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Derivative-based Metrics¤

Related to Sobolev Norms

exponax.metrics.H1_MSE ¤

H1_MSE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Optional[Float[Array, "C ... N"]] = None,
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None
) -> float

Compute the mean squared error associated with the H1 norm, i.e., the MSE across state and all its first derivatives.

Given the correct domain_extent, this is consistent with the squared norm in the H1 Sobolev space H^1 = W^(1,2): https://en.wikipedia.org/wiki/Sobolev_space#The_case_p_=_2

Tip

To apply this function to a state tensor with a leading batch axis, use jax.vmap. Then the batch axis can be reduced, e.g., by jnp.mean. As a helper for this, exponax.metrics.mean_metric is provided.

Warning

Not supplying domain_extent will have the result be orders of magnitude different.

Arguments:

  • u_pred: The state array. Must follow the Exponax convention with a leading channel axis, and either one, two, or three subsequent spatial axes.
  • u_ref: The reference state array. Must have the same shape as u_pred. If not specified, the MSE is computed against zero, i.e., the norm of u_pred.
  • domain_extent: The extent L of the domain Ω = (0, L)ᴰ.
  • low: The lower cutoff (inclusive) frequency for filtering. If not specified, it is set to 0, meaning start it starts (including) the mean/zero mode.
  • high: The upper cutoff (inclusive) frequency for filtering. If not specified, it is set to N//2 + 1, meaning it ends (including) at the Nyquist mode.
Source code in exponax/metrics/_derivative.py 
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def H1_MSE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Optional[Float[Array, "C ... N"]] = None,
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None,
) -> float:
    """
    Compute the mean squared error associated with the H1 norm, i.e., the MSE
    across state and all its first derivatives.

    Given the correct `domain_extent`, this is consistent with the squared norm
    in the H1 Sobolev space H^1 = W^(1,2):
    https://en.wikipedia.org/wiki/Sobolev_space#The_case_p_=_2

    !!! tip
        To apply this function to a state tensor with a leading batch axis, use
        `jax.vmap`. Then the batch axis can be reduced, e.g., by `jnp.mean`. As
        a helper for this, [`exponax.metrics.mean_metric`][] is provided.

    !!! warning
        Not supplying `domain_extent` will have the result be orders of
        magnitude different.

    **Arguments**:

    - `u_pred`: The state array. Must follow the `Exponax` convention
        with a leading channel axis, and either one, two, or three subsequent
        spatial axes.
    - `u_ref`: The reference state array. Must have the same shape as
        `u_pred`. If not specified, the MSE is computed against zero, i.e., the
        norm of `u_pred`.
    - `domain_extent`: The extent `L` of the domain `Ω = (0, L)ᴰ`.
    - `low`: The lower cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `0`, meaning start it starts (including) the
        mean/zero mode.
    - `high`: The upper cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `N//2 + 1`, meaning it ends (including) at the
        Nyquist mode.
    """
    regular_mse = fourier_MSE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=None,
    )
    first_derivative_mse = fourier_MSE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=1,
    )
    return regular_mse + first_derivative_mse

exponax.metrics.H1_MAE ¤

H1_MAE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Optional[Float[Array, "C ... N"]] = None,
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None
) -> float

Compute the mean abolute error associated with the H1 norm, i.e., the MAE across state and all its first derivatives.

This is not consistent with the H1 norm because it uses a Fourier-based approach.

Tip

To apply this function to a state tensor with a leading batch axis, use jax.vmap. Then the batch axis can be reduced, e.g., by jnp.mean. As a helper for this, exponax.metrics.mean_metric is provided.

Warning

Not supplying domain_extent will have the result be orders of magnitude different.

Arguments:

  • u_pred: The state array. Must follow the Exponax convention with a leading channel axis, and either one, two, or three subsequent spatial axes.
  • u_ref: The reference state array. Must have the same shape as u_pred. If not specified, the MAE is computed against zero, i.e., the norm of u_pred.
  • domain_extent: The extent L of the domain Ω = (0, L)ᴰ.
  • low: The lower cutoff (inclusive) frequency for filtering. If not specified, it is set to 0, meaning start it starts (including) the mean/zero mode.
  • high: The upper cutoff (inclusive) frequency for filtering. If not specified, it is set to N//2 + 1, meaning it ends (including) at the Nyquist mode.
Source code in exponax/metrics/_derivative.py 
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def H1_MAE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Optional[Float[Array, "C ... N"]] = None,
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None,
) -> float:
    """
    Compute the mean abolute error associated with the H1 norm, i.e., the MAE
    across state and all its first derivatives.

    This is **not** consistent with the H1 norm because it uses a Fourier-based
    approach.

    !!! tip
        To apply this function to a state tensor with a leading batch axis, use
        `jax.vmap`. Then the batch axis can be reduced, e.g., by `jnp.mean`. As
        a helper for this, [`exponax.metrics.mean_metric`][] is provided.

    !!! warning
        Not supplying `domain_extent` will have the result be orders of
        magnitude different.


    **Arguments**:

    - `u_pred`: The state array. Must follow the `Exponax` convention
        with a leading channel axis, and either one, two, or three subsequent
        spatial axes.
    - `u_ref`: The reference state array. Must have the same shape as
        `u_pred`. If not specified, the MAE is computed against zero, i.e., the
        norm of `u_pred`.
    - `domain_extent`: The extent `L` of the domain `Ω = (0, L)ᴰ`.
    - `low`: The lower cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `0`, meaning start it starts (including) the
        mean/zero mode.
    - `high`: The upper cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `N//2 + 1`, meaning it ends (including) at the
        Nyquist mode.
    """
    regular_mae = fourier_MAE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=None,
    )
    first_derivative_mae = fourier_MAE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=1,
    )
    return regular_mae + first_derivative_mae

exponax.metrics.H1_RMSE ¤

H1_RMSE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Optional[Float[Array, "C ... N"]] = None,
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None
) -> float

Compute the root mean squared error associated with the H1 norm, i.e., the RMSE across state and all its first derivatives.

Given the correct domain_extent, this is consistent with the norm in the H1 Sobolev space H^1 = W^(1,2): https://en.wikipedia.org/wiki/Sobolev_space#The_case_p_=_2

Tip

To apply this function to a state tensor with a leading batch axis, use jax.vmap. Then the batch axis can be reduced, e.g., by jnp.mean. As a helper for this, exponax.metrics.mean_metric is provided.

Warning

Not supplying domain_extent will have the result be orders of magnitude different.

Arguments:

  • u_pred: The state array. Must follow the Exponax convention with a leading channel axis, and either one, two, or three subsequent spatial axes.
  • u_ref: The reference state array. Must have the same shape as u_pred. If not specified, the RMSE is computed against zero, i.e., the norm of u_pred.
  • domain_extent: The extent L of the domain Ω = (0, L)ᴰ.
  • low: The lower cutoff (inclusive) frequency for filtering. If not specified, it is set to 0, meaning start it starts (including) the mean/zero mode.
  • high: The upper cutoff (inclusive) frequency for filtering. If not specified, it is set to N//2 + 1, meaning it ends (including) at the Nyquist mode.
Source code in exponax/metrics/_derivative.py 
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def H1_RMSE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Optional[Float[Array, "C ... N"]] = None,
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None,
) -> float:
    """
    Compute the root mean squared error associated with the H1 norm, i.e., the
    RMSE across state and all its first derivatives.

    Given the correct `domain_extent`, this is consistent with the norm in the
    H1 Sobolev space H^1 = W^(1,2):
    https://en.wikipedia.org/wiki/Sobolev_space#The_case_p_=_2

    !!! tip
        To apply this function to a state tensor with a leading batch axis, use
        `jax.vmap`. Then the batch axis can be reduced, e.g., by `jnp.mean`. As
        a helper for this, [`exponax.metrics.mean_metric`][] is provided.

    !!! warning
        Not supplying `domain_extent` will have the result be orders of
        magnitude different.

    **Arguments**:

    - `u_pred`: The state array. Must follow the `Exponax` convention
        with a leading channel axis, and either one, two, or three subsequent
        spatial axes.
    - `u_ref`: The reference state array. Must have the same shape as
        `u_pred`. If not specified, the RMSE is computed against zero, i.e., the
        norm of `u_pred`.
    - `domain_extent`: The extent `L` of the domain `Ω = (0, L)ᴰ`.
    - `low`: The lower cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `0`, meaning start it starts (including) the
        mean/zero mode.
    - `high`: The upper cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `N//2 + 1`, meaning it ends (including) at the
        Nyquist mode.
    """
    regular_rmse = fourier_RMSE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=None,
    )
    first_derivative_rmse = fourier_RMSE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=1,
    )
    return regular_rmse + first_derivative_rmse

exponax.metrics.H1_nMSE ¤

H1_nMSE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Float[Array, "C ... N"],
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None
) -> float

Compute the normalized mean squared error associated with the H1 norm, i.e., the nMSE across state and all its first derivatives.

Given the correct domain_extent, this is consistent with the relative squared norm in the H1 Sobolev space H^1 = W^(1,2): https://en.wikipedia.org/wiki/Sobolev_space#The_case_p_=_2

Tip

To apply this function to a state tensor with a leading batch axis, use jax.vmap. Then the batch axis can be reduced, e.g., by jnp.mean. As a helper for this, exponax.metrics.mean_metric is provided.

Warning

Not supplying domain_extent will have the result be orders of magnitude different.

Arguments:

  • u_pred: The state array. Must follow the Exponax convention with a leading channel axis, and either one, two, or three subsequent spatial axes.
  • u_ref: The reference state array. Must have the same shape as u_pred.
  • domain_extent: The extent L of the domain Ω = (0, L)ᴰ.
  • low: The lower cutoff (inclusive) frequency for filtering. If not specified, it is set to 0, meaning start it starts (including) the mean/zero mode.
  • high: The upper cutoff (inclusive) frequency for filtering. If not specified, it is set to N//2 + 1, meaning it ends (including) at the Nyquist mode.
Source code in exponax/metrics/_derivative.py 
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def H1_nMSE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Float[Array, "C ... N"],
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None,
) -> float:
    """
    Compute the normalized mean squared error associated with the H1 norm, i.e.,
    the nMSE across state and all its first derivatives.

    Given the correct `domain_extent`, this is consistent with the **relative**
    squared norm in the H1 Sobolev space H^1 = W^(1,2):
    https://en.wikipedia.org/wiki/Sobolev_space#The_case_p_=_2

    !!! tip
        To apply this function to a state tensor with a leading batch axis, use
        `jax.vmap`. Then the batch axis can be reduced, e.g., by `jnp.mean`. As
        a helper for this, [`exponax.metrics.mean_metric`][] is provided.

    !!! warning
        Not supplying `domain_extent` will have the result be orders of
        magnitude different.

    **Arguments**:

    - `u_pred`: The state array. Must follow the `Exponax` convention
        with a leading channel axis, and either one, two, or three subsequent
        spatial axes.
    - `u_ref`: The reference state array. Must have the same shape as
        `u_pred`.
    - `domain_extent`: The extent `L` of the domain `Ω = (0, L)ᴰ`.
    - `low`: The lower cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `0`, meaning start it starts (including) the
        mean/zero mode.
    - `high`: The upper cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `N//2 + 1`, meaning it ends (including) at the
        Nyquist mode.
    """
    regular_nmse = fourier_nMSE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=None,
    )
    first_derivative_nmse = fourier_nMSE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=1,
    )
    return regular_nmse + first_derivative_nmse

exponax.metrics.H1_nMAE ¤

H1_nMAE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Float[Array, "C ... N"],
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None
) -> float

Compute the normalized mean abolute error associated with the H1 norm, i.e., the nMAE across state and all its first derivatives.

This is not consistent with the H1 norm because it uses a Fourier-based approach.

Tip

To apply this function to a state tensor with a leading batch axis, use jax.vmap. Then the batch axis can be reduced, e.g., by jnp.mean. As a helper for this, exponax.metrics.mean_metric is provided.

Warning

Not supplying domain_extent will have the result be orders of magnitude different.

Arguments:

  • u_pred: The state array. Must follow the Exponax convention with a leading channel axis, and either one, two, or three subsequent spatial axes.
  • u_ref: The reference state array. Must have the same shape as u_pred.
  • domain_extent: The extent L of the domain Ω = (0, L)ᴰ.
  • low: The lower cutoff (inclusive) frequency for filtering. If not specified, it is set to 0, meaning start it starts (including) the mean/zero mode.
  • high: The upper cutoff (inclusive) frequency for filtering. If not specified, it is set to N//2 + 1, meaning it ends (including) at the Nyquist mode.
Source code in exponax/metrics/_derivative.py 
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def H1_nMAE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Float[Array, "C ... N"],
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None,
) -> float:
    """
    Compute the normalized mean abolute error associated with the H1 norm, i.e.,
    the nMAE across state and all its first derivatives.

    This is **not** consistent with the H1 norm because it uses a Fourier-based
    approach.

    !!! tip
        To apply this function to a state tensor with a leading batch axis, use
        `jax.vmap`. Then the batch axis can be reduced, e.g., by `jnp.mean`. As
        a helper for this, [`exponax.metrics.mean_metric`][] is provided.

    !!! warning
        Not supplying `domain_extent` will have the result be orders of
        magnitude different.

    **Arguments**:

    - `u_pred`: The state array. Must follow the `Exponax` convention
        with a leading channel axis, and either one, two, or three subsequent
        spatial axes.
    - `u_ref`: The reference state array. Must have the same shape as
        `u_pred`.
    - `domain_extent`: The extent `L` of the domain `Ω = (0, L)ᴰ`.
    - `low`: The lower cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `0`, meaning start it starts (including) the
        mean/zero mode.
    - `high`: The upper cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `N//2 + 1`, meaning it ends (including) at the
        Nyquist mode.
    """
    regular_nmae = fourier_nMAE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=None,
    )
    first_derivative_nmae = fourier_nMAE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=1,
    )
    return regular_nmae + first_derivative_nmae

exponax.metrics.H1_nRMSE ¤

H1_nRMSE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Float[Array, "C ... N"],
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None
) -> float

Compute the normalized root mean squared error associated with the H1 norm, i.e., the nRMSE across state and all its first derivatives.

Given the correct domain_extent, this is consistent with the relative norm in the H1 Sobolev space H^1 = W^(1,2): https://en.wikipedia.org/wiki/Sobolev_space#The_case_p_=_2

Tip

To apply this function to a state tensor with a leading batch axis, use jax.vmap. Then the batch axis can be reduced, e.g., by jnp.mean. As a helper for this, exponax.metrics.mean_metric is provided.

Warning

Not supplying domain_extent will have the result be orders of magnitude different.

Arguments:

  • u_pred: The state array. Must follow the Exponax convention with a leading channel axis, and either one, two, or three subsequent spatial axes.
  • u_ref: The reference state array. Must have the same shape as u_pred.
  • domain_extent: The extent L of the domain Ω = (0, L)ᴰ.
  • low: The lower cutoff (inclusive) frequency for filtering. If not specified, it is set to 0, meaning start it starts (including) the mean/zero mode.
  • high: The upper cutoff (inclusive) frequency for filtering. If not specified, it is set to N//2 + 1, meaning it ends (including) at the Nyquist mode.
Source code in exponax/metrics/_derivative.py 
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def H1_nRMSE(
    u_pred: Float[Array, "C ... N"],
    u_ref: Float[Array, "C ... N"],
    *,
    domain_extent: float = 1.0,
    low: Optional[int] = None,
    high: Optional[int] = None,
) -> float:
    """
    Compute the normalized root mean squared error associated with the H1 norm,
    i.e., the nRMSE across state and all its first derivatives.

    Given the correct `domain_extent`, this is consistent with the **relative**
    norm in the H1 Sobolev space H^1 = W^(1,2):
    https://en.wikipedia.org/wiki/Sobolev_space#The_case_p_=_2

    !!! tip
        To apply this function to a state tensor with a leading batch axis, use
        `jax.vmap`. Then the batch axis can be reduced, e.g., by `jnp.mean`. As
        a helper for this, [`exponax.metrics.mean_metric`][] is provided.

    !!! warning
        Not supplying `domain_extent` will have the result be orders of
        magnitude different.

    **Arguments**:

    - `u_pred`: The state array. Must follow the `Exponax` convention
        with a leading channel axis, and either one, two, or three subsequent
        spatial axes.
    - `u_ref`: The reference state array. Must have the same shape as
        `u_pred`.
    - `domain_extent`: The extent `L` of the domain `Ω = (0, L)ᴰ`.
    - `low`: The lower cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `0`, meaning start it starts (including) the
        mean/zero mode.
    - `high`: The upper cutoff (inclusive) frequency for filtering. If not
        specified, it is set to `N//2 + 1`, meaning it ends (including) at the
        Nyquist mode.
    """
    regular_nrmse = fourier_nRMSE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=None,
    )
    first_derivative_nrmse = fourier_nRMSE(
        u_pred,
        u_ref,
        domain_extent=domain_extent,
        low=low,
        high=high,
        derivative_order=1,
    )
    return regular_nrmse + first_derivative_nrmse