Let's Do Turbulence Right: In Space and Time!

Maximilian Plattner; Fabian Paischer; Johannes Brandstetter

physics.flu-dynOct 23, 2026

Let's Do Turbulence Right: In Space and Time!

Authors: Maximilian Plattner, Fabian Paischer, Johannes Brandstetter

Organizations: ELLIS Unit, LIT AI Lab, Institute for Machine Learning, JKU Linz · Mistral AI

Abstract

The three-dimensional Navier-Stokes equations govern fluid turbulence, a chaotic and non-linear physical phenomenon characterized by complex eddies interacting across vast spatial and temporal scales. A numerically accurate modeling of this multi-scale energy cascade requires fully resolving both the fine spatial structures and the continuous temporal evolution of the flow field. In this paper, we leverage massive, time-resolved direct numerical simulations, which exceed 28 TB per run, to assess how numerical choices govern the resolved (3+1)-dimensional physics of turbulent flows. Building on this analysis, we study how neural networks represent the recorded turbulent motion. Our results highlight that spatial and temporal assessments can disagree for both numerical records and learned representations. We therefore call on the machine learning community to treat (3+1)-dimensional turbulence as a four-dimensional phenomenon.

Figures & tables

Figure 1: The visualization shows turbulent structures from a single 28.9 TB time-resolved direct numerical simulation . The evolution of these structures (top) reveals coupled motion across space and time (bottom). How neural representations capture these coupled spatio–temporal dynamics remains an open question. To address this gap, we first use massive time-resolved simulation records to assess how numerical choices shape turbulent dynamics. We then examine how neural networks represent the recorded motion. Our analyses indicate that spatial and temporal assessments can disagree. We therefore call on the machine learning community to treat $(3+1)$ -dimensional turbulence as a four-dimensional phenomenon.

Figure 2: Turbulence diagnostics (schematic). The energy spectrum shows how energy is distributed across spatial scales, with small scales carrying little energy but producing strong velocity gradients. These gradients generate local rotation, quantified by enstrophy. Their moments then reveal asymmetry and rare extreme events. Correlation time finally measures how quickly motion at each scale loses memory. Together, these diagnostics describe spatial and temporal dynamics.

Figure 3: From DNS to neural representation. Computation and recording determine the velocity samples supplied to joint or independent neural fitting.

Figure 4: Space–time filtering. From left to right, the DNS record ${\bm{u}}_{1024,\mathrm{FP32}}$ is followed by its projections $\mathcal{P}_{M}$ at $M=512,256,128$ . Blue shows retained vorticity and yellow highlights the removed part. Each panel shows the same time and plane $z=\pi$ on a common intensity scale, with $0\leq x,y<2\pi$ . Lighter shades reveal stronger vorticity. We will release these filtered records and their construction metadata.

Figure 5: Numerical modeling and representation of the recorded flow.

Figure 6: Joint and independent neural representations of the recorded flow.

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

Quantity	Resolution comparison	Precision comparison
Grid size $N$	$32,64,128,256,512,1024$	$128$
Solver precision $\sigma$	FP32	FP64, FP32, FP16, FP8, FP4
Viscosity $\nu$	$0.00065$	$0.010421385668748454$
Nominal step $\Delta t$	$0.00040015018141313906$	$0.004701911987471931$
Steps between snapshots	$6$	$2$
Snapshot interval $\Delta t_{\mathrm{snap}}$	$0.0024009010884788345$	$0.009403823974943862$

Appendix

Table 1: Physical and recording setup.

Label	Encoding	Bits per value	Largest finite magnitude	Overflow rule
FP64	binary64	64	$1.798\times 10^{308}$	IEEE infinity
FP32	binary32	32	$3.403\times 10^{38}$	IEEE infinity
FP16	binary16	16	$65{,}504$	IEEE infinity
FP8	E4M3	8	$448$	Finite saturation
FP4	E2M1	4	$6$	Finite saturation

Appendix

Table 2: Floating-point formats used for computation and recording.

Record	$\overline{K}$	$\overline{\Omega}$	$\overline{S}$	$\overline{F}$	$d_{\nabla}$	$\tau_{e}(8)$	$\tau_{e}(24)$	TB
Resolution
$N=1024$	1.795	323.20	-0.562	7.005	$2.91\!\times\!10^{-6}$	0.206	0.058	28.954457
$N=512$	1.734	291.14	-0.532	7.055	$2.56\!\times\!10^{-6}$	0.213	0.059	3.608717
$N=256$	1.824	330.24	-0.354	5.160	$2.53\!\times\!10^{-6}$	0.203	0.058	0.456393
$N=128$	1.771	314.76	-0.143	3.676	$2.61\!\times\!10^{-6}$	0.235	0.056	0.056384
$N=64$	2.235	448.19	-0.037	3.126	$5.99\!\times\!10^{-6}$	0.171	—	0.007383

Appendix

Table 3: Measured records and stored payloads underlying Figure 5 .

Record	$u^{\prime}$	$\overline{{\epsilon}}$	$L$	$\mathit{Re}_{\lambda}$	$k_{\max}\eta$	$\Delta t_{\mathrm{snap}}/\tau_{\eta}$	$(J-1)\Delta t_{\mathrm{snap}}/T_{L}$
$N=1024$	1.094	0.4202	1.393	280.5	1.724	0.061	14.26
$N=512$	1.075	0.3785	1.426	285.5	0.882	0.058	13.68
$N=256$	1.103	0.4293	1.379	281.9	0.427	0.062	14.51
$N=128$	1.087	0.4092	1.411	280.5	0.214	0.060	13.98
$N=64$	1.221	0.5826	1.096	296.5	0.098	0.072	20.22
$N=32$	2.408	1.0767	0.487	848.6	0.040	0.098	89.77

Appendix

Table 4: Flow scales estimated on the observation window.

Contribution	TB
Resolution records, including the finite $N=32$ failure	33.084213
Precision records, including the frozen FP8 control	0.222906
Five filtered movies	0.879608
Additional quantized records, excluding the shared FP64 reference	0.111453
All distinct records	$\mathbf{34.298181}$
Computed records excluding the frozen FP8 and failed $N=32$ controls	33.291379

Appendix

Table 5: Distinct stored payloads of the records depicted in Figure 5 .

Figure 7: Joint spatial–temporal spectra of the numerical records. This adds the frequency distribution to the comparisons in Figure 5 . Resolution changes the distribution even at shared scales. Filtering restricts its spatial and temporal extent. For precision , the frozen FP8 record retains zero-frequency energy, while FP4 supplies no valid temporal spectrum. Coarse quantization broadens the small-scale temporal content. All panels use the same absolute color scale. Gray denotes unrepresented spatial or temporal frequencies. Spectra use up to 512 sampled Fourier modes per shell, scaled to the measured full-shell energy. The temporal mean is retained. The frequency axis is linear near zero and logarithmic above it.

Comparison	Supplied record ${\bm{u}}^{\mathrm{rec}}$	Settings
Resolution	$\mathcal{P}_{64}{\bm{u}}_{N,\mathrm{FP32}}$	$N=128,256,512,1024$
Filtering	$\mathcal{P}_{M}{\bm{u}}_{1024,\mathrm{FP32}}$	$M=32,64,128$
Precision	$\mathcal{P}_{64}{\bm{u}}_{128,\sigma}$	$\sigma=\mathrm{FP64},\mathrm{FP32},\mathrm{FP16}$
Quantization	$\mathcal{P}_{64}\mathcal{Q}_{q}{\bm{u}}_{128,\mathrm{FP64}}$	$q=\mathrm{FP64},\mathrm{FP32},\mathrm{FP16},\mathrm{FP8},\mathrm{FP4}$

Appendix

Table 6: Records supplied to neural fitting.

Figure 8: Recorded motion supplied to neural fitting.

Figure 9: Joint and independent neural representations of the recorded flow.

Figure 16

Record	Representation	$\overline{K}$	$\overline{\Omega}$	$\overline{S}$	$\overline{F}$	$d_{\nabla}$	$\tau_{e}(8)$	$\tau_{e}(24)$
$N=1024$	Supplied	1.722	81.89	-0.329	3.807	$5.70\!\times\!10^{-6}$	0.207	0.060
	SIREN, joint	1.697	64.42	-0.290	3.826	$2.02\!\times\!10^{-1}$	0.207	0.074
	Gaussian, joint	1.708	70.75	-0.364	4.053	$1.21\!\times\!10^{-1}$	0.207	0.063
	SIREN, indep.	1.660	45.96	-0.256	3.735	$2.57\!\times\!10^{-1}$	0.207	0.031
	Gaussian, indep.	1.662	46.03	-0.383	4.450	$1.76\!\times\!10^{-1}$	0.207	0.038
$N=512$	Supplied	1.668	76.54	-0.316	3.947	$4.91\!\times\!10^{-6}$	0.213	0.062

Appendix

Table 7: Absolute diagnostics of supplied records and neural fits: resolution.

Record	Representation	$\overline{K}$	$\overline{\Omega}$	$\overline{S}$	$\overline{F}$	$d_{\nabla}$	$\tau_{e}(8)$	$\tau_{e}(24)$
$M=128$	Supplied	1.780	191.02	-0.381	4.344	$3.78\!\times\!10^{-6}$	0.207	0.058
	SIREN, joint	1.760	139.82	-0.325	4.293	$2.13\!\times\!10^{-1}$	0.207	0.058
	Gaussian, joint	1.757	133.88	-0.399	4.675	$1.70\!\times\!10^{-1}$	0.207	0.058
	SIREN, indep.	1.738	105.78	-0.269	3.880	$2.58\!\times\!10^{-1}$	0.207	0.057
	Gaussian, indep.	1.735	99.52	-0.385	4.757	$2.14\!\times\!10^{-1}$	0.207	0.057
$M=64$	Supplied	1.722	81.89	-0.329	3.807	$5.70\!\times\!10^{-6}$	0.207	0.060

Appendix

Table 8: Absolute diagnostics of supplied records and neural fits: filtering.

Record	Representation	$\overline{K}$	$\overline{\Omega}$	$\overline{S}$	$\overline{F}$	$d_{\nabla}$	$\tau_{e}(8)$	$\tau_{e}(24)$
FP64	Supplied	1.617	19.20	-0.413	4.276	$1.33\!\times\!10^{-7}$	0.245	0.089
	SIREN, joint	1.617	19.15	-0.412	4.287	$2.08\!\times\!10^{-2}$	0.245	0.092
	Gaussian, joint	1.617	19.15	-0.408	4.270	$4.13\!\times\!10^{-2}$	0.245	0.091
	SIREN, indep.	1.616	18.72	-0.387	4.213	$7.03\!\times\!10^{-2}$	0.245	0.050
	Gaussian, indep.	1.616	19.00	-0.393	4.221	$7.69\!\times\!10^{-2}$	0.245	0.066
FP32	Supplied	1.612	19.12	-0.420	4.260	$1.77\!\times\!10^{-7}$	0.245	0.089

Appendix

Table 9: Absolute diagnostics of supplied records and neural fits: precision.

Record	Representation	$\overline{K}$	$\overline{\Omega}$	$\overline{S}$	$\overline{F}$	$d_{\nabla}$	$\tau_{e}(8)$	$\tau_{e}(24)$
FP64	Supplied	1.617	19.20	-0.413	4.276	$1.33\!\times\!10^{-7}$	0.245	0.089
	SIREN, joint	1.617	19.15	-0.412	4.287	$2.08\!\times\!10^{-2}$	0.245	0.092
	Gaussian, joint	1.617	19.15	-0.408	4.270	$4.13\!\times\!10^{-2}$	0.245	0.091
	SIREN, indep.	1.616	18.72	-0.387	4.213	$7.03\!\times\!10^{-2}$	0.245	0.050
	Gaussian, indep.	1.616	19.00	-0.393	4.221	$7.69\!\times\!10^{-2}$	0.245	0.066
FP32	Supplied	1.617	19.20	-0.413	4.276	$1.34\!\times\!10^{-7}$	0.245	0.089

Appendix

Table 10: Absolute diagnostics of supplied records and neural fits: quantization.

Parameters	$M=32$	$M=64$	$M=128$
SIREN weights, biases	$6\times 10^{-4}$	$3\times 10^{-4}$	$3\times 10^{-4}$
Gaussian centers	$0.024$	$0.006$	$0.003$
Gaussian log-diagonal and off-diagonal shape	$0.040$	$0.010$	$0.005$
Gaussian velocity coefficients	$0.080$	$0.020$	$0.010$

Appendix

Table 11: Initial learning rates within each restart cycle.

Family	Procedure	$M=32$	$M=64$	$M=128$
SIREN	Joint	$335{,}235$	$5{,}359{,}219$	$85{,}803{,}163$
	Independent	$335{,}254$	$5{,}358{,}988$	$85{,}804{,}416$
Gaussian	Joint	$334{,}611$	$5{,}353{,}776$	$85{,}660{,}416$
	Independent	$334{,}608$	$5{,}353{,}776$	$85{,}660{,}416$

Appendix

Table 12: Total learned scalar parameters per movie.

Fit	Spectrum error (%)	Enstrophy	$S$	$F$	$\langle\tau/\tau_{\rm rec}\rangle_{k}$
Record	0	81.89	-0.329	3.807	1
SIREN, Joint	$1.70\!\pm\!0.03$	$64.17\!\pm\!0.22$	$-0.289\!\pm\!0.002$	$3.824\!\pm\!0.003$	$1.116\!\pm\!0.001$
SIREN, Indep.	$3.63\!\pm\!0.01$	$46.12\!\pm\!0.14$	$-0.255\!\pm\!0.001$	$3.725\!\pm\!0.008$	$0.801\!\pm\!0.001$
Gaussian, Joint	$0.96\!\pm\!0.00$	$70.75\!\pm\!0.00$	$-0.364\!\pm\!0.000$	$4.052\!\pm\!0.001$	$1.029\!\pm\!0.000$
Gaussian, Indep.	$3.64\!\pm\!0.00$	$46.03\!\pm\!0.00$	$-0.383\!\pm\!0.000$	$4.448\!\pm\!0.004$	$0.846\!\pm\!0.000$

Appendix

Table 13: Physical diagnostics across three training seeds on one unquantized

N=1024

record at

M=64

		Spatial MSE ( $\times 10^{2}$ )		Change error: Indep./Joint
Family	Seed	Joint	Indep.	$\Delta t$	$2\Delta t$	$4\Delta t$
SIREN	0	4.544	4.463	1.358	0.939	0.933
SIREN	1	4.552	4.517	1.367	0.947	0.944
SIREN	2	5.467	5.470	1.391	0.984	0.942
Gaussian	0	3.457	3.427	1.229	0.917	1.003
Gaussian	1	3.456	3.426	1.229	0.916	1.003

Appendix

Table 14: Trained checkpoints at matched spatial error on the unquantized

N=1024

M=64

movie.

Figure 10: Joint spectra of supplied records and their neural fits. The resolution and filtering comparisons extend Figure 6 . Every column compares a record with its joint and independent SIREN and Gaussian fits. Independent fitting broadens small-scale temporal spectra, while joint fitting tends to narrow them. The maps display absolute energy on a common color scale, without normalizing each panel. The estimator follows Figure 7 . Gray denotes frequencies outside the supplied sampling range. The precision and quantization comparisons continue on the next page.

Figure 11: Matching gradient power does not remove the family difference. All fields are restricted to the same spatial band, and their gradient power is matched to the supplied record shell by shell. Gaussian fits still show more negative skewness and higher flatness than SIRENs under both fitting procedures. Points and error bars show means and sample standard deviations over three training seeds per architecture and fitting procedure.

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