CasEm: A Cascade Architecture for Long-Horizon Neural Emulation

Zhaoyi Li; Jingtao Ding; Shihua Li

cs.LGSep 28, 2026

CasEm: A Cascade Architecture for Long-Horizon Neural Emulation

Authors: Zhaoyi Li, Jingtao Ding, Shihua Li

Organizations: School of Automation, Southeast University · Beijing Zhongguancun Academy · Department of Earth System Science, Tsinghua University

Abstract

Autoregressive neural emulators can drift or diverge over long rollouts despite accurate short-term predictions. We introduce Cascaded Emulation (CasEm), a one-way rollout architecture that augments an existing full-state backbone with an independently evolving model of physically specified aggregates. Its forecasts guide corrections to full-state predictions, without feedback from the backbone to the aggregate model. Effective guidance requires aggregates that cover substantial backbone error, remain accurately predictable, and support useful full-state corrections. We derive a finite-horizon error bound that clarifies these three factors and use empirical diagnostics to guide subsystem selection. Across four ODE/PDE benchmarks, CasEm reduces long-horizon rollout errors across diverse backbones and suppresses the trend toward error divergence in both diffusion tasks using Fourier neural operator backbones. In global climate emulation, CasEm with a regional total-water subsystem reduces 10-year full-state time-mean error by 66.6% and 46.3% for frozen ACE and Spherical DYffusion backbones, respectively, while adding less than 3% to inference time.

Figures & tables

Figure 1: One-step architectures: (a) standard autoregression; (b) HINE’s joint full/coarse prediction, using prior-step predictions of spatially downsampled future states as coarse inputs; (c) CasEm’s independent subsystem guiding full-state correction.

Figure 3: Full-state NRMSE versus rollout steps. Lines show means; shaded bands indicate

\pm

SD.

		Method	One-step	Intermediate	Long	Extended
BVE	ResCNN	Base	$0.0117\pm 0.00171$	$0.703\pm 0.163$	$14.9\pm 2.54$	$65.7\pm 7.03$
		HINE	$0.0113\pm 0.00167$	$0.677\pm 0.157$	$13.4\pm 2.29$	$63.5\pm 6.76$
		CasEm	$\mathbf{0.00943\pm 0.00128}$	$\mathbf{0.488\pm 0.112}$	$\mathbf{2.17\pm 0.320}$	$\mathbf{4.96\pm 3.17}$
		Oracle	$0.00943\pm 0.00128$	$0.483\pm 0.115$	$2.14\pm 0.376$	$5.19\pm 3.22$
	SFNO	Base	$0.0250\pm 0.00312$	$0.159\pm 0.0401$	$1.36\pm 1.61$	$4.98\pm 3.99$
		HINE	$0.0241\pm 0.00309$	$0.161\pm 0.0403$	$1.39\pm 1.58$	$5.53\pm 4.05$

Table 1: Full-state endpoint NRMSE for BVE and alanine dipeptide (mean

\pm

sample SD across trajectories). The four horizons correspond to 1/50/500/1000 steps for BVE and 1/500/2000/10000 steps for Alanine. Within each backbone and horizon, bold highlights the lowest mean among Base, HINE, and CasEm unless differences are marginal. Oracle and backbone-independent subsystem-only (SO) rollouts serve as diagnostic references.

Rollout NRMSE
		Full state		Total water
	Method	TM	ST	TM	ST
ACE	Base	$0.1580$	$0.9409$	$0.4691$	$1.3832$
ACE	CasEm	$0.0527$ (66.6% $\uparrow$ )	$0.7913$ (15.9% $\uparrow$ )	$0.0840$ (82.1% $\uparrow$ )	$0.8671$ (37.3% $\uparrow$ )
S-DY	Base	$0.0785\pm 0.0028$	$0.8160\pm 0.0029$	$0.2134\pm 0.0095$	$0.9517\pm 0.0097$
	CasEm	$0.0421\pm 0.0016$	$0.7929\pm 0.0019$	$0.0653\pm 0.0014$	$0.8553\pm 0.0025$

Table 2: 10-year comparison of ACE, S-DY, and their water-guided CasEm variants. Both backbones are frozen. ACE: one rollout; S-DY: 25 paired members (mean

\pm

sample SD). Ensemble scores use 10-year mean fields. Parentheses show CasEm’s relative gains over the corresponding Base (

\uparrow

Time	ACE	S-DY
Base	12.14	37.03
CasEm	12.45 $(+2.6\%)$	37.54 $(+1.4\%)$

Table 3: 10-year inference time (min); overhead in parentheses.

Figure 5: Annual TM and ST NRMSE for 2021–2024 starts on the evaluation trajectory. Lines: means; shading:

\pm

sample SD across starts available each year.

Metric	Truth	S-DY	CasEm
Slow-anomaly SD (kg m -2 )	1.4881	2.7434	1.6826 (84.5% $\uparrow$ )
P95 exceedance frequency (%)	5.0000	30.5435	6.3043 (94.9% $\uparrow$ )
Regional slow RMSE (kg m -2 )	–	3.2665	1.9162 (41.3% $\uparrow$ )
Native-grid daily RMSE (kg m -2 )	–	8.7917	7.1828 (18.3% $\uparrow$ )
Transport RMSE (kg m -1 s -1 )	–	229.9891	194.3430 (15.5% $\uparrow$ )
Convergence RMSE (kg m -2 day -1 )	–	6.4330	5.5570 (13.6% $\uparrow$ )

Table 4: South Asian summer diagnostics. Parentheses: relative error or bias reductions versus Base S-DY (

\uparrow

). Final three rows: unfiltered daily regional means.

Figure 6: Area-weighted slow TWP anomalies over South Asia during June–August.

Panel of Table 5 — Table 5: TM error change (%) from full CasEm ( $12\!\times\!24$ , 8 vertical groups); positive is worse.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Block diagram of a continuous-time two-stage cascade interconnection. The upstream state

x^{(1)}(t)

drives the downstream dynamics

f^{(2)}

, with no feedback from the downstream system to the upstream system. Both systems receive the external input

u(t)

; integrators evolve the states from the indicated initial conditions.

Figure 8: Subsystem screening and CasEm validation. The qualitative criterion

\gamma\gtrsim 1

allows values slightly below one but requires positive ideal correction gain (

\gamma>0

Figure 9: Temporal state dependencies: (a) coarse-grid full-state rollout with per-time-step downscaling; (b) joint sequence generation through spatiotemporal refinement; (c) CasEm’s autonomous subsystem guiding recurrent full-state prediction. Physical time runs downward. Horizontal arrows denote downscaling in (a), whole-sequence refinement in (b), and subsystem guidance in (c); the refinement arrow in (b) is not a physical-time update.

Case	System type	Specified subsystem	$n$	Closure	Backbone
Heat	1D PDE	DC+Low-4 Fourier, $k=0{:}4$	9	Exact	ResCNN, FNO
Fisher–KPP	1D PDE	Joint low-frequency block, $k=0{:}16$	33	Approximate	ResCNN, FNO
BVE	Spherical PDE	Real $l=1$ spherical harmonics	3	Exact	ResCNN, SFNO
Alanine NVE	ODE	Mass-weighted COM position/velocity	6	Exact	GNS, EGNN-velocity

Appendix

Table 6: Benchmark systems and specified subsystems.

System	Spatial representation	Data generation	Integration step	Prediction interval
Heat	256 grid points	Custom solver	0.05	0.05
Fisher–KPP	2048 reference points; 512 emulator points	Custom solver	0.00125	0.02
BVE	$32\times 64$ spherical grid	SpeedyWeather.jl	900 s	3 hours
Alanine	22 atoms	OpenMM	0.1 fs	0.01 ps

Appendix

Table 7: Simulation settings and emulator prediction intervals.

Split	Heat	Fisher–KPP	BVE	Alanine
Training	64	64	128	96
Development	12	12	32	32

Appendix

Table 8: Numbers of training and development trajectories.

Method	Endpoint NRMSE at the indicated rollout step
Heat/ResCNN	$h=1$	$h=100$	$h=200$	$h=500$
Base	$.00949\pm.00358$	$.00602\pm.00130$	$.00739\pm.00235$	$.00964\pm.00353$
HINE	$.00947\pm.00357$	$.00583\pm.00132$	$.00727\pm.00233$	$.00944\pm.00327$
CasEm	$.00938\pm.00357$	$\mathbf{.00353\pm.00100}$	$\mathbf{.00376\pm.00124}$	$\mathbf{.00415\pm.00157}$
Oracle	$.00938\pm.00357$	$.00352\pm.00101$	$.00376\pm.00123$	$.00414\pm.00155$
SO	$.408\pm.152$	$.305\pm.166$	$.195\pm.109$	$.138\pm.0843$

Appendix

Table 9: Full-state endpoint NRMSE for Heat and Fisher–KPP at selected rollout steps (mean

\pm

sample SD).

$K$	CasEm	Oracle
Base	$1.616\pm 0.619$
0	$1.049\pm 0.569$	$1.049\pm 0.569$
1	$.4418\pm 0.208$	$.4418\pm 0.208$
2	$.1868\pm 0.106$	$.1868\pm 0.106$
3	$.1135\pm 0.0641$	$.1135\pm 0.0641$
4	$.1054\pm 0.0630$	$.1054\pm 0.0630$

Appendix

Table 10: Heat/FNO subsystem-size ablation at 1000 steps. Full-state NRMSE (

\times 10^{-3}

; mean

\pm

sample SD) over 12 development trajectories.

$K$	CasEm	Oracle
Base	$748.98\pm 244$
4	$175.95\pm 98.4$	$199.03\pm 95.8$
8	$15.075\pm 6.83$	$16.286\pm 8.15$
16	$.01911\pm.00602$	$.01476\pm.00741$
32	$.06342\pm.00848$	$.01188\pm.00681$

Appendix

Table 11: Fisher–KPP/FNO subsystem-size ablation at 1000 steps. Values are full-state NRMSE (mean

\pm

sample SD) over 12 development trajectories.

	Base	$[0,8)$	$[8,16)$	$[16,24)$	$[16,32)$	Joint $0{:}16$
NRMSE	$744.066$	$27.616$	$55.751$	$290.060$	$282.600$	$\mathbf{0.0181}$

Appendix

Table 12: Mean full-state NRMSE at 1000 steps for Fisher–KPP/FNO with different Fourier subsystems.

Figure 10: Spectral fractions of Base error for Heat/FNO and Fisher–KPP/FNO. Colors distinguish individual modes for Heat and disjoint frequency bands for Fisher–KPP.

Backbone	Method	$h=1$	$h=200$	$h=1000$	$h=2000$
MLP	Base	$(9.82\pm 7.03)\!\times 10^{-4}$	$.390\pm.326$	$1.32\pm.491$	$(1.24\pm 6.08)\!\times 10^{36}$
	HINE	$(9.82\pm 7.03)\!\times 10^{-4}$	$.392\pm.329$	$1.36\pm.560$	$(1.36\pm 6.68)\!\times 10^{27}$
	CasEm	$\mathbf{(5.58\pm 3.74)\!\times 10^{-4}}$	$\mathbf{.0729\pm.0501}$	$\mathbf{.312\pm.360}$	$\mathbf{.595\pm.481}$
	Oracle	$(5.33\pm 3.73)\!\times 10^{-4}$	$.0615\pm.0372$	$.172\pm.112$	$.338\pm.180$
HNN	Base	$(1.56\pm.930)\!\times 10^{-4}$	$.0225\pm.0157$	$.120\pm.0936$	$.271\pm.209$
	HINE	$.0170\pm.00635$	$1.33\pm.500$	$1.27\pm.250$	$1.11\pm.234$

Appendix

Table 13: Full-state endpoint NRMSE for central-force dynamics (mean

\pm

sample SD).

Figure 11: One-step bidirectional variant of Figure 1 (c). Pink elements denote the added residual network and connections. Hats and forcing inputs are omitted.

Figure 12: Ablation of unidirectional coupling on Heat and Fisher–KPP. Top: full-state NRMSE. Bottom: aggregate-component RMSE of the full-state predictions. Lines show means over 12 development trajectories; shaded bands show one sample SD for CasEm and the bidirectionally coupled variant. Horizontal axes show rollout steps.

Figure 13: Water-subsystem construction and one-way guidance of S-DY. (a) Background-subtracted water fields undergo low-mode DCT encoding, autoregressive prediction, and inverse DCT reconstruction. Restoring the fitted seasonal–trend background yields regional water predictions. (b) 3 subsystem updates of 4 days each match 8 S-DY rounds, each producing 6 states at 6-hour intervals. Every 12 days, the subsystem target corrects only the full-state water fields, without feedback to the subsystem. Here, $t$ denotes physical time.

Figure 14: South Asian case-study domain, outlined by the rectangle.

Backbone	Method	Full TM	Full ST	Water TM	Water ST
S-DY	Base	0.0786	0.8136	0.2096	0.9500
	CasEm	0.0444 (43.46%)	0.7879 (3.16%)	0.0756 (63.92%)	0.8447 (11.08%)
ACE	Base	0.1353	0.8967	0.3699	1.2280
	CasEm	0.0502 (62.86%)	0.7903 (11.86%)	0.0804 (78.26%)	0.8693 (29.21%)

Appendix

Table 14: 10-year climate errors averaged equally across 11 trajectories. Parentheses give reductions relative to the corresponding backbone.

Figure 15: Per-start annual full-state and total-water TM (a,b) and ST (c,d) NRMSE. Colors distinguish methods; markers distinguish 2021–2024 rollout starts on IC11.

IC11
	Full state		Total water		Energy
Method	ST	TM	ST	TM	ST	TM
Base	0.9409	0.1580	1.3832	0.4691	0.4710	0.1637
CasEm	0.8727 (7.2%)	0.1180 (25.3%)	1.1822 (14.5%)	0.3277 (30.1%)	0.4065 (13.7%)	0.0540 (67.0%)
IC1–11 mean
Base	0.8967	0.1353	1.2280	0.3699	0.4873	0.0835

Appendix

Table 15: ACE-Energy NRMSE on IC11 and averaged equally across IC1–11. Energy errors are computed from the full-state rollout. Parentheses give reductions relative to Base; reductions for IC1–11 use ratios of mean errors.

Group	Member RMSE	Ensemble-mean RMSE	Fair CRPS
Water	18.49% (8/8)	16.11% (6/8)	16.45% (5/8)
Temperature	8.90% (8/9)	11.01% (8/9)	7.33% (5/9)
Zonal wind	7.25% (8/8)	8.00% (8/8)	8.11% (6/8)
Meridional wind	1.47% (6/8)	1.99% (7/8)	0.88% (7/8)
Surface pressure	5.38% (1/1)	9.75% (1/1)	9.19% (1/1)
Other fields	5.97% (23/26)	7.26% (24/26)	5.66% (19/26)

Appendix

Table 16: 10-year-mean-field error reductions across physical variable groups. Each cell gives the mean per-variable reduction and the number of improved variables.

Figure 16: Relative reduction in grouped spatial time-mean error for (a) 8 water layers, (b) all 34 fields, and (c) the 26 non-water fields. Blue indicates improvement and orange degradation.

Quantity	Horizontal-grid configurations
Horizontal grid	$1\!\times\!1$	$3\!\times\!6$	$6\!\times\!12$	$12\!\times\!24$	$18\!\times\!36$
Parameters	88	1,472	5,792	23,072	51,872
Dynamic dimension	8	32	32	32	32

Appendix

Table 17: Target-model configurations for the horizontal-grid and vertical-group ablations.

	Full state		Total water
Configuration	ST	TM	ST	TM
Full CasEm	0.80338	0.04011	0.73852	0.06652
Monthly averaging	0.80195 ( $-0.2\%$ )	0.03830 ( $-4.5\%$ )	0.73685 ( $-0.2\%$ )	0.06521 ( $-2.0\%$ )
Quarterly averaging	0.80326 ( $0.0\%$ )	0.04171 ( $+4.0\%$ )	0.73712 ( $-0.2\%$ )	0.06475 ( $-2.7\%$ )
Annual averaging	0.81774 ( $+1.8\%$ )	0.04636 ( $+15.6\%$ )	0.75383 ( $+2.1\%$ )	0.07098 ( $+6.7\%$ )
10-year averaging	0.81983 ( $+2.0\%$ )	0.04576 ( $+14.1\%$ )	0.76813 ( $+4.0\%$ )	0.07205 ( $+8.3\%$ )

Appendix

Table 18: Temporal target interventions: ST/TM NRMSE. Parentheses give relative changes (%) from full CasEm; positive is worse.

	Full state		Total water
Grid	ST	TM	ST	TM
$1\!\times\!1$	0.80909 ( $+0.7\%$ )	0.06292 ( $+56.9\%$ )	0.75808 ( $+2.6\%$ )	0.13636 ( $+105.0\%$ )
$3\!\times\!6$	0.80427 ( $+0.1\%$ )	0.04783 ( $+19.2\%$ )	0.73847 ( $0.0\%$ )	0.08588 ( $+29.1\%$ )
$6\!\times\!12$	0.80452 ( $+0.1\%$ )	0.04631 ( $+15.5\%$ )	0.73727 ( $-0.2\%$ )	0.08053 ( $+21.1\%$ )
$12\!\times\!24$	0.80338	0.04011	0.73852	0.06652
$18\!\times\!36$	0.80612 ( $+0.3\%$ )	0.03994 ( $-0.4\%$ )	0.74116 ( $+0.4\%$ )	0.06669 ( $+0.3\%$ )

Appendix

Table 19: Retrained target grids: ST/TM NRMSE. Parentheses give relative changes (%) from the

12\!\times\!24

reference; positive is worse.

	Full state		Total water
Groups	ST	TM	ST	TM
1	32.19587 ( $+3907.5\%$ )	8.59269 ( $+21322.7\%$ )	82.51388 ( $+11072.9\%$ )	19.14808 ( $+28685.5\%$ )
2	4.40199 ( $+447.9\%$ )	1.31205 ( $+3171.1\%$ )	11.10586 ( $+1403.8\%$ )	2.78783 ( $+4091.0\%$ )
4	0.97867 ( $+21.8\%$ )	0.24401 ( $+508.3\%$ )	1.15386 ( $+56.2\%$ )	0.64185 ( $+864.9\%$ )
8	0.80338	0.04011	0.73852	0.06652
Layer mean	17.85571 ( $+2122.6\%$ )	4.76563 ( $+11781.3\%$ )	45.78716 ( $+6099.9\%$ )	10.09095 ( $+15069.8\%$ )

Appendix

Table 20: Vertical grouping and correction allocation: ST/TM NRMSE. Parentheses give relative changes (%) from the 8-group reference; positive is worse.

	Full state		Total water
Transformation	ST	TM	ST	TM
Full CasEm	0.80338	0.04011	0.73852	0.06652
Horizontally uniform	0.81495	0.05796	0.78065	0.11824
Mean over longitude	0.80673	0.03870	0.74329	0.06419
Cosine-weighted mean over latitude	0.81718	0.06227	0.77709	0.12481

Appendix

Table 21: Absolute ST and TM NRMSE for horizontal correction-field transformations. Full CasEm is the unmodified reference; lower is better. ST uses uniform spatial weights, whereas TM uses cosine-latitude area weights.

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Samudra 2: Scaling Ocean Emulators across Resolutions

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Courant Institute School of Mathematics, Computing, and Data Science, New York University, New York, NY, USA · Open Athena AI Foundation, Inc., New York, NY, USA · Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, NJ, USA

Ocean Reynolds-Averaged Navier-Stokes

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Ocean general circulation models (OGCMs) are essential to climate science but computationally expensive, limiting ensemble size and forcing scenarios. Neural emulators promise orders-of-magnitude speedups, yet existing ocean emulators have not combined fine spatial resolution with multi-year autoregressive rollouts. Samudra, the first autoregressive neural ocean emulator to produce multi-decade global rollouts, is limited to

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Abstract

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