# Temporal attention model in Mesh-Reduced space for transient vortex shedding

This example is an implementation of the paper “Predicting Physics in Mesh-reduced Space with Temporal Attention” in PyTorch. It demonstrates how to train a Graph Neural Network (GNN) as encoder to compress the high-dimensional physical state into latent space and apply a multi-head attention model for temporal predictions for the transient vortex shedding on parameterized geometries.

We use vortex shedding dataset for this example. The dataset includes 51 training, and 50 test samples that are simulated using OpenFOAM with irregular triangle 2D meshes, each for 401 time steps with a time step size of 0.5s. These samples vary in the Reynolds number. Each sample share the same mesh with 1699 nodes.

The model is auto-regressive. It first encodes the graph state into a latent vector via a Graph Nueral Network. Then a multi-head temporal model takes the initial condition tokens and pysical paramerters as the input and predicts the solution for the following sequence in the latent space just like a language model.

The model uses the input mesh to construct a bi-directional DGL graph for each sample. The node features include (3 in total):

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

Pressure at time step \(t\), \(p_t\)

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 for the following steps, i.e., \([\ldots, (u_{t}\), \(v_{t}), (u_{t+1}\), \(v_{t+1}), \ldots]\), as well as the pressure \([\ldots,p_{t},p_{t+1}\,\ldots]\).

For the PbGMR-GMUS, a hidden dimensionality of 128 is used in the encoder, and decoder. The encoder and decoder consist of two hidden layers. Batch size per GPU is set to 1 for the encoding-decoding process. Mean 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. Traing epochs is set as 300.

For the multi-head attention temporal model, the dimension for each token is \(3 \times 256 = 768\). The hidden dimension usded in the temporal model is \(4 \times 768 = 4072\). The number of head is 8. Batch size per GPU is set to 10 for the sequence model training. Traing epochs is set as 200000.

To download the data , run

```
```cd dataset
sh TODO: need a shell to download the raw dataset from a cloud storage.

This example requires the `torch-scatter`

and `torch-cluster`

library for the graph nodes agrregation. Install with

```
```conda install pytorch-scatter -c pyg
conda install pytorch-cluster -c pyg

To train the encoding-decoding model, run

```
```python train.py

To test the reconstruction error, run

```
```python test.py

To train the sequence model, run

```
```python train_sequence.py

Once the model is trained, run

```
```python test_sequence.py