neuralop.losses.differentiation.FourierDiff

class neuralop.losses.differentiation.FourierDiff(dim, L=None, use_fc=False, fc_degree=4, fc_n_additional_pts=50, low_pass_filter_ratio=None)[source]

A unified class for computing Fourier/spectral derivatives in 1D, 2D, 3D.

This class provides comprehensive methods for computing derivatives using Fourier/spectral methods with support for both periodic and non-periodic functions through Fourier continuation: - Periodic functions: Direct Fourier differentiation using FFT - Non-periodic functions: Fourier continuation (FC) is used to extend functions to a larger

domain on which the functions are periodic before applying Fourier differentiation with FFT.

The class also provides gradient, divergence, curl, and Laplacian operations.

Parameters:

dimint

Dimension of the input field. Must be 1, 2, or 3.

Lfloat or tuple, optional

Length of the domain for Fourier differentiation. By default 2*pi for each dimension.

use_fcstr, optional

Whether to use Fourier continuation for non-periodic functions. Options: False (no FC), ‘Legendre’, ‘Gram’. By default False.

fc_degreeint, optional

Degree of the Fourier continuation polynomial matching. This is the number of matching points on the left and right boundaries used for the Fourier continuation procedure. By default 4.

fc_n_additional_ptsint, optional

Number of additional points to add with the Fourier continuation layer. This extends the domain to handle non-periodic functions. By default 50.

low_pass_filter_ratiofloat, optional

If not None, apply a low-pass filter to the Fourier coefficients to reduce high-frequency noise. Should be between 0 and 1. By default None.

Available Methods

—————-

Derivative Methods:

- dx(u, order=1): Compute derivative with respect to x

- dy(u, order=1): Compute derivative with respect to y (2D/3D only)

- dz(u, order=1): Compute derivative with respect to z (3D only)

- derivative(u, order): Compute derivative with order tuple (e.g., (1,0) for ∂/∂x)

Vector Calculus Operators:

- laplacian(u): Compute the Laplacian ∇²f

- gradient(u): Compute the gradient ∇f (returns vector field)

- divergence(u): Compute the divergence ∇·u (for vector fields)

- curl(u): Compute the curl ∇×u (for vector fields, 2D/3D only)

Methods

compute_multiple_derivatives(u, derivatives)	Compute multiple derivatives in a single FFT/IFFT call for better performance.
curl(u)	Compute the curl ∇×u (for vector fields, 2D/3D only).
derivative(u, order)	Compute Fourier derivative of a given tensor.
divergence(u)	Compute the divergence ∇·u (for vector fields).
dx(u[, order])	Compute derivative with respect to x.
dy(u[, order])	Compute derivative with respect to y (2D/3D only).
dz(u[, order])	Compute derivative with respect to z (3D only).
gradient(u)	Compute the gradient ∇f (returns vector field).
laplacian(u)	Compute the Laplacian ∇²f.
partial(u[, direction, order])	Compute partial Fourier derivative along a specific direction.

Examples

>>> # 1D Fourier derivatives
>>> x = torch.linspace(0, 2*torch.pi, 100)
>>> u = torch.sin(x)
>>> fd1d = FourierDiff(dim=1, L=2*torch.pi, use_fc=False)
>>> du_dx = fd1d.dx(u)  # First derivative
>>> d2u_dx2 = fd1d.dx(u, order=2)  # Second derivative
>>>
>>> # 2D Fourier derivatives
>>> fd2d = FourierDiff(dim=2, L=(2*torch.pi, 2*torch.pi), use_fc=False)
>>> x = torch.linspace(0, 2*torch.pi, 50)
>>> y = torch.linspace(0, 2*torch.pi, 50)
>>> X, Y = torch.meshgrid(x, y, indexing='ij')
>>> u = torch.sin(X) * torch.cos(Y)
>>> du_dx = fd2d.dx(u)
>>> du_dy = fd2d.dy(u)
>>> grad = fd2d.gradient(u)  # Returns [du_dx, du_dy]
>>>
>>> # 3D Fourier derivatives
>>> fd3d = FourierDiff(dim=3, L=(2*torch.pi, 2*torch.pi, 2*torch.pi), use_fc=False)
>>> x = torch.linspace(0, 2*torch.pi, 20)
>>> y = torch.linspace(0, 2*torch.pi, 20)
>>> z = torch.linspace(0, 2*torch.pi, 20)
>>> X, Y, Z = torch.meshgrid(x, y, z, indexing='ij')
>>> u = torch.sin(X) * torch.cos(Y) * torch.sin(Z)  # 3D scalar field
>>> du_dx = fd3d.dx(u)
>>> du_dy = fd3d.dy(u)
>>> du_dz = fd3d.dz(u)
>>> laplacian = fd3d.laplacian(u)
>>>
>>> # Vector field operations
>>> vx = torch.sin(X) * torch.cos(Y) * torch.sin(Z)
>>> vy = torch.cos(X) * torch.sin(Y) * torch.cos(Z)
>>> vz = torch.sin(X) * torch.sin(Y) * torch.cos(Z)
>>> v = torch.stack([vx, vy, vz], dim=-4)
>>> div_v = fd3d.divergence(v)
>>> curl_v = fd3d.curl(v)

compute_multiple_derivatives(u, derivatives)[source]

Compute multiple derivatives in a single FFT/IFFT call for better performance.

Parameters:

utorch.Tensor

Input tensor.

derivativeslist

List of derivative specifications: - 1D: list of int (orders) - 2D: list of tuples (order_x, order_y) - 3D: list of tuples (order_x, order_y, order_z)

Returns:

list of torch.Tensor

List of computed derivatives in the same order as derivatives input

derivative(u, order)[source]

Compute Fourier derivative of a given tensor.

Parameters:

utorch.Tensor

Input tensor

ordertuple

Derivative orders: - 1D: (order_x,) - 2D: (order_x, order_y) - 3D: (order_x, order_y, order_z)

Returns:

torch.Tensor

The derivative of the input tensor

partial(u, direction='x', order=1)[source]

Compute partial Fourier derivative along a specific direction.

Parameters:

utorch.Tensor

Input tensor

directionstr, optional

Direction along which to compute the derivative, by default ‘x’ Options: ‘x’, ‘y’ (2D/3D only), ‘z’ (3D only)

orderint, optional

Order of the derivative, by default 1

Returns:

torch.Tensor

The partial derivative of the input tensor

dx(u, order=1)[source]

Compute derivative with respect to x.

dy(u, order=1)[source]

Compute derivative with respect to y (2D/3D only).

dz(u, order=1)[source]

Compute derivative with respect to z (3D only).

laplacian(u)[source]

Compute the Laplacian ∇²f.

gradient(u)[source]

Compute the gradient ∇f (returns vector field).

divergence(u)[source]

Compute the divergence ∇·u (for vector fields).

curl(u)[source]

Compute the curl ∇×u (for vector fields, 2D/3D only).