""" .. _fourier_continuation : Fourier Continuation ==================== This tutorial demonstrates Fourier continuation methods for extending non-periodic functions to periodic ones, enabling efficient spectral analysis and neural operator applications. Fourier continuation is crucial for: - Converting non-periodic data to periodic form - Enabling spectral methods on arbitrary domains - Improving convergence of Fourier-based neural operators - Handling boundary conditions in spectral computations The tutorial covers both FC-Legendre and FC-Gram methods for 1D, 2D, and 3D data. """ # %% # .. raw:: html # #
# # Import the library # ------------------ # We first import our `neuralop` library and required dependencies. import torch import matplotlib.pyplot as plt from matplotlib.lines import Line2D from neuralop.layers.fourier_continuation import FCLegendre, FCGram device = torch.device("cuda" if torch.cuda.is_available() else "cpu") # %% # .. raw:: html # #
# # Creating an example of 1D non-periodic function # ---------------------------------------------- # We consider f(x) = sin(16x) - cos(8x) on the interval [0,1]. # This function is not periodic on [0,1], making it a good test case # for Fourier continuation methods. length_signal = 101 # Length of the original 1D signal add_pts = 50 # Number of additional points for continuation batch_size = 3 # Batch size for processing multiple signals # Create the input signal x = torch.linspace(0, 1, length_signal).repeat(batch_size, 1) f = torch.sin(16 * x) - torch.cos(8 * x) # %% # .. raw:: html # #
# # Extending the signal # ----------------------------------------- # We use the FC-Legendre and FC-Gram Fourier continuation layers to extend the signal. # FC-Legendre: Uses Legendre polynomial basis for continuation Extension_Legendre = FCLegendre(d=2, n_additional_pts=add_pts) f_extend_Legendre = Extension_Legendre(f, dim=1) # FC-Gram: Uses Gram polynomial basis for continuation Extension_Gram = FCGram(d=4, n_additional_pts=add_pts) f_extend_Gram = Extension_Gram(f, dim=1) # %% # .. raw:: html # #
# # Visualizing the 1D Fourier continuation results # ---------------------------------------------- # We plot the original function and both continuation methods to compare # their effectiveness in creating smooth periodic extensions. # Define the extended coordinates x_extended = torch.linspace(-0.25, 1.25, 151) # Adjust the extended functions for visualization purposes f_extend_Legendre_adjusted = f_extend_Legendre - 0.6 f_extend_Gram_adjusted = f_extend_Gram + 0.6 plt.figure(figsize=(13, 6)) plt.plot(x[0], f[0], "k", label="Original Function", lw=2.2) plt.plot(x_extended, f_extend_Gram_adjusted[0], "b", label="FC-Gram Extension", lw=2.2) plt.plot(x_extended, f_extend_Legendre_adjusted[0], "g", label="FC-Legendre Extension", lw=2.2) plt.plot([0, 0], [-2.9, 1.9], "-", color="gray", lw=1.5) plt.plot([1, 1], [-2.9, 1.1], "-", color="gray", lw=1.5) plt.plot([-0.25, 1.25], [f_extend_Legendre_adjusted[0, 0], f_extend_Legendre_adjusted[0, 0]], "--", color="g", lw=1.6) plt.plot([-0.25, 1.25], [f_extend_Gram_adjusted[0, 0], f_extend_Gram_adjusted[0, 0]], "--", color="b", lw=1.6) legend_elements = [ Line2D([0], [0], color="k", lw=2.2, label="Original Function"), Line2D([0], [0], color="b", lw=2.2, label="FC-Gram Extension"), Line2D([0], [0], color="g", lw=2.2, label="FC-Legendre Extension"), ] legend = plt.legend(handles=legend_elements, fontsize=19) plt.xlim([-0.28, 1.31]) plt.ylim([-3.1, 2.6]) ax = plt.gca() ax.spines["top"].set_visible(False) ax.spines["right"].set_visible(False) ax.tick_params(axis="x", which="major", labelsize=19) ax.tick_params(axis="y", which="major", labelsize=19) plt.xticks([-0.25, 0, 1, 1.25], ["-0.25", "0", "1", "1.25"]) plt.yticks([-2, 2]) plt.tight_layout() plt.show() # %% # .. raw:: html # #
# # Creating an example of 2D non-periodic function # ------------------------------------------------ # We consider f(x,y) = sin(12x) - cos(14y) + 3xy on the domain [0,1]×[0,1]. # This function is not periodic on the unit square, making it suitable # for testing 2D Fourier continuation methods. length_signal = 101 # Length of the signal in each dimension add_pts = 50 # Number of additional points for continuation in each dimension batch_size = 3 # Batch size for processing multiple signals # Create the 2D coordinate grids x = torch.linspace(0, 1, length_signal).view(1, length_signal, 1).repeat(batch_size, 1, length_signal) y = torch.linspace(0, 1, length_signal).view(1, 1, length_signal).repeat(batch_size, length_signal, 1) # Define the 2D test function f = torch.sin(12 * x) - torch.cos(14 * y) + 3 * x * y # %% # .. raw:: html # #
# # Extending the signal # ----------------------------------------- # We use the FC-Legendre and FC-Gram Fourier continuation layers to extend the signal. # FC-Legendre: Uses Legendre polynomial basis for 2D continuation Extension_Legendre = FCLegendre(d=3, n_additional_pts=add_pts) f_extend_Legendre = Extension_Legendre(f, dim=2) # FC-Gram: Uses Gram polynomial basis for 2D continuation Extension_Gram = FCGram(d=3, n_additional_pts=add_pts) f_extend_Gram = Extension_Gram(f, dim=2) # %% # .. raw:: html # #
# # Visualizing the 2D Fourier continuation results # ---------------------------------------------- # We plot the original function and both continuation methods to compare # their effectiveness in creating smooth 2D periodic extensions. # Black lines delineate the original domain boundaries. fig, axs = plt.subplots(figsize=(15, 5), nrows=1, ncols=3) # Plot the original function axs[0].imshow(f[0]) axs[0].set_title(r"Original Function", fontsize=15.5) axs[1].imshow(f_extend_Legendre[0]) axs[1].plot([add_pts//2, length_signal + add_pts//2], [add_pts//2, add_pts//2], "-", color="k", lw=3) axs[1].plot([add_pts//2, add_pts//2], [add_pts//2, length_signal + add_pts//2], "-", color="k", lw=3) axs[1].plot([add_pts//2, length_signal + add_pts//2], [length_signal + add_pts//2, length_signal + add_pts//2], "-", color="k", lw=3) axs[1].plot([length_signal + add_pts//2, length_signal + add_pts//2], [add_pts//2, length_signal + add_pts//2], "-", color="k", lw=3) axs[1].set_title(r"FC-Legendre Extension", fontsize=15.5) axs[2].imshow(f_extend_Gram[0]) axs[2].plot([add_pts//2, length_signal + add_pts//2], [add_pts//2, add_pts//2], "-", color="k", lw=3) axs[2].plot([add_pts//2, add_pts//2], [add_pts//2, length_signal + add_pts//2], "-", color="k", lw=3) axs[2].plot([add_pts//2, length_signal + add_pts//2], [length_signal + add_pts//2, length_signal + add_pts//2], "-", color="k", lw=3) axs[2].plot([length_signal + add_pts//2, length_signal + add_pts//2], [add_pts//2, length_signal + add_pts//2], "-", color="k", lw=3) axs[2].set_title(r"FC-Gram Extension", fontsize=15.5) for ax in axs.flat: ax.set_xticks([]) ax.set_yticks([]) plt.subplots_adjust(wspace=0.05) # Reduce white space between plots plt.show() # %% # .. raw:: html # #
# # Creating an example of a 3D function # -------------------- # Here we consider f(x,y,z) = exp(-2z) + 2xz + sin(12xy) + y sin(10yz) # which is not periodic on [0,1]x[0,1]x[0,1] batch_size = 2 length_signal = 101 add_pts = 50 # Create 3D grid x = torch.linspace(0, 1, length_signal).view(1, length_signal, 1, 1).repeat(batch_size, 1, length_signal, length_signal) y = torch.linspace(0, 1, length_signal).view(1, 1, length_signal, 1).repeat(batch_size, length_signal, 1, length_signal) z = torch.linspace(0, 1, length_signal).view(1, 1, 1, length_signal).repeat(batch_size, length_signal, length_signal, 1) # Create 3D function f = torch.exp(-2 * z) + 2 * z * x + torch.sin(12 * x * y) + y * torch.sin(10 * y * z) # %% # .. raw:: html # #
# # Extending the signal # ----------------------------------------- # We use the FC-Legendre and FC-Gram Fourier continuation layers to extend the signal. Extension_Legendre = FCLegendre(d=3, n_additional_pts=add_pts) f_extend_Legendre = Extension_Legendre(f, dim=3) Extension_Gram = FCGram(d=3, n_additional_pts=add_pts) f_extend_Gram = Extension_Gram(f, dim=3) # %% # .. raw:: html # #
# # Plot the FC-Legendre and FC-Gram results for 3D # ---------------------- # We also add white lines to deliminate the original signal f_min = f.min().item() f_max = f.max().item() f_ext_legendre_min = f_extend_Legendre.min().item() f_ext_legendre_max = f_extend_Legendre.max().item() f_ext_gram_min = f_extend_Gram.min().item() f_ext_gram_max = f_extend_Gram.max().item() global_min = min(f_min, f_ext_legendre_min, f_ext_gram_min) global_max = max(f_max, f_ext_legendre_max, f_ext_gram_max) # %% # .. raw:: html # #
# # Figure for X slices fig = plt.figure(figsize=(24, 20)) slice_indices = [length_signal//4, length_signal//2, 3*length_signal//4] slice_names = ["First Quarter", "Middle", "Third Quarter"] for i, (idx, name) in enumerate(zip(slice_indices, slice_names)): # Original function - X-slice ax = fig.add_subplot(3, 3, i + 1) im = ax.imshow(f[0, idx, :, :].numpy(), cmap="viridis", aspect="auto", vmin=global_min, vmax=global_max) ax.set_title(f"Original: X-slice {name}", fontsize=22, fontweight="bold") ax.set_xlabel("Y", fontsize=20) ax.set_ylabel("Z", fontsize=20) ax.set_xticks([]) ax.set_yticks([]) cbar = plt.colorbar(im, ax=ax) cbar.ax.tick_params(labelsize=18) # FC-Legendre extension - X-slice ax = fig.add_subplot(3, 3, i + 4) ext_idx = idx + add_pts//2 im = ax.imshow(f_extend_Legendre[0, ext_idx, :, :].numpy(), cmap="viridis", aspect="auto", vmin=global_min, vmax=global_max) ax.set_title(f"FC-Legendre: X-slice {name}", fontsize=22, fontweight="bold") ax.set_xlabel("Y", fontsize=20) ax.set_ylabel("Z", fontsize=20) # Draw boundary lines ax.axhline(y=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axhline(y=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.set_xticks([]) ax.set_yticks([]) cbar = plt.colorbar(im, ax=ax) cbar.ax.tick_params(labelsize=18) # FC-Gram extension - X-slice ax = fig.add_subplot(3, 3, i + 7) im = ax.imshow(f_extend_Gram[0, ext_idx, :, :].numpy(), cmap="viridis", aspect="auto", vmin=global_min, vmax=global_max) ax.set_title(f"FC-Gram: X-slice {name}", fontsize=22, fontweight="bold") ax.set_xlabel("Y", fontsize=20) ax.set_ylabel("Z", fontsize=20) # Draw boundary lines ax.axhline(y=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axhline(y=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.set_xticks([]) ax.set_yticks([]) cbar = plt.colorbar(im, ax=ax) cbar.ax.tick_params(labelsize=18) plt.subplots_adjust(hspace=0.15, wspace=0.05, top=0.95) plt.show() # %% # .. raw:: html # #
# # Figure for Y-slices fig2 = plt.figure(figsize=(24, 20)) for i, (idx, name) in enumerate(zip(slice_indices, slice_names)): # Original function - Y-slice ax = fig2.add_subplot(3, 3, i + 1) im = ax.imshow(f[0, :, idx, :].numpy(), cmap="viridis", aspect="auto", vmin=global_min, vmax=global_max) ax.set_title(f"Original: Y-slice {name}", fontsize=22, fontweight="bold") ax.set_xlabel("X", fontsize=20) ax.set_ylabel("Z", fontsize=20) ax.set_xticks([]) ax.set_yticks([]) cbar = plt.colorbar(im, ax=ax) cbar.ax.tick_params(labelsize=18) # FC-Legendre extension - Y-slice ax = fig2.add_subplot(3, 3, i + 4) ext_idx = idx + add_pts//2 im = ax.imshow(f_extend_Legendre[0, :, ext_idx, :].numpy(), cmap="viridis", aspect="auto", vmin=global_min, vmax=global_max) ax.set_title(f"FC-Legendre: Y-slice {name}", fontsize=22, fontweight="bold") ax.set_xlabel("X", fontsize=20) ax.set_ylabel("Z", fontsize=20) # Draw boundary lines ax.axhline(y=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axhline(y=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.set_xticks([]) ax.set_yticks([]) cbar = plt.colorbar(im, ax=ax) cbar.ax.tick_params(labelsize=18) # FC-Gram extension - Y-slice ax = fig2.add_subplot(3, 3, i + 7) im = ax.imshow(f_extend_Gram[0, :, ext_idx, :].numpy(), cmap="viridis", aspect="auto", vmin=global_min, vmax=global_max) ax.set_title(f"FC-Gram: Y-slice {name}", fontsize=22, fontweight="bold") ax.set_xlabel("X", fontsize=20) ax.set_ylabel("Z", fontsize=20) # Draw boundary lines ax.axhline(y=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axhline(y=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.set_xticks([]) ax.set_yticks([]) cbar = plt.colorbar(im, ax=ax) cbar.ax.tick_params(labelsize=18) plt.subplots_adjust(hspace=0.15, wspace=0.05, top=0.95) plt.show() # %% # .. raw:: html # #
# # Figure for Z-slices fig3 = plt.figure(figsize=(24, 20)) for i, (idx, name) in enumerate(zip(slice_indices, slice_names)): # Original function - Z-slice ax = fig3.add_subplot(3, 3, i + 1) im = ax.imshow(f[0, :, :, idx].numpy(), cmap="viridis", aspect="auto", vmin=global_min, vmax=global_max) ax.set_title(f"Original: Z-slice {name}", fontsize=22, fontweight="bold") ax.set_xlabel("X", fontsize=20) ax.set_ylabel("Y", fontsize=20) ax.set_xticks([]) ax.set_yticks([]) cbar = plt.colorbar(im, ax=ax) cbar.ax.tick_params(labelsize=18) # FC-Legendre extension - Z-slice ax = fig3.add_subplot(3, 3, i + 4) ext_idx = idx + add_pts//2 im = ax.imshow(f_extend_Legendre[0, :, :, ext_idx].numpy(), cmap="viridis", aspect="auto", vmin=global_min, vmax=global_max) ax.set_title(f"FC-Legendre: Z-slice {name}", fontsize=22, fontweight="bold") ax.set_xlabel("X", fontsize=20) ax.set_ylabel("Y", fontsize=20) # Draw boundary lines ax.axhline(y=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axhline(y=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.set_xticks([]) ax.set_yticks([]) cbar = plt.colorbar(im, ax=ax) cbar.ax.tick_params(labelsize=18) # FC-Gram extension - Z-slice ax = fig3.add_subplot(3, 3, i + 7) im = ax.imshow(f_extend_Gram[0, :, :, ext_idx].numpy(), cmap="viridis", aspect="auto", vmin=global_min, vmax=global_max) ax.set_title(f"FC-Gram: Z-slice {name}", fontsize=22, fontweight="bold") ax.set_xlabel("X", fontsize=20) ax.set_ylabel("Y", fontsize=20) # Draw boundary lines ax.axhline(y=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axhline(y=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=add_pts//2, color="white", linewidth=2, linestyle="-") ax.axvline(x=length_signal + add_pts//2, color="white", linewidth=2, linestyle="-") ax.set_xticks([]) ax.set_yticks([]) cbar = plt.colorbar(im, ax=ax) cbar.ax.tick_params(labelsize=18) plt.subplots_adjust(hspace=0.15, wspace=0.05, top=0.95) plt.show()