sources/neuraloperator/examples/data/plot_darcy_flow_spectrum.py

"""
A simple Darcy-Flow spectrum analysis
=====================================
Using ``neuralop.utils.spectrum_2d`` to perform spectrum analysis on our small Darcy-Flow example.

This tutorial demonstrates how to analyze the spectral characteristics of the Darcy-Flow dataset,
which provides insights into the frequency content and energy distribution of the vector fields.

For more details on spectrum analysis, users can take a look at this reference: 
https://www.astronomy.ohio-state.edu/ryden.1/ast825/ch7.pdf

Short summary
--------------
Spectral analysis is useful because it allows researchers to study the distribution of energy 
across different scales in a fluid flow. The energy spectrum is analyzed through the Fourier 
transform, a mathematical tool that decomposes a function or signal into its constituent frequencies. 
In a fluid flow, it is used to analyze the distribution of energy across different scales in a flow. 

Specifically, the Fourier transform is applied to the velocity field of the flow, converting it 
into a frequency domain representation. Higher wavenumbers correspond to higher frequencies and 
higher energy, and are much harder to solve as we need higher modes to capture the high-frequency 
behavior of the flow. Overall, this allows researchers to study the energy spectrum, which 
provides insights into the behavior of turbulence and the underlying physical processes.

"""

# %%
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#
#    <div style="margin-top: 3em;"></div>
#
# Import dependencies
# --------------------
# We first import our `neuralop` library and required dependencies.
import numpy as np
import torch
import matplotlib
import matplotlib.pyplot as plt
from neuralop.utils import spectrum_2d
from neuralop.data.datasets import load_darcy_flow_small

font = {"size": 28}
matplotlib.rc("font", **font)

torch.manual_seed(0)
np.random.seed(0)

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#    <div style="margin-top: 3em;"></div>
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# Define analysis parameters
# --------------------------
# These parameters control the spectral analysis of our Darcy-Flow dataset
samples = 50  # Number of samples to analyze
s = 16  # Resolution of the dataset (16x16 grid)
dataset_name = "Darcy Flow"

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#    <div style="margin-top: 3em;"></div>
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# Loading the Darcy-Flow dataset
# --------------------------------
# We load the Darcy-Flow dataset with multiple resolutions for spectral analysis
train_loader, test_loaders, data_processor = load_darcy_flow_small(
    n_train=50,
    batch_size=50,
    test_resolutions=[16, 32],
    n_tests=[50, 50],
    test_batch_sizes=[32],
    encode_output=False,
)

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#    <div style="margin-top: 3em;"></div>
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# Preparing data for spectral analysis
# ------------------------------------
# The dataset structure is ['x', 'y'] where 'x' is the permeability field and 'y' is the pressure field
# We'll analyze the pressure fields ('y') for their spectral characteristics
print("Original dataset shape:", train_loader.dataset[:samples]["x"].shape)

# Extract the pressure fields (output) for spectral analysis
# We want the last two dimensions to represent the spatial dimensions
dataset_pred = train_loader.dataset[:samples][
    "y"
].squeeze()  # Remove empty channel dimension

# Shape of the dataset
shape = dataset_pred.shape
print(f"Pressure field shape: {shape}")

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# Creating coordinate grids for spectral analysis
# ------------------------------------------------
# We need to define the spatial grid for proper spectral analysis
# For 2D grids, we create x and y coordinate arrays
batchsize, size_x, size_y = 1, shape[1], shape[2]

# Create coordinate grids normalized to [-1, 1]
gridx = torch.tensor(np.linspace(-1, 1, size_x), dtype=torch.float)
gridx = gridx.reshape(1, size_x, 1).repeat([batchsize, 1, size_y])
gridy = torch.tensor(np.linspace(-1, 1, size_y), dtype=torch.float)
gridy = gridy.reshape(1, 1, size_y).repeat([batchsize, size_x, 1])
grid = torch.cat((gridx, gridy), dim=-1)

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#    <div style="margin-top: 3em;"></div>
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# Computing the energy spectrum
# -----------------------------
# We compute the 2D energy spectrum of the pressure fields using the spectrum_2d utility
# This gives us insight into how energy is distributed across different spatial frequencies

# Generate the spectrum of the dataset
# We reshape our samples into the form expected by ``spectrum_2d``: ``(n_samples, h, w)``
truth_sp = spectrum_2d(dataset_pred.reshape(samples * batchsize, s, s), s)

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#    <div style="margin-top: 3em;"></div>
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# Visualizing the energy spectrum
# --------------------------------
# The energy spectrum shows how much energy is contained in each wavenumber (spatial frequency)
# Higher wavenumbers correspond to smaller spatial scales and higher frequencies

# Configure pyplot and generate the plot
plt.style.use("classic")
fig, ax = plt.subplots(figsize=(10, 10), dpi=150)
ax.set_facecolor("white")
ax.set_yscale("log")  # Log scale for better visualization of energy decay
length = dataset_pred.shape[-1]  # The resolution length of the dataset
buffer = 4  # Add a buffer to the plot for better visualization
k = np.arange(length + buffer) * 1.0

# Plot the spectrum
ax.plot(truth_sp, color="navy", linestyle="--", linewidth=2.5, label="Energy Spectrum")

# Customizing the plot
ax.set_xlim(1, 16)
ax.set_ylim(0.01, 20)
ax.set_xlabel("Wavenumber", fontsize=16, fontweight="bold")
ax.set_ylabel("Energy", fontsize=16, fontweight="bold")
ax.set_title(
    f"Energy Spectrum of {dataset_name} Dataset", fontsize=18, fontweight="bold", pad=20
)
ax.tick_params(axis="both", which="major", labelsize=15)

plt.tight_layout()
plt.show()

# %%