sources/neuraloperator/examples/layers/plot_sinusoidal_embeddings.py

"""
Sinusoidal Embeddings
====================

Inputs to deep learning models often represent positions on a spatial, temporal, or
spatio-temporal grid. To enrich these coordinates, positional embeddings can be introduced
to improve a model's capacity to generalize across the domain. In this tutorial, we focus
on sinusoidal positional embeddings.

Sinusoidal embeddings encode inputs as periodic functions (sines and cosines), thereby
lifting low-dimensional coordinates into a richer spectral representation. This spectral
lifting enhances the model's ability to capture fine-scale variations and high-frequency
dynamics.
"""

# %%
# .. raw:: html
#
#    <div style="margin-top: 3em;"></div>
#
# Setup in 1D
# ----------------------
# To build intuition, consider a simple 1D example. Let :math:`x \in \mathbb{R}`
# be a single input, and define the embedding function
#
# .. math::
#    g: \mathbb{R} \rightarrow \mathbb{R}^{2 L}, \quad g(x)=[\sin (x), \  \cos (x), \ \sin (2 x), \ \cos (2 x), \ldots, \ \sin (L x), \ \cos (L x)],
#
# where :math:`L` defines the number of frequencies we wish to use for the embedding. Each
# pair of sine and cosine terms introduces a higher frequency, enriching how positional
# information is represented.
#
# This idea naturally extends to an entire 1D input. Let :math:`\vec{x} \in \mathbb{R}^N`
# denote a discretized domain of :math:`N` points. Then the embedding function becomes
#
# .. math::
#    g: \mathbb{R}^N \rightarrow \mathbb{R}^{N \times 2 L}, \quad g(\vec{x})=\operatorname{concat}(\sin (\vec{x}), \cos (\vec{x}), \sin (2 \vec{x}), \cos (2 \vec{x}), \ldots, \sin (L \vec{x}), \cos (L \vec{x})),
#
# In practice, both the original coordinate and its embedding are passed to the model:
#
# .. math::
#    \operatorname{input}(\vec{x})=\operatorname{concat}(\vec{x}, \ g(\vec{x})) \in \mathbb{R}^{N \times 2 L + 1},
#
# preserving the original input, while augmenting it with a hierarchy of frequency components.
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# Domain Normalization
# ~~~~~~~~~~~~~~~~~~~~
# When applying sinusoidal embeddings, it is often useful to normalize the input coordinates
# to a periodic interval that aligns with the natural period of the sine and cosine functions.
# For example, a 1D spatial domain :math:`\vec{x} \in[0,1]` of :math:`N` points can be rescaled to
#
# .. math::
#    \vec{x}^{\prime}=2 \pi \vec{x},
#
# so that :math:`\vec{x}^{\prime} \in[0,2 \pi]`.
#
# This mapping preserves the number of sampling points :math:`N` and the overall shape of the domain
# while ensuring that the lowest-frequency sine and cosine components complete exactly one
# full oscillation over the interval.
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# Choosing :math:`L` to Satisfy the Nyquist-Criterion
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# .. warning::
#    When choosing the number of frequency levels :math:`L`, it is important to ensure that the
#    highest frequency component in the embedding does not exceed the Nyquist limit imposed by
#    the discretisation of the input domain.
#
# For a domain of :math:`N` points, the Nyquist frequency is
#
# .. math::
#    f_{\text{Nyquist}} = \frac{N}{2}.
#
# For the sinusoidal embedding defined above, the Nyquist constraint becomes:
#
# .. math::
#    L < \frac{N}{2}.
#
# The Nyquist frequency represents the maximum frequency that can be correctly captured
# when sampling a signal, equal to half the sampling rate. If frequencies higher than this
# limit are used, they will not be represented as true high frequencies but will instead appear
# as lower ones, producing distortion known as aliasing. This is why we must ensure that
# the highest frequency in our embedding does not exceed the Nyquist limit.
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# Visualizing the Sinusoidal Embeddings
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Below, we visualize the sinusoidal embeddings for a spatial input domain
# :math:`\vec{x} \in[0,1]` consisting of 1000 equally spaced points, using :math:`L = 3` frequency levels.

# %%

# Import required libraries
import torch
import matplotlib.pyplot as plt
import numpy as np
from neuralop.layers.embeddings import SinusoidalEmbedding

# Set default font sizes for better readability
plt.rcParams.update(
    {
        "font.size": 14,
        "axes.titlesize": 18,
        "axes.labelsize": 16,
        "xtick.labelsize": 14,
        "ytick.labelsize": 14,
        "legend.fontsize": 14,
    }
)

device = "cpu"

# Define a spatial domain and number of frequencies
# Create 1000 equally spaced points in [0, 1]
#  and normalize to [0, 2π] for proper sinusoidal embedding
x = torch.linspace(0, 1, 1000)
x_normalized = torch.linspace(0, 2 * torch.pi, len(x))
# Number of frequency levels for the embedding
L = 3

# Check if the number of frequencies satisfies the Nyquist-Criterion
if L < len(x_normalized) / 2:
    print(f"Nyquist-Shannon sampling theorem is satisfied for the given number of frequencies {L}.")
else:
    print(f"Nyquist-Shannon sampling theorem is violated for the given number of frequencies {L}.")

# Build embedding: [sin(x), cos(x), sin(2x), cos(2x), ...]
# Each frequency level contributes a sine and cosine pair
g = []
for l in range(1, L + 1):
    g.append(torch.sin(l * x_normalized))
    g.append(torch.cos(l * x_normalized))

# Construct input by concatenating the original input and the embedding
# This preserves the original coordinates while adding spectral information
input_arr = np.asarray([x, *g])
input_tensor = torch.tensor(input_arr)

# Plot the embedding components
colors = plt.cm.tab10.colors

plt.figure(figsize=(10, 5))
for freq_idx in range(L):
    color = colors[freq_idx % len(colors)]
    sin_idx = 2 * freq_idx + 1
    cos_idx = 2 * freq_idx + 2

    plt.plot(x, input_tensor[sin_idx], color=color, label=f"Frequency {freq_idx + 1}")
    plt.plot(x, input_tensor[cos_idx], color=color)

plt.xlabel("x", fontsize=16)
plt.ylabel("Embedding value", fontsize=16)
plt.title("Sinusoidal Embedding Components (L = 3)", fontsize=18)
plt.legend(loc="lower left", framealpha=1.0, fontsize=14)
plt.locator_params(axis="y", nbins=5)
plt.tight_layout()
plt.show()

# %%
# .. raw:: html
#
#    <div style="margin-top: 4em;"></div>
#
# Encoding Constant Parameters
# ----------------------------
# A particularly useful extension of sinusoidal embeddings is their ability to encode constant
# parameters. Consider a setting where you have a scalar parameter :math:`m` (such as a material
# property, boundary condition, or physical constant) that you wish to feed into a model.
# Instead of treating :math:`m` as a fixed scalar input, we can represent it using periodic
# functions, either by modulating the amplitude or the frequency of the sinusoidal components.
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# **1. Amplitude Modulation:** To encode :math:`m` by scaling the amplitudes of the sinusoidal
# functions, we define the embedding as
#
# .. math::
#    m \rightarrow m g(\vec{x}),
#
# where each element of the embedding :math:`g(\vec{x})` is multiplied by :math:`m`.
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# **2. Frequency Modulation:** Alternatively, to encode :math:`m` by scaling the frequencies,
# we define
#
# .. math::
#    m \rightarrow g(m\vec{x})
#
# where :math:`m` multiplies the input argument of each sinusoidal component.
#
# When encoding constant parameters through frequency modulation, care must be taken to ensure
# that the Nyquist criterion is satisfied. In this case, where the modulation factor :math:`m`
# scales the frequencies, the Nyquist constraint becomes :math:`L < \frac{N}{2m}`.
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# Below, we demonstrate an example of encoding the parameter :math:`m = 2.5` through both
# amplitude and frequency modulation.

# %%

# Define a spatial domain and number of frequencies
x = torch.linspace(0, 1, 1000)
x_normalized = torch.linspace(0, 2 * torch.pi, len(x))
L = 3

# Define parameter to encode
m = 2.5
m_tensor = torch.tensor([m])

# Check if the number of frequencies and parameter satisfies the Nyquist-Criterion
if L <= len(x_normalized) / (2 * m):
    print(f"Nyquist-Shannon sampling theorem is satisfied for the given parameter {m} and number of frequencies {L}.")
else:
    print(f"Nyquist-Shannon sampling theorem is violated for the given parameter {m} and number of frequencies {L}.")

# Build amplitude-modulated embedding: m * g(x)
g_amplitude = []
for l in range(1, L + 1):
    g_amplitude.append(torch.sin(l * x_normalized) * m_tensor)
    g_amplitude.append(torch.cos(l * x_normalized) * m_tensor)

# Build frequency-modulated embedding: g(m * x)
g_frequency = []
for l in range(1, L + 1):
    g_frequency.append(torch.sin(l * x_normalized * m_tensor))
    g_frequency.append(torch.cos(l * x_normalized * m_tensor))

# Convert to arrays for visualization
input_amplitude = torch.tensor(np.asarray([x, *g_amplitude]))
input_frequency = torch.tensor(np.asarray([x, *g_frequency]))

# Plot both embeddings
colors = plt.cm.tab10.colors
fig, axes = plt.subplots(2, 1, figsize=(10, 9), sharex=True)

## Amplitude modulation
for freq_idx in range(L):
    color = colors[freq_idx % len(colors)]
    sin_idx, cos_idx = 2 * freq_idx + 1, 2 * freq_idx + 2
    axes[0].plot(
        x, input_amplitude[sin_idx], color=color, label=f"Channel {freq_idx + 1}"
    )
    axes[0].plot(x, input_amplitude[cos_idx], color=color)
axes[0].set_title("Amplitude Modulation", fontsize=18, pad=20)
axes[0].set_ylabel("Embedding value", fontsize=16)
axes[0].legend(loc="lower left", framealpha=1.0, fontsize=14)
axes[0].locator_params(axis="y", nbins=5)

## Frequency modulation
for freq_idx in range(L):
    color = colors[freq_idx % len(colors)]
    sin_idx, cos_idx = 2 * freq_idx + 1, 2 * freq_idx + 2
    axes[1].plot(x, input_frequency[sin_idx], color=color, label=f"Channel {freq_idx + 1}")
    axes[1].plot(x, input_frequency[cos_idx], color=color)
axes[1].set_title("Frequency Modulation", fontsize=18, pad=20)
axes[1].set_ylabel("Embedding value", fontsize=16)
axes[1].set_xlabel("x", fontsize=16)
axes[1].locator_params(axis="y", nbins=5)

plt.suptitle(f"Sinusoidal Embeddings with Parameter m = {m}", y=0.98, fontsize=20)
plt.tight_layout()
plt.show()

# %%
# .. raw:: html
#
#    <div style="margin-top: 4em;"></div>
#
# Neural Operator SinusoidalEmbedding Class
# -----------------------------------------
# The ``neuralop`` library provides a unified sinusoidal positional embedding class,
# ``neuralop.layers.embeddings.SinusoidalEmbedding``, with the following embedding techniques:
#
# - ``transformer`` - Vaswani, A. et al (2017), "Attention Is All You Need".
# - ``nerf`` - Mildenhall, B. et al (2020), "NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis".
#
# The `SinusoidalEmbedding` class expects inputs to be of shape
#
#              ``(batch_size, N, input_channels)`` or ``(N, input_channels)``
#
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# Embedding Variants
# ~~~~~~~~~~~~~~~~~~
# Let :math:`\vec{x} \in \mathbb{R}^N` denote a 1D input domain consisting of
# :math:`N` discretized points. The embedding function
# :math:`g: \mathbb{R}^N \rightarrow \mathbb{R}^{N \times 2L}` maps each input value
# :math:`x_n` to a :math:`2L`-dimensional vector composed of sine and cosine terms evaluated
# at different frequencies. Each embedding type defines these frequencies differently,
# leading to distinct representations.
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# **1. Transformer-style embedding:** For :math:`0 \leq k < L`:
#
# .. math::
#    g(\vec{x})_{:, 2 k}=\sin \left(\frac{\vec{x}}{\text { max_positions }^{k / L}}\right), \quad g(\vec{x})_{:, 2 k+1}=\cos \left(\frac{\vec{x}}{\text { max_positions }^{k / L}}\right) .
#
# Here, :math:`\text{max_positions}` controls the maximum position for the embedding.
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# **2. NeRF-style embedding:** For :math:`0 \leq k < L`:
#
# .. math::
#    g(\vec{x})_{:, 2 k}=\sin \left(2^k \pi \vec{x}\right), \quad g(\vec{x})_{:, 2 k+1}=\cos \left(2^k \pi \vec{x}\right) .
#
# In order to ensure that the Nyquist-Criterion is satisfied, for the Transformer-style
# embedding, the embedding frequencies should satisfy: :math:`f_{\max} < f_{\text{Nyquist}}`.
#
# For the NeRF-style embedding:
#
# .. math::
#    2^{L-1} < \frac{N}{2} \ \ \implies \ \  L < 1 + \log_2\left(\frac{N}{2}\right).
#
# .. raw:: html
#
#    <div style="margin-top: 3em;"></div>
#
# Below, we include examples of using the `SinusoidalEmbedding` class with both the
# transformer- and NeRF-style embeddings.

# %%

# Define a spatial domain and the number of frequencies
x = torch.linspace(0, 1, 1000)
x_normalized = torch.linspace(0, 2 * torch.pi, len(x)).reshape(-1, 1)
L = 3

# Check if the number of frequencies satisfies the Nyquist-Criterion
if L <= 1 + torch.log2(torch.tensor(len(x_normalized) / 2)):
    print(f"Nyquist-Shannon sampling theorem is satisfied for the given number of frequencies {L}.")
else:
    print(f"Nyquist-Shannon sampling theorem is violated for the given number of frequencies {L}.")

# Define the transformer embedding
# max_positions controls the frequency scaling in transformer-style embeddings
max_positions = 1000
transformer_embedder = SinusoidalEmbedding(
    in_channels=1,
    num_frequencies=L,
    embedding_type="transformer",
    max_positions=max_positions,
).to(device)

# Apply transformer-style embedding
transformer_embedding = transformer_embedder(x_normalized).permute(1, 0)

# Define the NeRF embedding
nerf_embedder = SinusoidalEmbedding(
    in_channels=1, num_frequencies=L, embedding_type="nerf"
).to(device)

# Apply NeRF-style embedding, with the domain [0, 1]
nerf_embedding = nerf_embedder(x.reshape(-1, 1)).permute(1, 0)

# Plot both embeddings
colors = plt.cm.tab10.colors
fig, axes = plt.subplots(2, 1, figsize=(10, 9), sharex=True)

## Transformer embedding
for freq_idx in range(L):
    color = colors[freq_idx % len(colors)]
    sin_idx, cos_idx = 2 * freq_idx, 2 * freq_idx + 1

    axes[0].plot(x, transformer_embedding[sin_idx], color=color, label=f"Channel {freq_idx + 1}")
    axes[0].plot(x, transformer_embedding[cos_idx], color=color)

axes[0].set_title("Transformer embedding", fontsize=18, pad=20)
axes[0].set_ylabel("Embedding value", fontsize=16)
axes[0].legend(loc="lower left", framealpha=1.0, fontsize=14)
axes[0].locator_params(axis="y", nbins=5)

## NeRF embedding
for freq_idx in range(L):
    color = colors[freq_idx % len(colors)]
    sin_idx, cos_idx = 2 * freq_idx, 2 * freq_idx + 1

    axes[1].plot(x, nerf_embedding[sin_idx], color=color, label=f"Channel {freq_idx + 1}")
    axes[1].plot(x, nerf_embedding[cos_idx], color=color)

axes[1].set_title("NeRF embedding", fontsize=18, pad=20)
axes[1].set_xlabel("x", fontsize=16)
axes[1].set_ylabel("Embedding value", fontsize=16)
axes[1].locator_params(axis="y", nbins=5)

plt.suptitle("Sinusoidal Embeddings using transformer and NeRF embedding types", y=0.98, fontsize=20)
plt.tight_layout()
plt.show()

# %%
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# Encoding Constant Parameters with NeuralOp Class
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Similar to the earlier illustrative examples, we can also encode
# a scalar parameter :math:`m` before passing it to a model. Once again, care must be taken to
# ensure that the Nyquist criterion is satisfied.
#
# In the Transformer-style embedding, to avoid aliasing, the embedding frequencies should still
# satisfy
#
# .. math::
#    f_{\max} < f_{\text{Nyquist}}.
#
# For the NeRF-style embedding, the modified constraint becomes:
#
# .. math::
#    2^{L-1}m < \frac{N}{2} \implies L < 1 + \log_2\left(\frac{N}{2m}\right).
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# Below, we demonstrate an example of encoding the parameter :math:`m = 2.5` through frequency
# modulation of the NeRF-style embedding.

# %%

# Define a spatial domain and the number of frequencies
x = torch.linspace(0, 1, 1000)
L = 3

# Define the parameter to encode
m = 2.5
m_tensor = torch.tensor([m])

# Check if the number of frequencies and parameter satisfies the Nyquist-Criterion
if L <= 1 + torch.log2(torch.tensor(len(x) / (2 * m))):
    print(f"Nyquist-Shannon sampling theorem is satisfied for the given parameter {m} and number of frequencies {L}.")
else:
    print(f"Nyquist-Shannon sampling theorem is violated for the given parameter {m} and number of frequencies {L}.")

# Define the NeRF embedding
nerf_embedder = SinusoidalEmbedding(
    in_channels=1, num_frequencies=L, embedding_type="nerf"
).to(device)

# Apply frequency modulation: multiply input, with the domain [0, 1], by parameter before embedding
# This scales all frequencies by the parameter m
nerf_embedding = nerf_embedder(x.reshape(-1, 1) * m_tensor).permute(1, 0)

# Plot the embedding
colors = plt.cm.tab10.colors

plt.figure(figsize=(10, 5))
for freq_idx in range(L):
    color = colors[freq_idx % len(colors)]
    sin_idx = 2 * freq_idx
    cos_idx = 2 * freq_idx + 1

    plt.plot(x, nerf_embedding[sin_idx], color=color, label=f"Channel {freq_idx}")
    plt.plot(x, nerf_embedding[cos_idx], color=color)

plt.xlabel("x", fontsize=16)
plt.ylabel("Embedding", fontsize=16)
plt.title("NeRF-style embedding with modulated frequency", fontsize=18, pad=20)
plt.legend(loc="lower left", framealpha=1.0, fontsize=14)
plt.locator_params(axis="y", nbins=5)
plt.tight_layout()
plt.show()

# %%
# Similarly, we can encode the parameter :math:`m = 2.5` through amplitude modulation, where we show
# an example using the NeRF-style embedding below.

# %%

# Define a spatial domain and the number of frequencies
x = torch.linspace(0, 1, 1000)
L = 3

# Define the parameter to encode
m = 2.5
m_tensor = torch.tensor([m])

# Check if the number of frequencies and parameter satisfies the Nyquist-Criterion
if L <= 1 + torch.log2(torch.tensor(len(x) / 2)):
    print(f"Nyquist-Shannon sampling theorem is satisfied for the given number of frequencies {L}.")
else:
    print(f"Nyquist-Shannon sampling theorem is violated for the given number of frequencies {L}.")

# Define the embedding
nerf_embedder = SinusoidalEmbedding(
    in_channels=1, num_frequencies=L, embedding_type="nerf"
).to(device)

# Apply amplitude modulation: multiply embedding, with the domain [0, 1], by parameter after computation
# This scales all embedding components by the parameter m
nerf_embedding = nerf_embedder(x.reshape(-1, 1)).permute(1, 0) * m_tensor

# Plot the embedding
colors = plt.cm.tab10.colors

plt.figure(figsize=(10, 5))
for freq_idx in range(L):
    color = colors[freq_idx % len(colors)]
    sin_idx = 2 * freq_idx
    cos_idx = 2 * freq_idx + 1

    plt.plot(x, nerf_embedding[sin_idx], color=color, label=f"Channel {freq_idx}")
    plt.plot(x, nerf_embedding[cos_idx], color=color)

plt.xlabel("x", fontsize=16)
plt.ylabel("Embedding", fontsize=16)
plt.title("NeRF-style embedding with amplitude modulation", fontsize=18, pad=20)
plt.legend(loc="lower left", framealpha=1.0, fontsize=14)
plt.locator_params(axis="y", nbins=5)
plt.tight_layout()
plt.show()

# %%
# .. raw:: html
#
#    <div style="margin-top: 4em;"></div>
#
# Application to Fourier Neural Operators (FNOs)
# ----------------------------------------------
# Fourier Neural Operators (FNOs) learn mappings between functions by operating in the frequency domain.
# They use the Fourier transform to express data as combinations of sine and cosine components,
# enabling them to capture complex, multi-scale interactions across frequencies.
# Given that sinusoidal embeddings also lift low-dimensional data into a richer spectral
# representation, they complement FNOs naturally.
# This synergy makes sinusoidal embeddings particularly effective for neural operator architectures.
#
# In the general setting for neural operators, we strongly recommend choosing the number of frequencies
# :math:`L` such that the Nyquist-Criterion is not violated. This can be done by following the guidelines
# we provided earlier for selecting :math:`L` in both transformer-style and NeRF-style embeddings.
#
# When dealing with FNOs with a specified number of Fourier modes, :math:`\text{n_modes}`, the
# highest embedded frequency should ideally also remain below :math:`\text{n_modes}`,
# as higher frequencies will be zeroed out and not acted upon by the spectral convolution operation.
#
# For the NeRF-style embedding, this condition leads to an explicit upper bound on :math:`L`:
#
# .. math::
#    2^{L-1} < \text{n_modes} \ \ \implies \ \  L < 1 + \log_2\left(\text{n_modes}\right).

# %%
# .. raw:: html
#
#    <div style="margin-top: 4em;"></div>
#
# Setup in Higher Dimensions
# --------------------------
# Let :math:`X \in \mathbb{R}^{d \times N}` denote a :math:`d`-dimensional input domain consisting of
# :math:`N` discretised points, where each row :math:`\vec{x}_{i} \in \mathbb{R}^N` corresponds to
# the sampled coordinates along the :math:`i`-th spatial or temporal dimension. Thus, each column
# of :math:`X` represents a single point :math:`\vec{x}_{:,j} \in \mathbb{R}^d` in the
# :math:`d`-dimensional domain.
#
# Building on the 1D embedding function :math:`g` introduced earlier, we define the
# multi-dimensional embedding
#
# .. math::
#    h: \mathbb{R}^{d \times N} \ \  \rightarrow \ \ \mathbb{R}^{N \times 2 L d}, \quad h(X)=\operatorname{concat}\left(g\left(\vec{x}_1\right), g\left(\vec{x}_2\right), \ldots, g\left(\vec{x}_d\right)\right),
#
# where each :math:`\vec{x}_i` denotes the sampled domain along the :math:`i`-th input dimension.
#
# The multi-dimensional embedding function :math:`h` applies the 1D embedding function :math:`g`
# independently to each coordinate dimension and concatenates the resulting embeddings
# along the feature axis. This approach allows the model to capture
# frequency patterns along each dimension separately while maintaining the overall structure.
#
# .. raw:: html
#
#    <div style="margin-top: 2em;"></div>
#
# Below, we include an example of using the `SinusoidalEmbedding` class to construct both
# transformer- and NeRF-style embeddings for a 3D input.

# %%

# Define a 1D spatial domain and construct 3D input by repeating the 1D domain
dim = 3
x_1d = torch.linspace(0, 1, 1000)
# For transformer: normalize to [0, 2π] and repeat for 3D input, shape (N, 3)
x_normalized_1d = torch.linspace(0, 2 * torch.pi, x_1d.size(0), device=x_1d.device).unsqueeze(1)
x_normalized = x_normalized_1d.repeat(1, dim)
# For NeRF: coordinates in [0, 1] per dimension, shape (N, 3)
x_3d = x_1d.unsqueeze(1).repeat(1, dim)

# Define the number of frequencies
L = 3

# Check if the number of frequencies satisfies the Nyquist-Criterion
# For multi-dimensional inputs, the constraint applies to each dimension independently
if L <= 1 + torch.log2(torch.tensor(len(x_3d) / 2)):
    print(f"Nyquist-Shannon sampling theorem is satisfied for the given number of frequencies {L}.")
else:
    print(f"Nyquist-Shannon sampling theorem is violated for the given number of frequencies {L}.")

# Transformer-style 3D embedding: use normalized coordinates [0, 2π]
max_positions = 1000
transformer_embedder = SinusoidalEmbedding(
    in_channels=dim,
    num_frequencies=L,
    embedding_type="transformer",
    max_positions=max_positions,
).to(device)
transformer_embedding = transformer_embedder(x_normalized).permute(1, 0)

# NeRF-style 3D embedding: use coordinates in [0, 1] per dimension
nerf_embedder = SinusoidalEmbedding(
    in_channels=dim, num_frequencies=L, embedding_type="nerf"
).to(device)
nerf_embedding = nerf_embedder(x_3d).permute(1, 0)