Skip to content

Fourier-based¤

exponax.metrics.fourier_MSE ¤

fourier_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,
    derivative_order: Optional[float] = None
) -> float

Compute the mean squared error in Fourier space.

∑_(channels) ∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|²

The channel axis is summed after the aggregation.

Under default settings with correctly specific domain_extent, this function (up to rounding errors) produces the identical result as exponax.metrics.MSE which is a consequence of Parseval's theorem. However, it additionally allows filtering specific frequency ranges and to take derivatives. In higher dimensions, the derivative contributions (i.e., the entries of the gradient) are summed up.

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.

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.
  • derivative_order: The order of the derivative to take. If not specified, no derivative is taken.
Source code in exponax/metrics/_fourier.py 
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
def fourier_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,
    derivative_order: Optional[float] = None,
) -> float:
    """
    Compute the mean squared error in Fourier space.

        ∑_(channels) ∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|²

    The channel axis is summed **after** the aggregation.

    Under default settings with correctly specific `domain_extent`, this
    function (up to rounding errors) produces the identical result as
    [`exponax.metrics.MSE`][] which is a consequence of Parseval's theorem.
    However, it additionally allows filtering specific frequency ranges and to
    take derivatives. In higher dimensions, the derivative contributions (i.e.,
    the entries of the gradient) are summed up.

    !!! 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.


    **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.
    - `derivative_order`: The order of the derivative to take. If not specified,
        no derivative is taken.
    """
    return fourier_norm(
        u_pred,
        u_ref,
        mode="absolute",
        domain_extent=domain_extent,
        inner_exponent=2.0,
        outer_exponent=1.0,
        low=low,
        high=high,
        derivative_order=derivative_order,
    )

exponax.metrics.fourier_MAE ¤

fourier_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,
    derivative_order: Optional[float] = None
) -> float

Compute the mean absolute error in Fourier space.

∑_(channels) ∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|

The channel axis is summed after the aggregation.

While conceptually similar to exponax.metrics.MAE, this function is not consistent with the L¹(Ω) functional norm. However, it additionally allows filtering specific frequency ranges and to take derivatives. In higher dimensions, the derivative contributions (i.e., the entries of the gradient) are summed up.

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.

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.
  • derivative_order: The order of the derivative to take. If not specified, no derivative is taken.
Source code in exponax/metrics/_fourier.py 
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
def fourier_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,
    derivative_order: Optional[float] = None,
) -> float:
    """
    Compute the mean absolute error in Fourier space.

        ∑_(channels) ∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|

    The channel axis is summed **after** the aggregation.

    While conceptually similar to [`exponax.metrics.MAE`][], this
    function is **not** consistent with the `L¹(Ω)` functional norm. However, it
    additionally allows filtering specific frequency ranges and to take
    derivatives. In higher dimensions, the derivative contributions (i.e., the
    entries of the gradient) are summed up.

    !!! 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.


    **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.
    - `derivative_order`: The order of the derivative to take. If not specified,
        no derivative is taken.
    """
    return fourier_norm(
        u_pred,
        u_ref,
        mode="absolute",
        domain_extent=domain_extent,
        inner_exponent=1.0,
        outer_exponent=1.0,
        low=low,
        high=high,
        derivative_order=derivative_order,
    )

exponax.metrics.fourier_RMSE ¤

fourier_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,
    derivative_order: Optional[float] = None
) -> float

Compute the root mean squared error in Fourier space.

∑_(channels) √(∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|²)

The channel axis is summed after the aggregation.

Under default settings with correctly specific domain_extent, this function (up to rounding errors) produces the identical result as exponax.metrics.RMSE which is a consequence of Parseval's theorem. However, it additionally allows filtering specific frequency ranges and to take derivatives. In higher dimensions, the derivative contributions (i.e., the entries of the gradient) are summed up.

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.

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.
  • derivative_order: The order of the derivative to take. If not specified, no derivative is taken.
Source code in exponax/metrics/_fourier.py 
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
def fourier_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,
    derivative_order: Optional[float] = None,
) -> float:
    """
    Compute the root mean squared error in Fourier space.

        ∑_(channels) √(∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|²)

    The channel axis is summed **after** the aggregation.

    Under default settings with correctly specific `domain_extent`, this
    function (up to rounding errors) produces the identical result as
    [`exponax.metrics.RMSE`][] which is a consequence of Parseval's theorem.
    However, it additionally allows filtering specific frequency ranges and to
    take derivatives. In higher dimensions, the derivative contributions (i.e.,
    the entries of the gradient) are summed up.

    !!! 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.

    **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.
    - `derivative_order`: The order of the derivative to take. If not specified,
        no derivative is taken.
    """
    return fourier_norm(
        u_pred,
        u_ref,
        mode="absolute",
        domain_extent=domain_extent,
        inner_exponent=2.0,
        outer_exponent=0.5,
        low=low,
        high=high,
        derivative_order=derivative_order,
    )

exponax.metrics.fourier_nMSE ¤

fourier_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,
    derivative_order: Optional[float] = None
) -> float

Compute the normalized mean squared error in Fourier space.

∑_(channels) (∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|² / ∑_(modi) (L/N)ᴰ
|fft(uₕʳ)|²)

The channel axis is summed after the aggregation.

Under default settings with correctly specific domain_extent, this function (up to rounding errors) produces the identical result as exponax.metrics.nMSE which is a consequence of Parseval's theorem. However, it additionally allows filtering specific frequency ranges and to take derivatives. In higher dimensions, the derivative contributions (i.e., the entries of the gradient) are summed up.

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.

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.
  • derivative_order: The order of the derivative to take. If not specified, no derivative is taken.
Source code in exponax/metrics/_fourier.py 
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
def fourier_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,
    derivative_order: Optional[float] = None,
) -> float:
    """
    Compute the normalized mean squared error in Fourier space.

        ∑_(channels) (∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|² / ∑_(modi) (L/N)ᴰ
        |fft(uₕʳ)|²)

    The channel axis is summed **after** the aggregation.

    Under default settings with correctly specific `domain_extent`, this
    function (up to rounding errors) produces the identical result as
    [`exponax.metrics.nMSE`][] which is a consequence of Parseval's theorem.
    However, it additionally allows filtering specific frequency ranges and to
    take derivatives. In higher dimensions, the derivative contributions (i.e.,
    the entries of the gradient) are summed up.

    !!! 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.

    **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.
    - `derivative_order`: The order of the derivative to take. If not specified,
        no derivative is taken.
    """
    return fourier_norm(
        u_pred,
        u_ref,
        mode="normalized",
        domain_extent=domain_extent,
        inner_exponent=2.0,
        outer_exponent=1.0,
        low=low,
        high=high,
        derivative_order=derivative_order,
    )

exponax.metrics.fourier_nMAE ¤

fourier_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,
    derivative_order: Optional[float] = None
) -> float

Compute the normalized mean absolute error in Fourier space.

∑_(channels) (∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)| / ∑_(modi) (L/N)ᴰ
|fft(uₕʳ)|)

The channel axis is summed after the aggregation.

While conceptually similar to exponax.metrics.nMAE, this function is not consistent with the L¹(Ω) functional norm. However, it additionally allows filtering specific frequency ranges and to take derivatives. In higher dimensions, the derivative contributions (i.e., the entries of the gradient) are summed up.

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.

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.
  • derivative_order: The order of the derivative to take. If not specified, no derivative is taken.
Source code in exponax/metrics/_fourier.py 
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
def fourier_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,
    derivative_order: Optional[float] = None,
) -> float:
    """
    Compute the normalized mean absolute error in Fourier space.

        ∑_(channels) (∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)| / ∑_(modi) (L/N)ᴰ
        |fft(uₕʳ)|)

    The channel axis is summed **after** the aggregation.

    While conceptually similar to [`exponax.metrics.nMAE`][], this
    function is **not** consistent with the `L¹(Ω)` functional norm. However, it
    additionally allows filtering specific frequency ranges and to take
    derivatives. In higher dimensions, the derivative contributions (i.e., the
    entries of the gradient) are summed up.

    !!! 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.


    **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.
    - `derivative_order`: The order of the derivative to take. If not specified,
        no derivative is taken.
    """
    return fourier_norm(
        u_pred,
        u_ref,
        mode="normalized",
        domain_extent=domain_extent,
        inner_exponent=1.0,
        outer_exponent=1.0,
        low=low,
        high=high,
        derivative_order=derivative_order,
    )

exponax.metrics.fourier_nRMSE ¤

fourier_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,
    derivative_order: Optional[float] = None
) -> float

Compute the normalized root mean squared error in Fourier space.

∑_(channels) (√(∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|²) / √(∑_(modi) (L/N)ᴰ
|fft(uₕʳ)|²))

The channel axis is summed after the aggregation.

Under default settings with correctly specific domain_extent, this function (up to rounding errors) produces the identical result as exponax.metrics.nRMSE which is a consequence of Parseval's theorem. However, it additionally allows filtering specific frequency ranges and to take derivatives. In higher dimensions, the derivative contributions (i.e., the entries of the gradient) are summed up.

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.

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.
  • derivative_order: The order of the derivative to take. If not specified, no derivative is taken.
Source code in exponax/metrics/_fourier.py 
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
def fourier_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,
    derivative_order: Optional[float] = None,
) -> float:
    """
    Compute the normalized root mean squared error in Fourier space.

        ∑_(channels) (√(∑_(modi) (L/N)ᴰ |fft(uₕ - uₕʳ)|²) / √(∑_(modi) (L/N)ᴰ
        |fft(uₕʳ)|²))

    The channel axis is summed **after** the aggregation.

    Under default settings with correctly specific `domain_extent`, this
    function (up to rounding errors) produces the identical result as
    [`exponax.metrics.nRMSE`][] which is a consequence of Parseval's theorem.
    However, it additionally allows filtering specific frequency ranges and to
    take derivatives. In higher dimensions, the derivative contributions (i.e.,
    the entries of the gradient) are summed up.

    !!! 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.

    **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.
    - `derivative_order`: The order of the derivative to take. If not specified,
        no derivative is taken.
    """
    return fourier_norm(
        u_pred,
        u_ref,
        mode="normalized",
        domain_extent=domain_extent,
        inner_exponent=2.0,
        outer_exponent=0.5,
        low=low,
        high=high,
        derivative_order=derivative_order,
    )

exponax.metrics.fourier_norm ¤

fourier_norm(
    state: Float[Array, "C ... N"],
    state_ref: Optional[Float[Array, "C ... N"]] = None,
    *,
    mode: Literal["absolute", "normalized"] = "absolute",
    domain_extent: float = 1.0,
    inner_exponent: float = 2.0,
    outer_exponent: Optional[float] = None,
    low: Optional[int] = None,
    high: Optional[int] = None,
    derivative_order: Optional[float] = None
) -> float

Compute norms of states via aggregation in Fourier space.

Each channel is treated separately and the results are summed up.

While conceptually similar to exponax.metrics.spatial_norm, this function additionally allows filtering specific frequency ranges and to take derivatives. In higher dimensions, the derivative contributions (i.e., the entries of the gradient) are summed up.

Tip

To operate on states 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.

If both low and high are None, the full spectrum is considered. In this case, this function with inner_exponent=2.0 (up to rounding errors) produces the same result as exponax.metrics.spatial_norm which is a consequence of Parseval's theorem.

Arguments:

  • state: The state tensor. Must follow the Exponax convention with a leading channel axis, and either one, two, or three subsequent spatial axes.
  • state_ref: The reference state tensor. Must have the same shape as state. If not specified, only the absolute norm of state is computed.
  • mode: The mode of the norm. Either "absolute" or "normalized".
  • domain_extent: The extent L of the domain Ω = (0, L)ᴰ.
  • inner_exponent: The exponent p each magnitude of a Fourier coefficient is raised to before aggregation.
  • outer_exponent: The exponent q the aggregated magnitudes are raised to. If not specified, it is set to 1/p.
  • 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.
  • derivative_order: The order of the derivative to take. If not specified, no derivative is taken.
Source code in exponax/metrics/_fourier.py 
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
def fourier_norm(
    state: Float[Array, "C ... N"],
    state_ref: Optional[Float[Array, "C ... N"]] = None,
    *,
    mode: Literal["absolute", "normalized"] = "absolute",
    domain_extent: float = 1.0,
    inner_exponent: float = 2.0,
    outer_exponent: Optional[float] = None,
    low: Optional[int] = None,
    high: Optional[int] = None,
    derivative_order: Optional[float] = None,
) -> float:
    """
    Compute norms of states via aggregation in Fourier space.

    Each channel is treated separately and the results are summed up.

    While conceptually similar to [`exponax.metrics.spatial_norm`][], this
    function additionally allows filtering specific frequency ranges and to take
    derivatives. In higher dimensions, the derivative contributions (i.e., the
    entries of the gradient) are summed up.

    !!! tip
        To operate on states 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.

    If both `low` and `high` are `None`, the full spectrum is considered. In
    this case, this function with `inner_exponent=2.0` (up to rounding errors)
    produces the same result as [`exponax.metrics.spatial_norm`][] which is a
    consequence of Parseval's theorem.


    **Arguments**:

    - `state`: The state tensor. Must follow the `Exponax` convention with a
        leading channel axis, and either one, two, or three subsequent spatial
        axes.
    - `state_ref`: The reference state tensor. Must have the same shape as
        `state`. If not specified, only the absolute norm of `state` is
        computed.
    - `mode`: The mode of the norm. Either `"absolute"` or `"normalized"`.
    - `domain_extent`: The extent `L` of the domain `Ω = (0, L)ᴰ`.
    - `inner_exponent`: The exponent `p` each magnitude of a Fourier coefficient
        is raised to before aggregation.
    - `outer_exponent`: The exponent `q` the aggregated magnitudes are raised
        to. If not specified, it is set to `1/p`.
    - `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.
    - `derivative_order`: The order of the derivative to take. If not specified,
        no derivative is taken.
    """
    if state_ref is None:
        if mode == "normalized":
            raise ValueError("mode 'normalized' requires state_ref")
        diff = state
    else:
        diff = state - state_ref

    diff_norm_per_channel = jax.vmap(
        lambda s: fourier_aggregator(
            s,
            domain_extent=domain_extent,
            inner_exponent=inner_exponent,
            outer_exponent=outer_exponent,
            low=low,
            high=high,
            derivative_order=derivative_order,
        ),
    )(diff)

    if mode == "normalized":
        ref_norm_per_channel = jax.vmap(
            lambda r: fourier_aggregator(
                r,
                domain_extent=domain_extent,
                inner_exponent=inner_exponent,
                outer_exponent=outer_exponent,
                low=low,
                high=high,
                derivative_order=derivative_order,
            ),
        )(state_ref)
        normalized_diff_per_channel = diff_norm_per_channel / ref_norm_per_channel
        norm_per_channel = normalized_diff_per_channel
    else:
        norm_per_channel = diff_norm_per_channel

    return jnp.sum(norm_per_channel)

exponax.metrics.fourier_aggregator ¤

fourier_aggregator(
    state_no_channel: Float[Array, "... N"],
    *,
    num_spatial_dims: Optional[int] = None,
    domain_extent: float = 1.0,
    num_points: Optional[int] = None,
    inner_exponent: float = 2.0,
    outer_exponent: Optional[float] = None,
    low: Optional[int] = None,
    high: Optional[int] = None,
    derivative_order: Optional[float] = None
) -> float

Aggregate over the spatial axes of a (channel-less) state array in Fourier space.

While conceptually similar to exponax.metrics.spatial_aggregator, this function additionally allows filtering specific frequency ranges and to take derivatives. In higher dimensions, the derivative contributions (i.e., the entries of the gradient) are summed up.

Info

The result of this function (under default settings) is (up to rounding errors) identical to exponax.metrics.spatial_aggregator for inner_exponent=2.0. As such, it can be a consistent counterpart for metrics based on the L²(Ω) functional norm.

Tip

To apply this function to a state tensor with a leading channel axis, use jax.vmap.

Arguments:

  • state_no_channel: The state tensor without a leading channel dimension.
  • num_spatial_dims: The number of spatial dimensions. If not specified, it is inferred from the number of axes in state_no_channel.
  • domain_extent: The extent L of the domain Ω = (0, L)ᴰ.
  • num_points: The number of points N in each spatial dimension. If not specified, it is inferred from the last axis of state_no_channel.
  • inner_exponent: The exponent p each magnitude of a Fourier coefficient is raised to before aggregation.
  • outer_exponent: The exponent q the aggregated magnitudes are raised to. If not specified, it is set to 1/p.
  • 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.
  • derivative_order: The order of the derivative to take. If not specified, no derivative is taken.
Source code in exponax/metrics/_fourier.py 
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
def fourier_aggregator(
    state_no_channel: Float[Array, "... N"],
    *,
    num_spatial_dims: Optional[int] = None,
    domain_extent: float = 1.0,
    num_points: Optional[int] = None,
    inner_exponent: float = 2.0,
    outer_exponent: Optional[float] = None,
    low: Optional[int] = None,
    high: Optional[int] = None,
    derivative_order: Optional[float] = None,
) -> float:
    """
    Aggregate over the spatial axes of a (channel-less) state array in Fourier
    space.

    While conceptually similar to [`exponax.metrics.spatial_aggregator`][], this
    function additionally allows filtering specific frequency ranges and to take
    derivatives. In higher dimensions, the derivative contributions (i.e., the
    entries of the gradient) are summed up.

    !!! info
        The result of this function (under default settings) is (up to rounding
        errors) identical to [`exponax.metrics.spatial_aggregator`][] for
        `inner_exponent=2.0`. As such, it can be a consistent counterpart for
        metrics based on the `L²(Ω)` functional norm.

    !!! tip
        To apply this function to a state tensor with a leading channel axis,
        use `jax.vmap`.

    **Arguments**:

    - `state_no_channel`: The state tensor **without a leading channel
        dimension**.
    - `num_spatial_dims`: The number of spatial dimensions. If not specified,
        it is inferred from the number of axes in `state_no_channel`.
    - `domain_extent`: The extent `L` of the domain `Ω = (0, L)ᴰ`.
    - `num_points`: The number of points `N` in each spatial dimension. If not
        specified, it is inferred from the last axis of `state_no_channel`.
    - `inner_exponent`: The exponent `p` each magnitude of a Fourier coefficient
        is raised to before aggregation.
    - `outer_exponent`: The exponent `q` the aggregated magnitudes are raised
        to. If not specified, it is set to `1/p`.
    - `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.
    - `derivative_order`: The order of the derivative to take. If not specified,
        no derivative is taken.
    """
    if num_spatial_dims is None:
        num_spatial_dims = state_no_channel.ndim
    if num_points is None:
        num_points = state_no_channel.shape[-1]

    if outer_exponent is None:
        outer_exponent = 1 / inner_exponent

    # Transform to Fourier space
    state_no_channel_hat = fft(state_no_channel, num_spatial_dims=num_spatial_dims)

    # Remove small values that occured due to rounding errors, can become
    # problematic for "normalized" norms
    state_no_channel_hat = jnp.where(
        jnp.abs(state_no_channel_hat) < 1e-5,
        jnp.zeros_like(state_no_channel_hat),
        state_no_channel_hat,
    )

    # Filtering out if desired
    if low is not None or high is not None:
        if low is None:
            low = 0
        if high is None:
            high = (num_points // 2) + 1

        low_mask = low_pass_filter_mask(
            num_spatial_dims,
            num_points,
            cutoff=low - 1,  # Need to subtract 1 because the cutoff is inclusive
        )
        high_mask = low_pass_filter_mask(
            num_spatial_dims,
            num_points,
            cutoff=high,
        )

        mask = jnp.invert(low_mask) & high_mask

        state_no_channel_hat = state_no_channel_hat * mask

    # Taking derivatives if desired
    if derivative_order is not None:
        derivative_operator = build_derivative_operator(
            num_spatial_dims, domain_extent, num_points
        )
        state_with_derivative_channel_hat = (
            state_no_channel_hat * derivative_operator**derivative_order
        )
    else:
        # Add singleton derivative axis to have subsequent code work
        state_with_derivative_channel_hat = state_no_channel_hat[None]

    # Scale coefficients to extract the correct form, this is needed because we
    # use the rfft
    scaling_array_recon = build_scaling_array(
        num_spatial_dims,
        num_points,
        mode="reconstruction",
    )

    scale = (domain_extent / num_points) ** num_spatial_dims

    def aggregate(s):
        scaled_coefficient_magnitude = (
            jnp.abs(s) ** inner_exponent / scaling_array_recon
        )
        aggregated = jnp.sum(scaled_coefficient_magnitude)
        return (scale * aggregated) ** outer_exponent

    aggregated_per_derivative = jax.vmap(aggregate)(state_with_derivative_channel_hat)

    return jnp.sum(aggregated_per_derivative)