Generative Forecasting from Coarse-Grained Observations: When Does Cascading Help?

Alexandra Spitzer; Giovanni Abati; Lorenzo Baldassari; Till Muser; Cheng Shi; Ivan Dokmanić

cs.LGOct 31, 2026

Generative Forecasting from Coarse-Grained Observations: When Does Cascading Help?

Authors: Alexandra Spitzer, Giovanni Abati, Lorenzo Baldassari, Till Muser, Cheng Shi, Ivan Dokmanić

Organizations: Department of Mathematics and Computer Science, University of Basel, Switzerland · Département d’Informatique, École Normale Supérieure, PSL University, Paris, France

Abstract

We study probabilistic forecasting of dynamical systems such as fluids and waves from coarse observations. Since the unseen fine scales influence the dynamics, the same coarse initial state can evolve into a whole distribution of futures. Generative forecasters may produce visually plausible samples, but it is hard to tell whether they sample the right conditional distribution, since trajectory data contain a single future for each initial state. We address this problem by generating reference data in a hierarchical way: for each coarse initial state, we generate many different fine-scale components and numerically evolve the full states. Using these conditional ensembles, we show that standard diffusion models and stochastic interpolants make substantial distributional errors. We then show that cascading shorter forecasts greatly reduces these errors for the turbulent flows and waves we study, but we also find and characterize cases in which it hurts. Since the coarse process is not Markov, cascading discards information about the unresolved fine scales even with exact transitions. To understand this, we propose a solvable analytic model which shows that the optimal cascade depth decreases as the coupling between the unresolved and observed scales gets stronger, and increases as the dependence of the forecast on the observation becomes more oscillatory. Consistent with our experiments, cascading is detrimental when the direct forecast is ``easy'' to learn, for example Gaussian as in the passive scalar case. We release our datasets as a first benchmark for probabilistic forecasting with ground-truth conditional ensembles.

Figures & tables

Figure 1: Hierarchical data generation. Many unobserved fine initial states $z_{0}^{(i)}$ are combined with the same observed coarse state $x_{0}$ . The solver evolves each full initial state $Y_{0}^{(i)}$ , and observing the result at time $T$ gives a reference ensemble $x_{T}^{(1)},x_{T}^{(2)},\ldots$ for $P(X_{T}\mid X_{0}=x_{0})$ .

Figure 2: Direct forecasters can sample the wrong conditional distribution. One held-out Euler observation (top left). Each row shows three samples of the vorticity at time $T$ , followed by the conditional mean and the pointwise standard deviation over $10{,}000$ samples. Reference: the simulator, run from $10{,}000$ fine-scale completions of the observation. Direct SI and EDM: generative forecasters trained on pairs $(X_{0},X_{T})$ . Cascaded SI: the cascaded forecaster of Section 3 at the validation-selected depth $S^{\star}=10$ , discussed in Section 5 .

System	Dynamics	Linear	Chaotic	Unobserved initial component	Distribution of $X_{T}$ given $x_{0}$	Cascading helps ( $S^{\star}$ )
Euler	transport by own velocity	no	expected	normalized Gaussian, independent	nonlinear in $x_{0}$	yes (10)
SQG	as Euler, stronger fine-scale coupling	no	expected	normalized Gaussian, independent	nonlinear in $x_{0}$	yes (10)
Het. waves	damped waves, scattering	yes	no	depends on $x_{0}$ ; velocity kicks	affine mean, cov. depends on $x_{0}$	yes (5)
NL Klein–Gordon	dispersive waves, cubic force	no	expected	normalized, independent	nonlinear in $x_{0}$	yes (20)
Passive scalar	fixed-flow transport, diffusion	yes	no	Gaussian, independent	Gaussian, affine mean, fixed cov.	no (1)

Table 1: Properties of the five systems relevant for forecasting. “Independent” means independent of

x_{0}

. The nonlinear systems are expected to be chaotic, but we do not show it formally or experimentally. Cascading helps when

S^{\star}>1

Figure 3: Experimental systems. Each row shows one observed initial state and four simulated futures from its conditioning ensemble; the futures differ only in their unobserved initial fine scales. The fields shown are the observed vorticity, buoyancy, displacement, displacement, and concentration.

	csFID-10k
System	Direct SI	EDM	Cascaded SI	$S^{\star}$
Euler	$57.26\pm 3.35$	$56.32\pm 3.74$	$\mathbf{16.32\pm 1.14}$	10
SQG	$138.48\pm 23.09$	$118.62\pm 21.67$	$\mathbf{51.24\pm 8.32}$	10
Heterogeneous waves	$115.86\pm 6.76$	$123.04\pm 7.07^{\dagger}$	$\mathbf{36.66\pm 1.78}$	5
Nonlinear Klein–Gordon	$98.48\pm 6.18^{\dagger}$	$92.37\pm 5.43^{\dagger}$	$\mathbf{16.24\pm 1.55}$	20
Passive scalar	$8.67\pm 0.28$	$\mathbf{7.88\pm 0.31}$	$8.67\pm 0.28$	1

Table 2: Conditional forecasting error. Test csFID-10k (lower is better), mean

\pm

SEM (standard error of the mean) over the ten test observations, for one training seed.

S^{\star}

is selected on validation data; for the passive scalar,

S^{\star}=1

, so Direct SI and Cascaded SI are the same fitted model. † The dagger signifies that the best value of one hyperparameter lay at the edge of the searched range, so a wider search might improve these baselines; Nonetheless, the last step of the search changed their validation score by much less than their gap to Cascaded SI (Appendix B.2 ).

Figure 4: Intermediate-depth cascades improve conditional forecasting in four of five systems. Test csFID-10k versus cascade depth

S

(values in Table 6 ).

S=1

is Direct SI; outlined markers: validation-selected

S^{\star}

; dashed lines: EDM; error bars:

\pm 1

SEM over the ten test observations.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Simulation grid	Horizon $T$	Observed variables
euler	$1024^{2}$	1	Vorticity (196)
sqg	$256^{2}$	20	Buoyancy (196)
het-waves	$512^{2}$	1	Displacement and velocity proxy (392)
nonlinear-kg	$128^{2}$	12	Displacement and velocity (392)
passive-scalar	$512^{2}$	1	Concentration (196)

Appendix

Table 3: Benchmark dataset specifications. Parentheses give the number of real coordinates observed by the model. Each benchmark trajectory contains 41 equally spaced snapshots. Horizons are reported in physical time.

		Displacement and velocity proxy
$S$	Displacement only	Initial proxy supplied	Initial proxy zero
1	0.731	0.760	0.647
5	0.108	0.949	0.871
10	0.094	0.953	0.888
20	0.024	0.946	0.848

Appendix

Table 4: Wave state diagnostic. Mean correlation with the true conditional endpoint mean over ten validation conditions. Both augmented columns use the same fitted model. The benchmark initializes the proxy to zero and subsequently carries generated values.

System	Method	$S$	Learning rate	Noise parameter	Update	Sampler steps	NFE
euler	Direct SI	1	$10^{-4}$	$\sigma=8$	5,000	40	40
	EDM	1	$10^{-3}$	$P_{\rm mean}=-1.6$	300	40	79
	Cascaded SI	10	$3\times 10^{-4}$	$\sigma=8$	17,000	80	800
sqg	Direct SI	1	$10^{-4}$	$\sigma=16$	4,000	40	40
	EDM	1	$3\times 10^{-4}$	$P_{\rm mean}=-3.2$	4,000	40	79
	Cascaded SI	10	$10^{-3}$	$\sigma=16$	11,000	160	1,600

Appendix

Table 5: Selected configurations and sampling cost.

Segments $S$	euler	sqg	het-waves	nonlinear-kg	passive-scalar
1 (Direct SI)	$57.26\pm 3.35$	$138.48\pm 23.09$	$115.86\pm 6.76$	$98.48\pm 6.18$	$\mathbf{8.67\pm 0.28}$
2	$53.60\pm 3.28$	$83.25\pm 16.31$	$71.23\pm 3.91$	$99.75\pm 5.89$	$18.75\pm 1.17$
5	$22.94\pm 2.55$	$\mathbf{43.57\pm 8.79}$	$\mathbf{36.66\pm 1.78}$	$53.10\pm 7.30$	$32.97\pm 1.61$
10	$\mathbf{16.32\pm 1.14}$	$51.24\pm 8.32$	$44.51\pm 2.34$	$22.50\pm 5.40$	$41.14\pm 2.36$
20	$17.86\pm 0.87$	$89.08\pm 7.56$	$58.28\pm 2.58$	$\mathbf{16.24\pm 1.55}$	$56.95\pm 2.79$
40	$23.29\pm 1.50$	$86.98\pm 12.00$	$62.81\pm 3.25$	$18.23\pm 2.05$	$75.21\pm 4.74$

Appendix

Table 6: Conditional forecasting performance across cascade depths. Test csFID-10k, mean

\pm

SEM across ten conditioning ensembles for one training seed. Bold indicates the lowest SI score within each system; EDM is shown for comparison. For SQG at

S=20

, one nonfinite generated endpoint was excluded from one conditioning ensemble, leaving 9,999 generated samples for that score.

Metric	Cascaded SI ( $S=10$ )	Direct SI	EDM
cFID	$8.96\pm 0.89$	$24.30\pm 2.08$	$14.94\pm 1.21$
csFID	$16.32\pm 1.14$	$57.26\pm 3.35$	$56.32\pm 3.74$
Sliced $W_{2}$	$0.188\pm 0.012$	$0.390\pm 0.026$	$0.389\pm 0.025$
Conditional-mean error	$1.94\pm 0.17$	$5.22\pm 0.34$	$5.17\pm 0.32$
Spread ratio	$0.931\pm 0.013$	$1.060\pm 0.023$	$1.065\pm 0.021$

Appendix

Table 7: Euler metric comparison. Mean

\pm

SEM over ten test conditions. Feature metrics use

10{,}000

true and generated endpoints per condition. State-space diagnostics use 200 of each in normalized Fourier coordinates: sliced

W_{2}

uses 100 fixed random projections, mean error is the Euclidean distance between ensemble means, and spread is the ratio of mean pairwise distances (generated/true, excluding self-pairs). Lower is better except for spread, whose target is one.

	euler		passive-scalar
Transformation	cFID	csFID	cFID	csFID
Translation $(2,2)$ px	$1.04$	$1.16$	$2.52$	$4.57$
Translation $(16,16)$ px	$2.62$	$5.68$	$28.50$	$30.80$
Rotation $90^{\circ}$	$3.65$	$7.55$	$11.17$	$26.07$
Reflection in $x$	$2.84$	$7.65$	$9.21$	$25.24$

Appendix

Table 8: Response to spatial perturbations. Each entry is the ratio of the perturbed score to the truth-to-truth score, using means over ten conditioning states. Passive scalar has only two distinct outputs because Cascaded SI selects

S^{\star}=1

Figure 5: Heterogeneous waves: direct and cascaded forecasts of one held-out observation. The displayed field is the displacement; the forecasters also predict a velocity proxy. Cascaded SI uses the validation-selected depth $S^{\star}=5$ . Means and standard deviations are estimated from $10{,}000$ samples.

Figure 6: Additional conditional endpoint distributions. Test conditions for SQG, nonlinear Klein–Gordon, and passive scalar. For each method, three endpoint samples are shown together with the conditional mean and pointwise standard deviation estimated from 10,000 generated endpoints; reference statistics are computed from 10,000 simulator-generated endpoints from the same conditioning state.

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