List of Spectral Convolutions

All spectral convolutions perform the same computation for a signal of different dimensions.

Formula

Using the notation from Li et al. [40], the spectral convolution is defined by

\[SC(x) = \mathcal{F}^{-1}( R ( \mathcal{F}(x) ) )\]

Note that these functions do not construct \(R\), allowing them to be used in both the deterministic and Bayesian cases.

Spectral Conv 1d

The function spectral_conv1d() is imported using the following command:

>>> from UQpy.scientific_machine_learning.functional import spectral_conv1d
spectral_conv1d(x, weights, out_channels, modes)[source]

Compute the 1d spectral convolution \(\mathcal{F}^{-1}(R (\mathcal{F}x) )\)

Parameters:

  • x (Tensor) – Tensor of shape \((N, C_\text{in}, L)\)

  • weights (Tensor) – Tensor of shape \((C_\text{in}, C_\text{out}, \text{modes})\). Weight tensor must have complex entries.

  • out_channels (int) – \(C_\text{out}\), Number of channels in the output signal

  • modes (int) – Number of Fourier modes to keep, at most \(\lfloor L / 2 \rfloor + 1\)

Return type:

Tensor

Returns:

Tensor \(\mathcal{F}^{-1}(R (\mathcal{F}x) )\) of shape \((N, C_\text{out}, L)\)

Spectral Conv 2d

The function spectral_conv2d() is imported using the following command:

>>> from UQpy.scientific_machine_learning.functional import spectral_conv2d
spectral_conv2d(x, weights, out_channels, modes)[source]

Compute the 2d spectral convolution \(\mathcal{F}^{-1}(R (\mathcal{F}x) )\)

Parameters:

  • x (Tensor) – Tensor of shape \((N, C_\text{in}, H, W)\)

  • weights (Tensor) – Tensors of shape \((2, C_\text{in}, C_\text{out}, \text{modes[0]}, \text{modes[1]})\). Tensors must have complex entries.

  • out_channels (int) – \(C_\text{out}\), Number of channels in the output signal

  • modes (tuple[int, int]) – Tuple of Fourier modes to keep. At most \((\lfloor H / 2 \rfloor + 1, \lfloor W / 2 \rfloor + 1)\)

Return type:

Tensor

Returns:

Tensor \(\mathcal{F}^{-1}(R (\mathcal{F}x) )\) of shape \((N, C_\text{out}, H, W)\)

Spectral Conv 3d

The function spectral_conv3d() is imported using the following command:

>>> from UQpy.scientific_machine_learning.functional import spectral_conv3d
spectral_conv3d(x, weights, out_channels, modes)[source]

Compute the 3d spectral convolution \(\mathcal{F}^{-1}(R (\mathcal{F}x) )\)

Parameters:

  • x (Tensor) – Tensor of shape \((N, C_\text{in}, D, H, W)\)

  • weights (Tensor) – Tensor of shape \((4, C_\text{in}, C_\text{out}, \text{modes[0]}, \text{modes[1]}, \text{modes[2]})\). Must have complex entries.

  • modes (tuple[int, int, int]) – Tuple of Fourier modes to keep. At most \((\lfloor D / 2 \rfloor + 1, \lfloor H / 2 \rfloor + 1, \lfloor W / 2 \rfloor + 1)\)

  • out_channels (int) – \(C_\text{out}\), Number of channels in the output signal

Return type:

Tensor

Returns:

Tensor \(\mathcal{F}^{-1}(R (\mathcal{F}x) )\) of shape \((N, C_\text{out}, D, H, W)\)