Monotone-Constrained Diffusion Models for Long-Horizon Production Forecasting

Temesgen Mikael Abraha; Yves Lucet

cs.LGSep 18, 2026

Monotone-Constrained Diffusion Models for Long-Horizon Production Forecasting

Authors: Temesgen Mikael Abraha, Yves Lucet

Organizations: Department of Computer Science, I. K. Barber Faculty of Science University of British Columbia Kelowna, BC V1V 1V7, Canada

Abstract

Forecasting a long horizon from only the first observations of a sequence is ill-posed: many trajectories are consistent with the same short history. We study this problem in oil and gas production forecasting, where forecasts made after roughly the first fifth of a well's producing life drive development and abandonment decisions, and where a usable forecast must describe a monotone decline. We present Physics-SIMS-TS, a conditional diffusion forecaster that combines negative guidance against synthetic artifacts, decline-curve constraints and an isotonic projection applied during sampling, spatial training augmentation, and an ensembled stochastic sampler yielding a full predictive distribution. Across three jurisdictions and more than 35,000 wells, under a shared-space, validation-frozen protocol, Physics-SIMS-TS is the most accurate diffusion forecaster in the comparison and is competitive with, but not superior to, ensembled transformer forecasters. Its forecasts are monotone by construction at a cost of at most 0.5% in mean squared error, and its trajectory ensemble yields calibrated intervals after one dispersion factor is fitted per jurisdiction. On six standard benchmarks a reversible-instance-normalization variant of the backbone is the leading diffusion baseline. We also quantify four protocol choices on which the measured ranking depends. Code and evaluation artifacts are released.

Figures & tables

Figure 1: Data augmentation comparison on 201 Montney wells. Left: MSE across methods for random forest and gradient boosting, with the augmentation factor in parentheses. Right: MSE improvement over the no-augmentation baseline. IDW outperforms TimeGAN and TimeVAE for both regressors.

Model	Method	1.5 $\times$	2 $\times$	3 $\times$	5 $\times$	8 $\times$	10 $\times$	12 $\times$	20 $\times$
RF	IDW	0.0071	0.0067	0.0070	0.0065	0.0067	0.0068	0.0067	0.0066
	TimeGAN	0.0066	0.0068	0.0070	0.0069	0.0075	0.0071	0.0077	0.0079
	TimeVAE	0.0069	0.0071	0.0069	0.0068	0.0075	0.0072	0.0077	0.0077
GB	IDW	0.0080	0.0078	0.0073	0.0075	0.0072	0.0070	0.0074	0.0075
	TimeGAN	0.0075	0.0074	0.0077	0.0075	0.0076	0.0077	0.0078	0.0077
	TimeVAE	0.0078	0.0078	0.0079	0.0077	0.0078	0.0078	0.0077	0.0080

Table 1: MSE across augmentation factors for the random forest (RF) and gradient boosting (GB) regressors. The best result per method is shown in italics . Baseline MSE without augmentation is 0.0069 for random forest and 0.0080 for gradient boosting. Percentage reductions reported in the text are computed from unrounded MSE.

Formation	Wells	Test	IDW+RF	SIMS-TS	Physics-SIMS-TS	scratch	pretrained
						PatchTST
Wilrich	51	11	0.0098	0.0067	0.0066	0.0158	0.0075
Golata	53	11	0.0380	0.0271	0.0299	0.0404	0.0278
Debolt	188	36	0.0222	0.0230	0.0236	0.0246	0.0224
Baldonnel	293	58	0.0227	0.0209	0.0218	0.0237	0.0199
Charlie Lake	690	138	0.0205	0.0180	0.0181	0.0192	0.0168

Table 2: Mean squared error across the British Columbia formation clusters (per-curve space, frozen configuration of Section 4.4 ); every column is a five-instance ensemble. Clusters below 2,000 wells use members fine-tuned from pretraining models whose training wells exclude every cluster’s test wells. The two PatchTST columns are the same architecture trained on the formation alone and given the identical pretrain-and-fine-tune pipeline. The best value per row is shown in italics .

Figure 2: Formation clusters, ordered by size. Left: MSE per method per cluster. Right: Physics-SIMS-TS relative to PatchTST at matched training budget, with negative values favouring Physics-SIMS-TS. The light bars compare against a PatchTST trained on the formation alone and show a large apparent advantage on the smallest formations; the dark bars give PatchTST the same pretrain-and-fine-tune pipeline that Physics-SIMS-TS uses, and the advantage disappears on six of the seven formations. The all-wells row is the full dataset, where the two treatments coincide.

Five-instance ensembles at matched training budget:
	BC (16,216)		Alberta (6,747)		PA Marc. (12,360)
Method	MSE	MAE	MSE	MAE	MSE	MAE
Physics-SIMS-TS (ours)	0.0171	0.0851	0.0172	0.0855	0.00916	0.0623
IDW+RF	0.0172	0.0879	0.0173	0.0885	0.00917	0.0633
PatchTST	0.0162	0.0847	0.0163	0.0856	0.00910	0.0630
iTransformer	0.0169	0.0860	0.0172	0.0878	0.00966	0.0650

Table 3: Cross-jurisdiction results on the all-wells cluster of each jurisdiction, all methods trained and evaluated in the shared per-curve space. The upper block is the matched comparison every claim in this article is stated against: five-instance ensembles trained for up to 200 epochs at patience 30. The lower block reports the remaining baselines as single trained models under their fixed configurations; Table 6 compares the diffusion baselines against a single Physics-SIMS-TS model. The best value per column within each block is shown in italics . BC denotes British Columbia and PA Marc. Pennsylvania Marcellus.

		squared error		absolute error
	Baseline (5-instance)	win	$p$	win	$p$
BC	IDW+RF	54.2%	$<0.001$	55.9%	$<0.001$
	PatchTST	50.1%	$0.015^{\,\blacktriangledown}$	51.7%	$0.34^{\,\circ}$
	iTransformer	52.6%	$0.43^{\,\circ}$	54.7%	$<0.001$
Alberta	IDW+RF	53.6%	$0.010$	53.9%	$<0.001$
	PatchTST	51.9%	$0.65^{\,\circ}$	52.5%	$0.057^{\,\circ}$

Table 4: Paired per-well comparison of Physics-SIMS-TS against the matched-budget baseline ensembles, all trained for 200 epochs. Win rate is the fraction of individual wells on which Physics-SIMS-TS attains the lower error;

p

is a two-sided Wilcoxon signed-rank test on per-well differences and is therefore undirected. Entries that do not resolve at the 5% level are marked

\circ

; entries that resolve against Physics-SIMS-TS are marked

\blacktriangledown

Figure 3: Cross-jurisdiction MSE on the all-wells cluster, logarithmic scale, showing Table 3 : Physics-SIMS-TS, IDW+RF, PatchTST, and iTransformer at the matched five-instance budget, and the remaining baselines as single trained models. Methods are ordered by mean rank across the three jurisdictions, so the order is shared by the panels. Ensembled PatchTST attains the lowest mean squared error on British Columbia and Alberta, and its Pennsylvania margin over Physics-SIMS-TS is not resolved; the ordering of the baselines below them is nearly identical across jurisdictions. The classical Arps prefix fit trails every method that learns across wells while remaining ahead of both zero-shot foundation models.

Method	One instance	Five instances
Physics-SIMS-TS (ours)	48.7 min	243 min
IDW+RF	8.7 min	43.4 min
PatchTST	4.1 min	20.3 min
iTransformer	2.3 min	11.4 min
CSDI	6.1 min	–
Diffusion-TS	15.7 min	–

Table 5: Compute cost on Pennsylvania Marcellus (12,360 wells; 9,888 train / 2,472 test), single-GPU wall clock on an RTX 6000 Ada. The middle column is training plus inference for one trained instance at that method’s sampling budget of Section 4.4 ; the last column gives the cost of the five-instance configurations compared in Table 3 , where those were run. The Chronos models are applied zero-shot, so their cost is inference only.

	BC (3,244 test)			Alberta (1,350 test)			PA Marcellus (2,472 test)
Method	MSE	MAE	win	MSE	MAE	win	MSE	MAE	win
Physics-SIMS-TS (one model)	0.0181	0.0875	–	0.0177	0.0873	–	0.0096	0.0636	–
CSDI	0.0191	0.0912	59.9%	0.0232	0.0989	69.1%	0.0118	0.0709	66.6%
Diffusion-TS	0.0280	0.1073	71.3%	0.0254	0.1058	72.1%	0.0152	0.0816	71.4%
TSDiff	0.0280	0.1068	69.9%	0.0301	0.1228	78.8%	0.0188	0.1043	82.2%
TimeGrad	0.0350	0.1200	75.0%	0.0330	0.1191	76.3%	0.0189	0.0907	76.1%

Table 6: Diffusion forecasters at matched budgets: one trained model each, multi-sample decoding, evaluated on the shared test wells. The Physics-SIMS-TS column is a single ensemble member, not the

R{=}5

ensemble reported elsewhere. Win rate is the fraction of individual wells on which Physics-SIMS-TS attains the lower squared error;

p

is a two-sided Wilcoxon signed-rank test on per-well differences. The best value per column is shown in italics .

Model	Train $\downarrow$ / Test $\rightarrow$	BC	Alberta	PA Marc.
Physics-SIMS-TS	BC	0.0171	0.0173 ( $+0.0000$ )	0.0099 ( $+0.0008$ )
	Alberta	0.0179 ( $+0.0007$ )	0.0172	0.0100 ( $+0.0008$ )
	PA Marcellus	0.0180 ( $+0.0009$ )	0.0178 ( $+0.0006$ )	0.0092
PatchTST	BC	0.0163	0.0163 ( $-0.0001$ )	0.0098 ( $+0.0005$ )
	Alberta	0.0170 ( $+0.0007$ )	0.0164	0.0100 ( $+0.0006$ )
	PA Marcellus	0.0174 ( $+0.0011$ )	0.0172 ( $+0.0009$ )	0.0093

Table 7: Cross-jurisdiction transfer MSE at matched training budget. Diagonal entries, shown in italics , are within-jurisdiction results for that model. Parenthesized deltas compare each transfer cell against the target jurisdiction’s own model of the same family, so both quantities share a test set. All Physics-SIMS-TS forecasts are monotone by construction; the transformer forecasts are not.

Model	$\to$ BC	$\to$ Alberta	$\to$ PA Marc.
	(Alberta+PA)	(BC+PA)	(BC+Alberta)
Physics-SIMS-TS	0.0177	0.0172	0.0099
iTransformer	0.0175	0.0170	0.0097
PatchTST	0.0167	0.0163	0.0096

Table 8: Leave-one-domain-out MSE at matched training budget: each model trains on the pooled training wells of two jurisdictions and predicts the held-out jurisdiction’s test wells. The best value per fold is shown in italics . Physics-SIMS-TS forecasts are monotone on every fold.

Rank	Model	ETTh1	ETTh2	ETTm1	ETTm2	Exchange	Weather	Overall
1	PatchTST	0.451	0.379	0.383	0.283	0.371	0.253	0.353
2	iTransformer	0.450	0.380	0.413	0.302	0.361	0.261	0.361
3	SIMS-TS (ours)	0.750	0.459	0.506	0.317	0.469	0.325	0.471
4	CSDI	0.558	0.661	0.519	0.432	0.658	0.285	0.519
5	Chronos-Bolt	0.525	0.425	1.033	0.352	0.368	0.413	0.520
6	Chronos-2	0.587	0.452	0.998	0.390	0.374	0.356	0.526

Table 9: Per-dataset and overall mean MSE across the four horizons of the standard benchmark suite. The best value per column is shown in italics . The RevIN SIMS-TS variant is the leading diffusion baseline overall, driven by the non-stationary datasets ETTh2, ETTm2, and Exchange.

Figure 4: Standard benchmark suite. (a) Overall mean MSE per model, colored by family; the RevIN SIMS-TS variant is the leading diffusion baseline, behind the two transformers. (b) Per-dataset mean MSE on a logarithmic scale, with the per-dataset best outlined. The diffusion baselines Diffusion-TS, TimeGrad, and TSDiff are unstable on the non-stationary datasets.

Rank	Model	Mean MSE	Std. across seeds
1	PatchTST	0.357	0.003
2	iTransformer	0.362	0.001
3	SIMS-TS (ours)	0.474	0.016
4	Chronos-Bolt	0.520	0.000 ‡
5	Chronos-2	0.526	0.000 ‡
6	CSDI	0.604	0.061

Table 10: Overall benchmark MSE averaged over the 24 settings and three seeds (42, 43, and 44). The best value is shown in italics . SIMS-TS is the most accurate generative model, ahead of the Chronos foundation models and CSDI. ‡ Chronos-Bolt and Chronos-2 are applied zero-shot with deterministic inference, so their predictions are identical across seeds and their standard deviation is exactly zero.

Model	ETTh1	ETTh2	ETTm1	ETTm2	Exchange	Weather
PatchTST	0.454	0.379	0.385	0.284	0.384	0.254
iTransformer	0.449	0.382	0.412	0.299	0.368	0.262
SIMS-TS (ours)	0.732	0.451	0.496	0.317	0.523	0.323
Chronos-Bolt	0.525	0.425	1.033	0.352	0.368	0.413
Chronos-2	0.587	0.452	0.998	0.390	0.374	0.356
CSDI	0.570	0.853	0.512	0.425	0.980	0.286

Table 11: Per-dataset MSE averaged over three seeds and the four horizons. The best value per column is shown in italics ; on Exchange, Chronos-Bolt and iTransformer differ only beyond the reported precision. Among the diffusion models, SIMS-TS is the more accurate on ETTh2, ETTm1, ETTm2, and Exchange, and CSDI on ETTh1 and Weather.

	BC		Alberta		PA Marcellus
Variant	MSE	Viol.	MSE	Viol.	MSE	Viol.
No physics (plain SIMS-TS)	0.01713	99.9%	0.01724	100%	0.00913	100%
Guidance only	0.01713	99.9%	0.01721	100%	0.00913	100%
Projection only	0.01714	0%	0.01726	0%	0.00917	0%
Guidance + projection (Physics-SIMS-TS)	0.01714	0%	0.01723	0%	0.00916	0%

Table 12: Physics-component ablation at the frozen configuration, evaluated at full ensemble level on each jurisdiction’s test wells. “Viol.” is the fraction of wells whose point forecast increases at any step, at zero tolerance; the projection guarantees zero by construction.

	BC			Alberta			PA Marcellus
Method (5-instance)	MSE	$+$ PAV	Viol.	MSE	$+$ PAV	Viol.	MSE	$+$ PAV	Viol.
IDW+RF	0.01721	0.01722	98.5%	0.01728	0.01723	99.6%	0.00917	0.00921	99.6%
PatchTST	0.01619	0.01620	100.0%	0.01629	0.01631	100.0%	0.00910	0.00910	100.0%
iTransformer	0.01686	0.01688	100.0%	0.01720	0.01721	99.9%	0.00966	0.00967	100.0%

Table 13: The same trade offered to the baselines: matched-budget baseline ensembles with the PAV projection applied to their finished forecasts, on the same test wells and in the same space. “Viol.” is the unprojected violation rate at zero tolerance; after projection it is zero for every method and jurisdiction.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

	ETTh1				ETTh2
Model	96	192	336	720	96	192	336	720
PatchTST	0.384	0.437	0.486	0.496	0.291	0.371	0.419	0.434
iTransformer	0.393	0.442	0.477	0.486	0.299	0.382	0.414	0.423
Chronos-Bolt	0.488	0.511	0.564	0.536	0.337	0.432	0.465	0.468
Chronos-2	0.468	0.544	0.633	0.702	0.340	0.452	0.490	0.525
CSDI	0.500	0.499	0.636	0.597	0.308	0.592	0.496	1.247

Appendix

Table 14: Per-horizon MSE on ETTh1 and ETTh2 (seed 42). The best value per column is shown in italics . CSDI leads the diffusion block on ETTh1, and SIMS-TS leads it on ETTh2 at horizons 192, 336, and 720.

	ETTm1				ETTm2
Model	96	192	336	720	96	192	336	720
PatchTST	0.324	0.359	0.392	0.457	0.177	0.246	0.308	0.402
iTransformer	0.340	0.384	0.419	0.507	0.187	0.260	0.340	0.421
Chronos-Bolt	1.007	1.081	0.984	1.062	0.242	0.315	0.372	0.480
Chronos-2	0.919	0.974	0.987	1.111	0.246	0.336	0.409	0.570
CSDI	0.376	0.402	0.545	0.755	0.215	0.491	0.486	0.534

Appendix

Table 15: Per-horizon MSE on ETTm1 and ETTm2 (seed 42). The best value per column is shown in italics . SIMS-TS leads the diffusion block on every ETTm2 horizon and on ETTm1 at horizon 720.

	Exchange				Weather
Model	96	192	336	720	96	192	336	720
PatchTST	0.082	0.169	0.352	0.882	0.171	0.218	0.272	0.351
iTransformer	0.090	0.174	0.325	0.854	0.176	0.228	0.282	0.359
Chronos-Bolt	0.087	0.184	0.342	0.858	0.345	0.407	0.407	0.494
Chronos-2	0.090	0.187	0.338	0.882	0.288	0.312	0.362	0.462
CSDI	0.152	0.260	0.767	1.451	0.220	0.242	0.315	0.362

Appendix

Table 16: Per-horizon MSE on Exchange and Weather (seed 42). The best value per column is shown in italics . SIMS-TS leads the diffusion block on every Exchange horizon and on Weather at horizon 96.

Method (5-instance)	MSE	MAE	Physics-SIMS-TS win rate	$p$
Physics-SIMS-TS	0.00868	0.0613	–	–
IDW+RF	0.00869	0.0619	49.9%	$0.91^{\,\circ}$
PatchTST	0.00886	0.0623	52.2%	$0.002$
iTransformer	0.00914	0.0631	53.0%	$<0.001$

Appendix

Table 17: Frozen-configuration results on the second Pennsylvania partition (2,472 held-out wells). Win rate: fraction of wells where Physics-SIMS-TS attains the lower per-well MSE;

p

: two-sided Wilcoxon signed-rank.

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