# Darcy Flow with Fourier Neural Operator¶

## Introduction¶

In this tutorial, you will use Modulus to set up a data-driven model for a 2D Darcy flow using the Fourier Neural Operator (FNO) architecture inside of Modulus. In this tutorial, you will learn the following:

1. How to load grid data and set up data-driven constraints

2. How to create a grid validator node

3. How to use Fourier Neural Operator architecture in Modulus

Note

This tutorial assumes that you are familiar with the basic functionality of Modulus and understand the FNO architecture. Please see the Introductory Example and Fourier Neural Operator sections for additional information.

Warning

The Python package gdown is required for this example if you do not already have the example data downloaded and converted. Install using pip install gdown.

## Problem Description¶

The Darcy PDE is a second order, elliptic PDE with the following form:

(172)$-\nabla \cdot \left(k(\textbf{x})\nabla u(\textbf{x})\right) = f(\textbf{x}), \quad \textbf{x} \in D,$

in which $$u(\textbf{x})$$ is the flow pressure, $$k(\textbf{x})$$ is the permeability field and $$f(\cdot)$$ is the forcing function. The Darcy flow can parameterize a variety of systems including flow through porous media, elastic materials and heat conduction. Here you will define the domain as a 2D unit square $$D=\left\{x,y \in (0,1)\right\}$$ with the boundary condition $$u(\textbf{x})=0, \textbf{x}\in\partial D$$. Recall that FNO requires a structured Euclidean input s.t. $$D = \textbf{x}_{i}$$ where $$i \in \mathbb{N}_{N\times N}$$. Thus both the permeability and flow fields are discretized into a 2D matrix $$\textbf{K}, \textbf{U} \in \mathbb{R}^{N \times N}$$.

This problem develops a surrogate model that learns the mapping between a permeability field and the pressure field, $$\textbf{K} \rightarrow \textbf{U}$$, for a distribution of permeability fields $$\textbf{K} \sim p(\textbf{K})$$. This is a key distinction of this problem from other examples, you are not learning just a single solution but rather a distribution.

## Case Setup¶

This example is a data-driven problem. This means that before starting any coding you need to make sure you have both the training and validation data. The training and validation datasets for this example can be found on the Fourier Neural Operator Github page. Here is an automated script for downloading and converting this dataset. This requires the package gdown which can easily installed through pip install gdown.

Note

The python script for this problem can be found at examples/darcy/darcy_FNO_lazy.py.

### Configuration¶

The configuration for this problem is fairly standard within Modulus. Note that we have two architectures in the config: one is the pointwise decoder for FNO and the other is the FNO model which will eventually ingest the decoder. The most important parameter for FNO models is dimension which tells Modulus to load a 1D, 2D or 3D FNO architecture. nr_fno_layers are the number of Fourier convolution layers in the model. The size of the latent features in FNO are determined based on the decoders input key z, in this case the embedded feature space is 32.

defaults :
- modulus_default
- /arch/conv_fully_connected_cfg@arch.decoder
- /arch/fno_cfg@arch.fno
- scheduler: tf_exponential_lr
- loss: sum
- _self_

arch:
decoder:
input_keys: [z, 32]
output_keys: sol
nr_layers: 1
layer_size: 32

fno:
input_keys: coeff
dimension: 2
nr_fno_layers: 4
fno_modes: 12

scheduler:
decay_rate: 0.95
decay_steps: 1000

training:
rec_results_freq : 1000
max_steps : 10000

batch_size:
grid: 32
validation: 32


Note

Modulus configs can allow users to define keys inside the YAML file. In this instance, input_keys: [z,32] will create a single key of size 32 and input_keys: coeff creates a single input key of size 1.

For this data-driven problem the first step is to get the training data into Modulus. Prior to loading data, set any normalization value that you want to apply to the data. For this dataset, calculate the scale and shift parameters for both the input permeability field and output pressure. Then, set this normalization inside Modulus by providing a shift/scale to each key, Key(name, scale=(shift, scale)).

    # load training/ test data
input_keys = [Key("coeff", scale=(7.48360e00, 4.49996e00))]
output_keys = [Key("sol", scale=(5.74634e-03, 3.88433e-03))]

train_path = to_absolute_path(
"datasets/Darcy_241/piececonst_r241_N1024_smooth1.hdf5"
)
test_path = to_absolute_path(
"datasets/Darcy_241/piececonst_r241_N1024_smooth2.hdf5"
)


    # make datasets
train_dataset = HDF5GridDataset(
train_path, invar_keys=["coeff"], outvar_keys=["sol"], n_examples=1000
)
test_dataset = HDF5GridDataset(
test_path, invar_keys=["coeff"], outvar_keys=["sol"], n_examples=100
)


This data is in HDF5 format which is ideal for lazy loading using the HDF5GridDataset object.

Note

The key difference when setting up eager versus lazy loading is the object passed in the variable dictionaries invar_train and outvar_train. In eager loading these dictionaries should be of the type Dict[str: np.array], where each variable is a numpy array of data. Lazy loading uses dictionaries of the type Dict[str: DataFile], consisting of DataFile objects which are classes that are used to map between example index and the datafile.

### Initializing the Model¶

FNO initialization allows users to define their own pointwise decoder model. Thus we first initialize the small fully-connected decoder network, which we then provide to the FNO model as a parameter.

    # make list of nodes to unroll graph on
decoder_net = instantiate_arch(
cfg=cfg.arch.decoder,
output_keys=output_keys,
)
fno = instantiate_arch(
cfg=cfg.arch.fno,
input_keys=input_keys,
decoder_net=decoder_net,
)
nodes = [fno.make_node('fno')]


For the physics-informed problems in Modulus, you typically need to define a geometry and constraints based on boundary conditions and governing equations. Here the only constraint is a SupervisedGridConstraint which performs standard supervised training on grid data. This constraint supports the use of multiple workers, which are particularly important when using lazy loading.

    # make domain
domain = Domain()

supervised = SupervisedGridConstraint(
nodes=nodes,
dataset=train_dataset,
batch_size=cfg.batch_size.grid,
num_workers=4,  # number of parallel data loaders
)


Note

Grid data refers to data that can be defined in a tensor like an image. Inside Modulus this grid of data typically represents a spatial domain and should follow the standard dimensionality of [batch, channel, xdim, ydim, zdim] where channel is the dimensionality of your state variables. Both Fourier and convolutional models use grid-based data to efficiently learn and predict entire domains in one forward pass, which contrasts to the pointwise predictions of standard PINN approaches.

The validation data is then added to the domain using GridValidator which should be used when dealing with structured data. Recall that unlike the training constraint, you will use eager loading for the validator. Thus, a dictionary of numpy arrays are passed to the constraint.

    # add validator
val = GridValidator(
nodes,
dataset=test_dataset,
batch_size=cfg.batch_size.validation,
plotter=GridValidatorPlotter(n_examples=5),
)

python darcy_FNO_lazy.py

The checkpoint directory is saved based on the results recording frequency specified in the rec_results_freq parameter of its derivatives. See Results Frequency for more information. The network directory folder (in this case 'outputs/darcy_fno/validators') contains several plots of different validation predictions. Several are shown below, and you can see that the model is able to accurately predict the pressure field for permeability fields it had not seen previously.