Spectral and physics-oriented metrics¶

These metrics evaluate whether a surrogate preserves physically important structure rather than only minimizing pointwise error.

Spectral relative error¶

Let \(\mathcal F\) denote a spatial Fourier transform:

\[ E_{\mathrm{spec}}= \frac{\|\,|\mathcal F(\hat y)|-|\mathcal F(y)|\,\|_2} {\|\,|\mathcal F(y)|\,\|_2+\varepsilon}. \]

This compares Fourier magnitudes globally. It does not localize errors to specific frequency bands.

Binned spectral MSE¶

For a wavenumber bin \(B\):

\[ \mathrm{SMSE}(B)= \frac{1}{N} \sum_{\boldsymbol\omega\in B} \left|\mathcal F(\hat y-y)(\boldsymbol\omega)\right|^2. \]

NAVIER-CFD uses an orthonormal FFT and radial low/middle/high bins by default. When all bins cover the spectrum, their sum matches spatial-domain MSE up to numerical precision.

Fourier RMSE (fRMSE)¶

Following the RealPDEBench interpretation, the transform can include time and space. For frequency bin \(B\):

\[ \mathrm{fRMSE}(B)= \sqrt{ \frac{1}{K|B|} \sum_{k=1}^{K} \sum_{\boldsymbol\omega\in B} \left\|\hat{\mathbf Y}_k(\boldsymbol\omega)-\mathbf Y_k(\boldsymbol\omega)\right\|_2^2 }. \]

Reports low-, middle-, and high-frequency errors.
Lower is better; \(0\) is perfect.
FFT axes, normalization and bin edges must be recorded.

Temporal frequency error¶

Let the spatially integrated signal be:

\[ s_k(t)=\sum_i y_k(t,\mathbf x_i). \]

Then:

\[ \mathrm{FE}= \frac{1}{KT} \sum_{k=1}^{K} \sum_{\omega} \left| \mathcal F_t(s_k)(\omega)- \mathcal F_t(\hat s_k)(\omega) \right|. \]

FE is useful for vortex shedding, oscillatory reactors and other periodic dynamics.

Divergence error¶

For velocity \(\mathbf u=(u_1,\ldots,u_d)\):

\[ \nabla\cdot\mathbf u= \sum_{j=1}^{d}\frac{\partial u_j}{\partial x_j}, \qquad D_{\mathrm{RMS}}= \sqrt{\mathbb E[(\nabla\cdot\mathbf u)^2]}. \]

NAVIER-CFD reports:

\[ E_{\mathrm{div}}= \left|D_{\mathrm{RMS}}(\hat{\mathbf u})- D_{\mathrm{RMS}}(\mathbf u)\right|. \]

This requires velocity-channel and grid-spacing metadata. It is not a substitute for checking local divergence fields or discrete solver conservation.

Kinetic-energy relative error¶

For constant-density flow:

\[ e_k=\frac{1}{2}\sum_{j=1}^{d}u_j^2. \]

For variable density:

\[ e_k=\frac{1}{2}\rho\sum_{j=1}^{d}u_j^2. \]

The relative field error is:

\[ E_{\mathrm{KE}}= \frac{\|\hat e_k-e_k\|_2}{\|e_k\|_2+\varepsilon}. \]

Turbulent kinetic-energy error¶

For two-dimensional velocity fluctuations:

\[ k=\frac{1}{2} \left( \overline{(u-\bar u)^2}+ \overline{(v-\bar v)^2} \right). \]

NAVIER-CFD computes the mean absolute difference between predicted and target fluctuation-energy fields. This follows the RealPDEBench-style long-time consistency interpretation.

Mean velocity profile error¶

At selected probes \(\mathbf x_j\):

\[ \mathrm{MVPE}= \frac{1}{KN_p} \sum_{k=1}^{K}\sum_{j=1}^{N_p} \left| \bar u_k(\mathbf x_j)- \bar{\hat u}_k(\mathbf x_j) \right|. \]

When probes are not supplied, NAVIER-CFD can average over all non-profile spatial axes and compare full mean profiles. The selected profile axis and probe coordinates must be reported.

Vorticity RMSE¶

For two-dimensional flow:

\[ \omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}, \qquad E_\omega= \sqrt{\mathbb E[(\hat\omega-\omega)^2]}. \]

This metric is sensitive to gradients and small-scale rotational structure.

Required context¶

context = MetricContext(
    time_axis=1,
    spatial_axes=(2, 3),
    channel_axis=-1,
    velocity_channels=(0, 1),
    spacing=(dx, dy),
    profile_axis=2,
    evaluation_space="physical",
)

Metrics that cannot be supported by the supplied context are returned with valid=False and an explanation.