ktongue/docker_container/sciann-applications/SciANN-DataGenerator - Storage Bucket

A guide for generating collocation grid for PINN solvers

Training PINNs is not a trivial optimization problem because it consists of multiple objectives (PDE, IC, BC). The optimizer performs best when each loss term is sampled correctly and equally during each mini-batch training. This implies that our fed collocation grid should contain enough sample points for each loss term, e.g. on the boundary or in the domain, so that each mini-batch contains information for all loss terms.

The code sciann-datagenerator.py includes the following classes:

• DataGeneratorX: Generate 1D collocation grid.
• DataGeneratorXY: Generate 2D collocation grid for a rectangular domain.
• DataGeneratorXT: Generate 1D time-dependent collocation grid.
• DataGeneratorXYT: Generate 2D time-dependent collocation grid for a rectangular domain.

For more complex geometries/problems, you should follow a similar approach to generate a proper collocation grid for training PINNs. As an example, for a complex 3D geometry, if you have a mesh that contains N points inside the domain, generate a second finer mesh that contains roughly the same number of nodes on the boundary. Then, concatenate them all in one large data set and assign proper ids for domain and boundary nodes.

How to use this code:

Let us assume you want to solve a 1D time-dependent problem with $x\in[0, 1]$ and $t\in[0, 10]$. Additonally, let us assume that your SciModel is defined as follows:

m = sn.SciModel([x, t], [PDE, IC, BC_left, BC_right])

You can then use DataGeneratorXT to generate a proper collocation grid for training your PINN model, as:

dg = DataGeneratorXT(
    X=[0., 1.],
    T=[0., 10.],
    targets=["domain", "ic", "bc-left", "bc-right"],
    num_sample=1_000,
    logT=False
)

This code generates roughly 1,000 collocation points for training your PINN model. Note that the order and type of targets defined in SciModel should be the same for the DataGeneratorXT. You can test your generated collocation grid as:

dg.plot_data()

For some problems, the training may converge faster if you generate time samples in a logarithmic distribution. In that case, we have:

dg = DataGeneratorXT(
    X=[0., 1.],
    T=[0., 10.],
    targets=["domain", "ic", "bc-left", "bc-right"],
    num_sample=1_000,
    logT=True
)
dg.plot_data()

Useful?

If you found this document useful, please support us by staring our repository and citing our SciANN paper:

@article{haghighat2021sciann,
  title={SciANN: A Keras/TensorFlow wrapper for scientific computations and physics-informed deep learning using artificial neural networks},
  author={Haghighat, Ehsan and Juanes, Ruben},
  journal={Computer Methods in Applied Mechanics and Engineering},
  volume={373},
  pages={113552},
  year={2021},
  publisher={Elsevier}
}