Learning the flow field of Stokes flow

This example demonstrates how to train the MeshGraphNet model to learn the flow field of Stokes flow and further improve the accuracy of the model predictions by physics-informed inference. This example also demonstrates how to use physics utilities from PhysicsNeMo-Sym to introduce physics-based constraints.

Problem overview

The partial differential equation is defined as

$$\begin{aligned} -\nu \Delta \mathbf{u} +\nabla p=0, \\ \nabla \cdot \mathbf{u} = 0, \end{aligned}$$

where $\mathbf{u} = (u, v)$ defines the velocity and $p$ the pressure, and $\nu$ is the kinematic viscosity. The underlying geometry is a pipe without a polygon. On the inlet $\Gamma_3=0 \times[0,0.4]$, a parabolic inflow profile is prescribed,

$$\begin{aligned} \mathbf{u}(0, y)= \mathbf{u}_{\mathrm{in}} = \left(\frac{4 U y(0.4-y)}{0.4^2}, 0\right) \end{aligned}$$

with a maximum velocity $U=0.3$. On the outlet $\Gamma_4=2.2 \times[0,0.4]$, we define the outflow condition

$$\begin{aligned} \nu \partial_\mathbf{n} \mathbf{u}-p \mathbf{n}=0, \end{aligned}$$

where $\mathbf{n}$ denotes the outer normal vector.

Our goal is to train a MeshGraphNet to learn the map from the polygon geometry to the velocity and pressure field. However, sometimes data-driven models may not be able to yield reasonable predictive accuracy due to network capacity or limited dataset. We can fine-tune our results using PINNs when the PDE is available. The fine-tuning during inference is much faster than training the PINN model from the scratch as the model has a better initialization from the data-driven training.

For the fine-tuning step, we formulate two losses. First loss is to match the predictions of the original MeshGraphNet model. Second loss includes the physics losses, i.e. the PDE residuals and the boundary conditions. Having the data loss helps the PINN model converge faster than training from scratch.

Dataset

Our dataset provides numerical simulations of Stokes flow in a pipe domain obstructed by a random polygon. It contains 1000 random samples and all the simulations were performed using Fenics. For each sample, the numerical solution contains the mesh and the flow information about velocity, pressure, and markers identifying different boundaries within the domain.

To download the full dataset, please run the bash script in raw_dataset

bash download_dataset.sh

Model overview and architecture

The inputs of our MeshGraphNet model is:

mesh

Output of the MeshGraphNet model are:

velocity field pressure
pressure field

The input to the model is in form of a .vtp file and is then converted to bi-directional graphs in the dataloader. The final results are also written in the form of .vtp files in the inference code. A hidden dimensionality of 256 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 1e-4 is used, which decays exponentially to 1e-6 over 500 epochs.

Prerequisites

Install the requirements using:

pip install -r requirements.txt
pip install nvidia-physicsnemo.sym --no-build-isolation

Getting Started

The dataset for this example is not publicly available. To get access, please reach out to the NVIDIA PhysicsNeMo team.

Once you've obtained the dataset, follow these steps to preprocess it:

Unzip the Dataset: If the dataset is compressed, make sure to extract its contents.
Run the Preprocessing Script: Execute the provided script to process the dataset. This will distribute the data randomly across three directories: training, validation, and test.

python preprocess.py

To train the model, run

python train.py

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

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 config.yaml file or via the command line. 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

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

python inference.py

To further fine-tune the model using physics-informed learning, run

python pi_fine_tuning.py

Note

The fine-tuning step involves training of a PINN model to first refine the predictions of the MeshGraphNet model followed by an inference of the PINN model.

If you are running this fine-tuning outside of the PhysicsNeMo container, install PhysicsNeMo Sym using the instructions from here

This will save the predictions for the test dataset in .vtp format in the results directory. Use ParaView to open and explore the results.

References

Learning Mesh-Based Simulation with Graph Networks