MeshGraphNet for transient vortex shedding

This example is a re-implementation of the DeepMind’s vortex shedding example https://github.com/deepmind/deepmind-research/tree/master/meshgraphnets in PyTorch. It demonstrates how to train a Graph Neural Network (GNN) for evaluation of the transient vortex shedding on parameterized geometries.

Mesh-based simulations play a central role in modeling complex physical systems across various scientific and engineering disciplines. They offer robust numerical integration methods and allow for adaptable resolution to strike a balance between accuracy and efficiency. Machine learning surrogate models have emerged as powerful tools to reduce the cost of tasks like design optimization, design space exploration, and what-if analysis, which involve repetitive high-dimensional scientific simulations.

However, some existing machine learning surrogate models, such as CNN-type models, are constrained by structured grids, making them less suitable for complex geometries or shells. The homogeneous fidelity of CNNs is a significant limitation for many complex physical systems that require an adaptive mesh representation to resolve multi-scale physics.

Graph Neural Networks (GNNs) present a viable approach for surrogate modeling in science and engineering. They are data-driven and capable of handling complex physics. Being mesh-based, GNNs can handle geometry irregularities and multi-scale physics, making them well-suited for a wide range of applications.

We rely on DeepMind’s vortex shedding dataset for this example. The dataset includes 1000 training, 100 validation, and 100 test samples that are simulated using COMSOL with irregular triangle 2D meshes, each for 600 time steps with a time step size of 0.01s. These samples vary in the size and the position of the cylinder. Each sample has a unique mesh due to geometry variations across samples, and the meshes have 1885 nodes on average. Note that the model can handle different meshes with different number of nodes and edges as the input.

The model is free-running and auto-regressive. It takes the initial condition as the input and predicts the solution at the first time step. It then takes the prediction at the first time step to predict the solution at the next time step. The model continues to use the prediction at time step \(t\) to predict the solution at time step \(t+1\), until the rollout is complete. Note that the model is also able to predict beyond the simulation time span and extrapolate in time. However, the accuracy of the prediction might degrade over time and if possible, extrapolation should be avoided unless the underlying data patterns remain stationary and consistent.

The model uses the input mesh to construct a bi-directional DGL graph for each sample. The node features include (6 in total):

  • Velocity components at time step \(t\), i.e., \(u_t\), \(v_t\)

  • One-hot encoded node type (interior node, no-slip node, inlet node, outlet node)

The edge features for each sample are time-independent and include (3 in total):

  • Relative \(x\) and \(y\) distance between the two end nodes of an edge

  • L2 norm of the relative distance vector

The output of the model is the velocity components at time step t+1, i.e., \(u_{t+1}\), \(v_{t+1}\), as well as the pressure \(p_{t+1}\).

vortex_shedding.gif

Fig. 3 Comparison between the MeshGraphNet prediction and the ground truth for the horizontal velocity for different test samples.

A hidden dimensionality of 128 is used in the encoder, processor, and decoder. The encoder and decoder consist of two hidden layers, and the processor includes 15 message passing layers. Batch size per GPU is set to 1. Summation aggregation is used in the processor for message aggregation. A learning rate of 0.0001 is used, decaying exponentially with a rate of 0.9999991. Training is performed on 8 NVIDIA A100 GPUs, leveraging data parallelism for 25 epochs.

This example requires the tensorflow library to load the data in the .tfrecord format. Install with

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pip install tensorflow

To download the data from DeepMind’s repo, run

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cd raw_dataset sh download_dataset.sh cylinder_flow

To train the model, run

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python train.py

Data parallelism is also supported with multi-GPU runs. To launch a multi-GPU training, run

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mpirun -np <num_GPUs> python train.py

If running in a docker container, you may need to include the --allow-run-as-root in the multi-GPU run command.

Progress and loss logs can be monitored using Weights & Biases. To activate that, set wandb_mode to online in the constants.py. This requires to have an active Weights & Biases account. You also need to provide your API key. There are multiple ways for providing the API key but you can simply export it as an environment variable

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export WANDB_API_KEY=<your_api_key>

The URL to the dashboard will be displayed in the terminal after the run is launched. Alternatively, the logging utility in train.py can be switched to MLFlow.

Once the model is trained, run

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python inference.py

This will save the predictions for the test dataset in .gif format in the animations directory.

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