than 5% error.


Investigation of Applying Physics Informed Neural Networks (Pinn) and Variants on 2d Aerodynamics Problems

Division / Sponsors: 
Heat Transfer Division (HTD), Fluids Engineering Division (FED)

Heat Transfer Summer Conference
Paper Proceedings: https://asmedigitalcollection.asme.org/HT/HT2020/volume/83709

18th International Conference on Nanochannels, Microchannels, and Minichannels
Paper Proceedings: https://asmedigitalcollection.asme.org/ICNMM/ICNMM2020/volume/83693

Fluids Engineering Division Summer Meeting
Paper Proceedings: https://asmedigitalcollection.asme.org/FEDSM/FEDSM2020/volume/83716

HT-ICNMM2020 - Virtual Conference, SHTC - Virtual Conference, FEDSM2020 - Virtual Conference

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