Fourier Differentiation

An example of usage of our Fourier Differentiation Function on 1d data.

Import the library

We first import our neuralop library and required dependencies.

import torch
import numpy as np
import matplotlib.pyplot as plt
from neuralop.losses.fourier_diff import fourier_derivative_1d

device = torch.device("cuda" if torch.cuda.is_available() else "cpu")

Creating an example of periodic 1D curve

Here we consider sin(x) and cos(x), which are periodic on the interval [0,2pi]

L = 2*torch.pi
x = torch.linspace(0, L, 101)[:-1]
f = torch.stack([torch.sin(x), torch.cos(x)], dim=0)
x_np = x.cpu().numpy()

Differentiate the signal

We use the Fourier differentiation function to differentiate the signal

dfdx = fourier_derivative_1d(f, order=1, L=L)
df2dx2 = fourier_derivative_1d(f, order=2, L=L)
df3dx3 = fourier_derivative_1d(f, order=3, L=L)

/home/runner/work/neuraloperator/neuraloperator/neuralop/losses/fourier_diff.py:76: UserWarning: Consider using Fourier continuation if the input is not periodic (use_FC=True).
  warnings.warn("Consider using Fourier continuation if the input is not periodic (use_FC=True).", category=UserWarning)

Plot the results for sin(x)

plt.figure()
plt.plot(x_np, dfdx[0].squeeze().cpu().numpy(), label='Fourier dfdx')
plt.plot(x_np, np.cos(x_np), '--', label='dfdx')
plt.plot(x_np, df2dx2[0].squeeze().cpu().numpy(), label='Fourier df2dx2')
plt.plot(x_np, -np.sin(x_np), '--', label='df2dx2')
plt.plot(x_np, df3dx3[0].squeeze().cpu().numpy(), label='Fourier df3dx3')
plt.plot(x_np, -np.cos(x_np), '--', label='df3dx3')
plt.xlabel('x')
plt.legend()
plt.show()

Plot the results for cos(x)

plt.figure()
plt.plot(x_np, dfdx[1].squeeze().cpu().numpy(), label='Fourier dfdx')
plt.plot(x_np, -np.sin(x_np), '--', label='dfdx')
plt.plot(x_np, df2dx2[1].squeeze().cpu().numpy(), label='Fourier df2dx2')
plt.plot(x_np, -np.cos(x_np), '--', label='df2dx2')
plt.plot(x_np, df3dx3[1].squeeze().cpu().numpy(), label='Fourier df3dx3')
plt.plot(x_np, np.sin(x_np), '--', label='df3dx3')
plt.xlabel('x')
plt.legend()
plt.show()

Creating an example of non-periodic 1D curve

Here we consider sin(16x)-cos(8x) and exp(-0.8x)+sin(x)

L = 2*torch.pi
x = torch.linspace(0, L, 101)[:-1]
f = torch.stack([torch.sin(3*x) - torch.cos(x), torch.exp(-0.8*x)+torch.sin(x)], dim=0)
x_np = x.cpu().numpy()

Differentiate the signal

We use the Fourier differentiation function with Fourier continuation to differentiate the signal

dfdx = fourier_derivative_1d(f, order=1, L=L, use_FC='Legendre', FC_d=4, FC_n_additional_pts=30, FC_one_sided=False)
df2dx2 = fourier_derivative_1d(f, order=2, L=L, use_FC='Legendre', FC_d=4, FC_n_additional_pts=30, FC_one_sided=False)

Plot the results for sin(16x)-cos(8x)

plt.figure()
plt.plot(x_np, dfdx[0].squeeze().cpu().numpy(), label='Fourier dfdx')
plt.plot(x_np, 3*torch.cos(3*x) + torch.sin(x), '--', label='dfdx')
plt.plot(x_np, df2dx2[0].squeeze().cpu().numpy(), label='Fourier df2dx2')
plt.plot(x_np, -9*torch.sin(3*x) + torch.cos(x), '--', label='df2dx2')
plt.xlabel('x')
plt.legend()
plt.show()

Plot the results for exp(-0.8x)+sin(x)

plt.figure()
plt.plot(x_np, dfdx[1].squeeze().cpu().numpy(), label='Fourier dfdx')
plt.plot(x_np, -0.8*torch.exp(-0.8*x)+torch.cos(x), '--', label='dfdx')
plt.plot(x_np, df2dx2[1].squeeze().cpu().numpy(), label='Fourier df2dx2')
plt.plot(x_np, 0.64*torch.exp(-0.8*x)-torch.sin(x), '--', label='df2dx2')
plt.xlabel('x')
plt.legend()
plt.show()

Total running time of the script: (0 minutes 0.282 seconds)

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