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dev/_downloads/02a7a82499cf14d2702c9234035c44c4/plot_diffusion_advection_solver.zip

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+"""
+.. _fourier_diff :
+
+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)
+
+
+# %%
+# 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()