Real-space losses

Real-space losses measure the difference between functions sampled on matching grids. Uniform-grid losses weight each point by its volume element; quadrature losses use the supplied integration weights.

The losses return integrated absolute or squared errors for each sample and sum over additional value components. Mean reduction averages these sample integrals without dividing by volume, total quadrature weight, or component count. The corresponding physical objects are described under Grids and volumetric quantities.

Uniform grids

uniform_volumetric_mae_loss

uniform_volumetric_mae_loss(
    input: UniformVolumetricData,
    target: UniformVolumetricData,
    *,
    reduction: _Reduction = "mean",
) -> Tensor

Return the integrated L1 error on each uniform grid.

For sample \(b\),

\[ \ell_b = \Delta V^{(b)}\sum_{p,c} \left|\widehat x^{(b)}_{pc}-x^{(b)}_{pc}\right|, \qquad \Delta V^{(b)}=\left|\det\mathsf H^{(b)}\right|. \]

The rows of \(\mathsf H^{(b)}\) are the grid step vectors. Additional value components are summed as part of the L1 norm. reduction="mean" averages the B integrated sample losses; it does not divide by grid volume or component count.

Parameters:

input (UniformVolumetricData) –

values_real: [sum(num_values), *extra_shape]. num_values: [B]. step_vectors: [B, 3, 3].
target (UniformVolumetricData) –

Target data with the same grid layout and field shapes as input.
reduction (_Reduction, default: 'mean' ) –

"none" returns integrated sample losses with shape [B]; "mean" averages them; "sum" sums them.

Returns:

Tensor –

Loss tensor with the shape determined by reduction.

uniform_volumetric_mse_loss

uniform_volumetric_mse_loss(
    input: UniformVolumetricData,
    target: UniformVolumetricData,
    *,
    reduction: _Reduction = "mean",
) -> Tensor

Return the integrated squared L2 error on each uniform grid.

For sample \(b\),

\[ \ell_b = \Delta V^{(b)}\sum_{p,c} \left(\widehat x^{(b)}_{pc}-x^{(b)}_{pc}\right)^2, \qquad \Delta V^{(b)}=\left|\det\mathsf H^{(b)}\right|. \]

The rows of \(\mathsf H^{(b)}\) are the grid step vectors. Additional value components are summed as part of the squared L2 norm. reduction="mean" averages the B integrated sample losses; it does not divide by grid volume or component count. No factor of \(1/2\) is applied.

Parameters:

input (UniformVolumetricData) –

values_real: [sum(num_values), *extra_shape]. num_values: [B]. step_vectors: [B, 3, 3].
target (UniformVolumetricData) –

Target data with the same grid layout and field shapes as input.
reduction (_Reduction, default: 'mean' ) –

"none" returns integrated sample losses with shape [B]; "mean" averages them; "sum" sums them.

Returns:

Tensor –

Loss tensor with the shape determined by reduction.

Quadrature grids

quadrature_volumetric_mae_loss

quadrature_volumetric_mae_loss(
    input: QuadratureVolumetricData,
    target: QuadratureVolumetricData,
    *,
    reduction: _Reduction = "mean",
) -> Tensor

Return the quadrature-integrated L1 error of each sample.

For sample \(b\),

\[ \ell_b = \sum_{p,c}w^{(b)}_p \left|\widehat x^{(b)}_{pc}-x^{(b)}_{pc}\right|. \]

The quadrature weights are taken from input and should be nonnegative. Additional value components are summed as part of the L1 norm. reduction="mean" averages the B integrated sample losses; it does not divide by the sum of weights or component count.

Parameters:

input (QuadratureVolumetricData) –

values_real: [sum(num_values), *extra_shape]. num_values: [B]. weights: [sum(num_values)].
target (QuadratureVolumetricData) –

Target data with the same grid layout and field shapes as input.
reduction (_Reduction, default: 'mean' ) –

"none" returns integrated sample losses with shape [B]; "mean" averages them; "sum" sums them.

Returns:

Tensor –

Loss tensor with the shape determined by reduction.

quadrature_volumetric_mse_loss

quadrature_volumetric_mse_loss(
    input: QuadratureVolumetricData,
    target: QuadratureVolumetricData,
    *,
    reduction: _Reduction = "mean",
) -> Tensor

Return the quadrature-integrated squared L2 error of each sample.

For sample \(b\),

\[ \ell_b = \sum_{p,c}w^{(b)}_p \left(\widehat x^{(b)}_{pc}-x^{(b)}_{pc}\right)^2. \]

The quadrature weights are taken from input and should be nonnegative. Additional value components are summed as part of the squared L2 norm. reduction="mean" averages the B integrated sample losses; it does not divide by the sum of weights or component count. No factor of \(1/2\) is applied.

Parameters:

input (QuadratureVolumetricData) –

values_real: [sum(num_values), *extra_shape]. num_values: [B]. weights: [sum(num_values)].
target (QuadratureVolumetricData) –

Target data with the same grid layout and field shapes as input.
reduction (_Reduction, default: 'mean' ) –

"none" returns integrated sample losses with shape [B]; "mean" averages them; "sum" sums them.

Returns:

Tensor –

Loss tensor with the shape determined by reduction.