Target-Constrained Pooling for Trajectory Extrapolation

Ning Wang; Claire Cardie; Sainyam Galhotra

cs.LGOct 9, 2026

Target-Constrained Pooling for Trajectory Extrapolation

Authors: Ning Wang, Claire Cardie, Sainyam Galhotra

Organizations: Cornell University

Abstract

Trajectory extrapolation from a partial target trajectory is underdetermined, but completed related trajectories provide information about the unresolved continuation. We propose target-constrained pooling (TCP), which separates the finite-horizon endpoint from continuation shape and estimates that endpoint from two complementary source signals. Uniform pooling captures final response levels, while matched-increment pooling captures the remaining change after aligning source and target states. TCP selects representative sources, projects both endpoint estimates onto a target-consistent region, converts them into corrections of the free forecast, and controls their influence through strength calibration and degradation-aware weighting. Across 25,245 targets from five domains and both forecasting directions, TCP reduces mean RMSE by 35.8% to 42.1% for all ten target-only forecasters and improves all 100 forecaster, dataset, and direction comparisons. The two endpoint signals consistently outperform either alone, while representative selection remains within 0.64% to 1.02% of the full donor pool using 12.7x fewer donor inputs and 138.8x fewer matching distance computations.

Figures & tables

Figure 1: TCP workflow.

	RMSE		Gain (%)		Forward gain (%)					Reverse gain (%)
Forecaster	free	TCP	Pooled	Avg.	OLMo 2	LCDB	gCSI	CTRP	GDSC	OLMo 2	LCDB	gCSI	CTRP	GDSC
Michaelis–Menten	0.389	0.246	36.7	40.8	15.6	22.5	43.4	22.4	20.0	34.9	59.0	74.7	68.1	47.6
exp. saturation	0.383	0.246	35.8	41.2	29.0	23.9	43.2	22.4	19.8	34.5	58.0	74.1	62.5	44.1
Gompertz	0.380	0.244	35.8	39.1	28.8	12.3	42.1	21.3	27.8	26.3	39.0	72.9	66.6	53.7
Weibull cure	0.428	0.248	42.1	42.9	31.9	9.9	49.7	25.5	36.9	27.8	40.8	77.1	70.7	58.5
log–log cure	0.405	0.246	39.3	41.3	36.0	5.4	46.4	23.6	33.3	28.1	40.2	75.5	68.4	56.3

Table 1: Mean RMSE over the held-out continuation for each target-only forecaster and its TCP corrections.

Figure 2: Gain decomposition and source access.

Figure 3: Endpoint-source contributions.

Figure 4: Endpoint construction.

Figure 5: Direction constraint.

Figure 6: Calibration and weighting.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Metric	Definition and aggregation
Trajectory RMSE	$\ell_{i}^{v}=\bigl[H_{i}^{-1}\sum_{j=1}^{H_{i}}(\widehat{y}_{ij}^{v}-y_{ij})^{2}\bigr]^{1/2}$ , where $H_{i}$ is the number of held-out observations. Interior RMSE excludes the terminal observation and uses denominator $H_{i}-1$ .
Mean RMSE	$L_{v}(I)=\|I\|^{-1}\sum_{i\in I}\ell_{i}^{v}$ . Each trajectory has equal weight. The reported pooled RMSE is the mean trajectory RMSE over the corresponding targets.
RMSE gain	$100[1-L_{v}(I)/L_{\mathrm{ref}}(I)]$ . Positive values favor $v$ . Main-table Pooled uses $I=E$ , while Avg. is the equal-weight mean of the ten dataset and direction gains.
Positive degradation	$H_{v}(I)=\sum_{i\in I}\max\{\ell_{i}^{v}-\ell_{i}^{\mathrm{free}},0\}$ . Only increases relative to the free forecast contribute. Mean positive degradation is $H_{v}(I)/\|I\|$ , and calibration uses $G_{C}=H_{p}(C)/\|C\|$ .
Targets with increased error and negative cells	$n_{\mathrm{harm}}^{v}(I)=\sum_{i\in I}{\bf 1}\{\ell_{i}^{v}>\ell_{i}^{\mathrm{free}}\}$ , with rate $100n_{\mathrm{harm}}^{v}/\|I\|$ . A negative cell instead satisfies $L_{v}(I_{\mathrm{cell}})>L_{\mathrm{free}}(I_{\mathrm{cell}})$ .
Degradation reduction	$100[1-H_{v}(I)/H_{\mathrm{ref}}(I)]$ for a positive reference total. Component tables use TCP as $v$ , while expanded-pool tables use Full. Each caption specifies the reference.

Appendix

Table 2: Definitions of the reported metrics.

Figure 7: Selected trajectory examples illustrating TCP correction mechanisms.

	T: target only			K: knee pooling set			F: all sources
Forecaster	free	TCP	gain	free	TCP	gain	free	TCP	gain
Michaelis–Menten	0.296	0.193	34.6	0.295	0.194	34.4	0.295	0.194	34.3
exp. saturation	0.290	0.193	33.5	0.290	0.193	33.4	0.293	0.194	33.9
Gompertz	0.322	0.193	40.1	0.276	0.196	29.1	0.271	0.193	28.8
Weibull cure	0.367	0.198	46.2	0.295	0.195	33.9	0.275	0.194	29.3
log–log cure	0.344	0.197	42.8	0.291	0.198	32.0	0.276	0.196	29.1

Appendix

Table 3: Matched subset of 1,032 evaluation targets (40 case-folds from 20 cohorts, covering all five domains and both directions), drawn by a frozen outcome-independent rule. Unweighted mean tail RMSE of the free forecast and of TCP, and TCP’s gain (%), for base forecasters fitted on the target alone (T), on the knee-selected pooling set (K) and on all permitted source trajectories (F). Every TCP uses the same pooled endpoints, with strengths and weight refitted on

C

within each regime. NSL-PFN has no pooled variant (N/A). ∗ Mean over three seeds of the iterative Gaussian-process fit (gain: mean of the per-seed gains,

1.65\pm 0.08

	Tail			Interior tail
Forecaster	$G$	$A$	$C$	$G$	$A$	$C$
Michaelis–Menten	36.5	21.0	15.5	33.9	16.7	17.2
exp. saturation	35.9	19.4	16.5	33.0	14.7	18.3
Gompertz	31.7	14.5	17.2	27.1	6.6	20.5
Weibull cure	35.4	18.7	16.7	31.5	12.0	19.5
log–log cure	34.0	17.0	17.0	29.9	9.9	20.0

Appendix

Table 4: Endpoint-source intervention on the 23,486 targets with at least one eligible alternative cohort. Cohort-balanced means over 67 cohorts of the per-cohort ratios (%):

G=(L_{F}-L_{B})/L_{F}

is TCP’s gain over the free forecast,

A=(L_{F}-L_{P})/L_{F}

the gain when the endpoints are pooled from other cohorts with the same frozen strengths and weight, and

C=(L_{P}-L_{B})/L_{F}=G-A

the additional value of the target’s own cohort. Here

C

is measured in percentage points of gain over free. Interior: tail RMSE without the terminal point. All 152,296 alternative endpoint pairs use the fit-free common operator.

Forecaster	$G$	$A$	$C$	$G$	$A$	$C$	Full
	TW			CML
Michaelis–Menten	37.0	21.5	15.5	36.8	22.6	14.2	36.7
exp. saturation	36.1	20.3	15.8	36.3	21.3	14.9	35.8
Gompertz	35.7	17.9	17.8	31.4	15.5	15.9	35.8
Weibull cure	42.5	25.3	17.2	35.9	20.2	15.7	42.1
log–log cure	39.5	22.0	17.5	34.2	18.4	15.8	39.3

Appendix

Table 5: Endpoint-source decomposition of the tail loss (%) under two further estimands on the 23,486 paired targets: target-weighted (TW) ratios of mean losses and ratios of cohort mean losses (CML). Full: TCP’s target-weighted gain on all 25,245 targets, which equals the pooled gain of Table 1 .

	TCP	Full pool	Random full	Random cal	All full	All cal
Michaelis–Menten	0.246218	0.244073	$0.314195\pm 0.000861$	$0.268289\pm 0.000209$	0.285807	0.252563
exp. saturation	0.246135	0.244021	$0.314181\pm 0.000861$	$0.268219\pm 0.000182$	0.285786	0.252479
Gompertz	0.244201	0.242637	$0.312624\pm 0.000855$	$0.267773\pm 0.000752$	0.283956	0.251681
Weibull cure	0.247682	0.245716	$0.313251\pm 0.000803$	$0.279238\pm 0.000761$	0.284375	0.259519
log–log cure	0.245606	0.243876	$0.312322\pm 0.000829$	$0.273917\pm 0.000765$	0.283530	0.255825
arctangent	0.246036	0.243981	$0.314150\pm 0.000862$	$0.268088\pm 0.000206$	0.285760	0.252427

Appendix

Table 6: Pooled mean tail RMSE for the retained calibrated TCP/Full-pool and full/calibrated ensemble controls (lower is better). Random variants report mean plus/minus sample SD over the three seeds.

	TCP	Full pool	Random full	Random cal	All full	All cal
Michaelis–Menten	533.838	539.080	$2275.654\pm 17.496$	$524.620\pm 8.502$	1917.077	651.147
exp. saturation	535.267	540.403	$2278.717\pm 17.492$	$525.262\pm 9.428$	1918.047	653.004
Gompertz	566.091	564.000	$2341.523\pm 14.918$	$609.393\pm 11.992$	1911.796	686.564
Weibull cure	588.878	587.641	$2290.648\pm 17.127$	$720.359\pm 5.442$	1896.276	770.165
log–log cure	585.241	591.866	$2318.720\pm 17.298$	$669.517\pm 10.496$	1912.822	735.323
arctangent	533.539	538.838	$2280.384\pm 17.874$	$524.853\pm 9.547$	1919.371	654.375

Appendix

Table 7: Total positive RMSE degradation over the shared free forecast for the same variants (lower is better). Random variants report mean plus/minus sample SD over the three seeds.

Forecaster	TCP full RMSE	Full pool RMSE	Gain (%)	TCP full degr.	Full pool degr.
MM	0.250283	0.246352	1.571	888.739	826.821
Exp. sat.	0.250244	0.246260	1.592	890.238	827.553
Gompertz	0.247835	0.243874	1.598	922.001	844.066
Weibull cure	0.248267	0.244250	1.618	913.652	837.991
Log-log cure	0.247362	0.243347	1.623	924.248	852.631
Arctan	0.250159	0.246159	1.599	888.823	825.463

Appendix

Table 8: Full-strength selected-pool and Full-pool mixtures use the same

C

-only weighting rule with both strengths fixed to one.

Forecaster	free	TCP	Full	(%)	TCP	Full	(%)	TCP	Full
	Mean tail RMSE			Full gain	Total RMSE degradation $H$		Degrad. red.	Harmed count
Michaelis–Menten	0.388869	0.246218	0.244073	0.87	533.838	539.080	-0.98	4938	4902
exp. saturation	0.383382	0.246135	0.244021	0.86	535.267	540.403	-0.96	5082	5085
Gompertz	0.380124	0.244201	0.242637	0.64	566.091	564.000	0.37	7052	6856
Weibull cure	0.427775	0.247682	0.245716	0.79	588.878	587.641	0.21	6672	6444
log–log cure	0.404516	0.245606	0.243876	0.70	585.241	591.866	-1.13	7321	7149

Appendix

Table 9: Complete pooled endpoint-pool comparison in the fixed evaluation coordinate. Gain is

100(1-L_{\rm full}/L_{\rm TCP})

; excess reduction is

100(1-H_{\rm full}/H_{\rm TCP})

, so negative entries mean more error or more total RMSE degradation for full. Total RMSE degradation and harmed counts use the identical free forecast.

	Forward gain (%)					Reverse gain (%)
Forecaster	OLMo 2	LCDB	gCSI	CTRP	GDSC	OLMo 2	LCDB	gCSI	CTRP	GDSC
Michaelis–Menten	8.30	7.17	0.98	0.44	0.88	2.57	6.71	1.27	1.05	0.24
exp. saturation	4.81	5.69	0.86	0.42	0.92	2.58	6.66	1.55	1.38	0.66
Gompertz	4.35	2.04	0.28	0.25	1.37	2.37	6.68	1.36	0.81	0.28
Weibull cure	4.77	-0.63	0.54	0.33	1.71	2.36	0.96	1.22	1.55	0.63
log–log cure	5.49	0.17	0.18	0.27	2.01	2.38	0.95	1.19	1.64	0.17

Appendix

Table 10: All 100 domain–direction gains from expanding the endpoint pool, relative to TCP. Positive means lower mean tail RMSE for full; three negative entries favor the selected pool.

Forecaster	error	degradation	with more error	and direction	(%)
	Domain cells with more		Cohort cells	Worst cohort	Gain
Michaelis–Menten	0	2	25	lcdb_005—fwd	-17.65
exp. saturation	0	2	28	olmo_027—fwd	-47.07
Gompertz	0	3	38	olmo_027—fwd	-48.21
Weibull cure	1	2	39	olmo_027—fwd	-42.58
log–log cure	0	3	45	lcdb_008—bwd	-30.52

Appendix

Table 11: Adverse cell counts for full relative to TCP and the worst cohort–direction gain. Counts cover ten domain–direction and 146 cohort–direction cells; every cohort is retained in the data ledger.

Bank	C donor inputs	E donor inputs	C+E donor inputs	Mean C+E
TCP	167,438	525,756	693,194	20.293
Full pool	2,071,471	6,702,630	8,774,101	256.853
Random-two	17,830	50,490	68,320	2.000
All-endpoints	2,071,471	6,702,630	8,774,101	256.853

Appendix

Table 12: Donor inputs consumed after selection across the 34,160 calibration/evaluation packets. TCP/Full-pool use original donor curves for U/M construction; the ensembles use selected donor endpoints, per seed for Random-two.

Bank	C distance elements	E distance elements	C+E total	Mean C+E
TCP	20,045,722	61,577,816	81,623,538	2389.448
Full pool	2,663,962,946	8,666,404,606	11,330,367,552	331685.233
Random-two	0	0	0	0.000 (not used)
All-endpoints	0	0	0	0.000 (not used)

Appendix

Table 13: Exact candidate-stage distance elements in N1 M matching across the same calibration/evaluation packets. Zero for an ensemble means that matching is not used, not that all computation is absent.

Bank	C terms	E terms	C+E terms	Mean C	Mean E	Mean C+E
TCP	17,830	50,490	68,320	2.000	2.000	2.000
Full pool	17,830	50,490	68,320	2.000	2.000	2.000
Random-two	17,830	50,490	68,320	2.000	2.000	2.000
All-endpoints	2,071,471	6,702,630	8,774,101	232.358	265.503	256.853

Appendix

Table 14: Exact logical endpoint response terms per base forecaster and run, before cache sharing. Full/cal variants share each Random-two or All-endpoints bank; the displayed means use the stated packet denominators.

	All	Holds	Fails	Holds/active	Holds/inactive	Fails/active	Fails/inactive
Michaelis–Menten	+0.0509	+0.6353	-5.6695	+10.7668	-0.0014	-10.0895	-1.1765
exp. saturation	+0.0330	+0.6247	-5.7334	+10.5737	+0.0008	-10.2599	-1.1402
Gompertz	-0.0632	+0.4467	-4.3230	+7.4438	+0.0301	-8.8601	+0.3162
Weibull cure	-0.0360	+0.5261	-4.5663	+7.2954	+0.1101	-9.2418	+0.4581
log–log cure	+0.0096	+0.5231	-4.2053	+7.5345	+0.0963	-8.5958	+0.4546
arctangent	+0.0307	+0.6163	-5.6714	+10.3765	+0.0064	-10.3415	-0.9078

Appendix

Table 15: Signed percentage change in mean tail RMSE after removing projection and recalibrating, relative to TCP. Positive favors TCP; all primary and activity groups are retained.

	All	Holds	Fails	Holds/active	Holds/inactive	Fails/active	Fails/inactive
Michaelis–Menten	-16.586	-4.648	-11.938	+5.304	-9.952	-8.105	-3.833
exp. saturation	-13.994	-2.763	-11.231	+6.563	-9.326	-7.563	-3.668
Gompertz	+2.204	+5.516	-3.312	+11.677	-6.161	-1.380	-1.932
Weibull cure	+6.398	+10.041	-3.643	+13.184	-3.143	-2.290	-1.353
log–log cure	+3.558	+7.895	-4.338	+12.453	-4.558	-3.218	-1.120
arctangent	-14.034	-3.142	-10.892	+5.812	-8.954	-7.600	-3.292

Appendix

Table 16: No-direction total RMSE degradation minus TCP total degradation within each group. Positive favors TCP; improvements and regressions both remain visible.

		Gain (%)
Variant	Better	min	max	Degrad.	Neg.	Worst
hard U	10/10	12.9	14.1	10/10	27	$-80.9$
hard M	10/10	1.8	4.2	10/10	2	$-5.6$
full strength	9/10	$-$ 0.15	2.0	10/10	8	$-16.7$
U only	10/10	2.7	4.5	10/10	3	$-6.3$
M only	10/10	1.7	2.4	3/10	0	$5.8$

Appendix

Table 17: Exact values for Figure 6 and the calibrated branch comparisons of RQ2, from the target-only comparison of Table 1 ; U and M denote uniform endpoint pooling and matched-increment pooling. Each variant is compared with TCP within the same base forecaster (10 forecasters, 100 cells). Gain: TCP’s pooled RMSE gain over the variant, minimum and maximum over forecasters (positive favors TCP). Better and Degrad.: forecasters where TCP has lower RMSE and lower total RMSE degradation. Neg.: the variant’s cells worse than the free forecast. Worst: the variant’s worst cell gain (%) against the free forecast (TCP: no negative cells). † One strength on the equal-weight hard forecast, which changes both the strengths and the weight.

	hard U		hard M		full strength		shared †
Forecaster	gain	degrad.	gain	degrad.	gain	degrad.	gain	degrad.
Michaelis–Menten	14.0	71.6	3.6	34.0	1.6	39.9	0.56	15.5
exp. saturation	14.0	71.5	3.6	34.0	1.6	39.9	0.54	15.5
Gompertz	14.1	70.5	3.6	34.2	1.5	38.6	0.57	13.6
Weibull cure	13.1	68.8	2.3	31.0	0.24	35.5	1.3	19.9
log–log cure	13.5	69.4	2.9	32.2	0.71	36.7	1.0	16.9

Appendix

Table 18: Strength-control variants: TCP RMSE gain (%) and total RMSE degradation reduction (%) over each variant. Positive favors TCP. Both are zero for the TCP reference.

	U only		M only
Forecaster	gain	degrad.	gain	degrad.
Michaelis–Menten	2.9	17.2	2.0	$-$ 0.85
exp. saturation	2.9	17.3	2.0	$-$ 0.61
Gompertz	2.7	17.4	2.3	1.6
Weibull cure	4.5	22.0	2.2	2.6
log–log cure	3.8	19.5	2.3	2.9

Appendix

Table 19: Calibrated branch variants: TCP RMSE gain (%) and total RMSE degradation reduction (%) over each variant. Positive favors TCP. Both are zero for the TCP reference.

	equal		RMSE only		degradation only
Forecaster	gain	degrad.	gain	degrad.	gain	degrad.
Michaelis–Menten	0.40	2.5	$-$ 0.65	10.8	1.8	$-$ 9.9
exp. saturation	0.39	2.6	$-$ 0.66	10.8	1.8	$-$ 9.9
Gompertz	0.49	1.8	$-$ 0.28	10.6	1.6	$-$ 6.4
Weibull cure	0.98	5.5	$-$ 0.13	10.3	1.5	$-$ 8.2
log–log cure	0.78	4.0	$-$ 0.22	11.2	1.6	$-$ 8.0

Appendix

Table 20: Weight-objective variants: TCP RMSE gain (%) and total RMSE degradation reduction (%) over each variant. Positive favors TCP. Both are zero for the TCP reference.

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