CIDM-Foundation-A1-R1-R2-FIX1-Results / sources /neuraloperator /examples /layers /plot_fourier_continuation.py
| """ | |
| .. _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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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 | |
| # | |
| # <div style="margin-top: 3em;"></div> | |
| # | |
| # 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() | |