System-Anchored Knee Estimation for Low-Cost Context Window Selection in PDE Forecasting

Wenshuo Wang; Fan Zhang

cs.AIMar 26, 2026

System-Anchored Knee Estimation for Low-Cost Context Window Selection in PDE Forecasting

Authors: Wenshuo Wang, Fan Zhang

Organizations: School of Future Technology, South China University of Technology, China · State Key Laboratory of Ocean Sensing & Ocean College, Zhejiang University, China · Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, USA

Abstract

Autoregressive neural PDE simulators predict the evolution of physical fields one step at a time from a finite history, but low-cost context-window selection for such simulators remains an unformalized problem. Existing approaches to context-window selection in time-series forecasting include exhaustive validation, direct low-cost search, and system-theoretic memory estimation, but they are either expensive, brittle, or not directly aligned with downstream rollout performance. We formalize explicit context-window selection for fixed-window autoregressive neural PDE simulators as an independent low-cost algorithmic problem, and propose System-Anchored Knee Estimation (SAKE), a two-stage method that first identifies a small structured candidate set from physically interpretable system anchors and then performs knee-aware downstream selection within it. Across all eight PDEBench families evaluated under the shared $L\in\{1,\dots,16\}$ protocol, SAKE is the strongest overall matched-budget low-cost selector among the evaluated methods, achieving 67.8% Exact, 91.7% Within-1, 6.1% mean regret@knee, and a cost ratio of 0.051 (94.9% normalized search-cost savings).

Figures & tables

Dataset	$L_{\text{core}}$	$L_{\text{plateau}}$	System band	Knee-in-band coverage	Covered oracle-knee models
DiffReact	3	6	[3, 4, 5, 6]	25.0%	UNet
RDB	2	6	[2, 3, 4, 5, 6]	75.0%	ConvLSTM, FNO, Transformer

Table 3: Dataset-level outputs of the system anchor estimator. Its role is to provide anchors and brackets for the downstream selector, rather than to directly serve as the final recommender.

Appendix figures & tables38 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Setting used in this appendix
Candidate grid	$\mathcal{L}=\{1,\dots,16\}$ , matching the shared main protocol
Additional PDEBench families	1D advection, 1D Burgers, 1D diffusion–sorption, 2D Darcy flow, 2D incompressible Navier–Stokes, 2D compressible Navier–Stokes
Backbones / seeds	Same four one-step autoregressive backbones and same three-seed protocol as the main paper
Methods	System-core, Direct-4-Shortlist, ASHA, and SAKE under matched downstream budgets
Oracle construction	Full sweep over $\{1,\dots,16\}$ for each dataset–backbone–seed case
Reported metrics	Exact, Within-1, mean $\|\Delta L\|$ , $\mathrm{Regret}_{\mathrm{knee}}$ , $\mathrm{Regret}_{\mathrm{best}}$ , mean unique $L$ evaluations, and cost ratio

Appendix

Table 4: Protocol for the broader PDEBench extension. The representative main-text pair is kept for compact case-wise discussion, while this appendix tests whether the same matched-budget ordering survives on the full PDE family suite under the shared main protocol.

Method	Exact	Within-1	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Mean unique $L$ eval.	Cost ratio
System-core	12.5%	40.6%	28.0%	0.0	0.000
Direct-4-Shortlist	50.0%	84.4%	9.4%	7.3	0.072
ASHA	57.3%	88.0%	8.0%	9.6	0.092
SAKE	67.8%	91.7%	6.1%	5.3	0.051

Appendix

Table 5: Overall aggregate effectiveness across all eight PDEBench families evaluated under the shared

\mathcal{L}=\{1,\dots,16\}

protocol. The representative main-text pair is included in this aggregate; the per-family tables below report only the six additional families not shown in the body.

Method	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Mean $\mathrm{Regret}_{\mathrm{best}}$	Mean unique $L$ eval.	Cost ratio
System-core	25.0%	58.3%	1.25	17.6%	18.1%	0.0	0.000
Direct-4-Shortlist	58.3%	91.7%	0.50	5.8%	6.4%	7.1	0.070
ASHA	66.7%	100.0%	0.42	4.9%	5.5%	9.2	0.089
SAKE	75.0%	100.0%	0.25	3.8%	4.4%	5.0	0.048

Appendix

Table 6: Aggregate effectiveness on the additional PDEBench family Advection .

Backbone	System-core	D4-Shortlist	ASHA	SAKE	Oracle knee	Oracle best
ConvLSTM	4	4	4	4	4	5
FNO	4	5	5	5	5	6
Transformer	4	4	4	3	3	4
U-Net	5	6	6	6	6	7

Appendix

Table 7: Case-level selected windows on the additional PDEBench family Advection .

Method	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Mean $\mathrm{Regret}_{\mathrm{best}}$	Mean unique $L$ eval.	Cost ratio
System-core	16.7%	50.0%	1.58	24.5%	25.1%	0.0	0.000
Direct-4-Shortlist	50.0%	83.3%	0.67	8.9%	9.6%	7.2	0.071
ASHA	58.3%	91.7%	0.58	7.4%	8.1%	9.3	0.090
SAKE	66.7%	91.7%	0.42	5.8%	6.5%	5.2	0.050

Appendix

Table 8: Aggregate effectiveness on the additional PDEBench family Burgers .

Backbone	System-core	D4-Shortlist	ASHA	SAKE	Oracle knee	Oracle best
ConvLSTM	3	5	4	4	4	5
FNO	4	5	5	5	5	6
Transformer	4	4	4	4	4	5
U-Net	4	4	4	3	3	4

Appendix

Table 9: Case-level selected windows on the additional PDEBench family Burgers .

Method	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Mean $\mathrm{Regret}_{\mathrm{best}}$	Mean unique $L$ eval.	Cost ratio
System-core	8.3%	33.3%	2.08	31.8%	32.4%	0.0	0.000
Direct-4-Shortlist	41.7%	83.3%	0.92	10.9%	11.7%	7.1	0.070
ASHA	50.0%	83.3%	0.75	8.8%	9.6%	9.5	0.092
SAKE	75.0%	91.7%	0.42	5.6%	6.3%	5.1	0.049

Appendix

Table 10: Aggregate effectiveness on the additional PDEBench family DiffusionSorption .

Backbone	System-core	D4-Shortlist	ASHA	SAKE	Oracle knee	Oracle best
ConvLSTM	2	3	3	3	3	4
FNO	4	6	5	5	5	6
Transformer	4	5	5	4	4	5
U-Net	3	4	3	3	2	4

Appendix

Table 11: Case-level selected windows on the additional PDEBench family DiffusionSorption .

Method	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Mean $\mathrm{Regret}_{\mathrm{best}}$	Mean unique $L$ eval.	Cost ratio
System-core	16.7%	41.7%	1.67	23.4%	24.0%	0.0	0.000
Direct-4-Shortlist	50.0%	83.3%	0.75	9.1%	9.8%	7.3	0.072
ASHA	58.3%	91.7%	0.58	7.8%	8.5%	9.4	0.091
SAKE	66.7%	91.7%	0.42	6.3%	7.0%	5.3	0.051

Appendix

Table 12: Aggregate effectiveness on the additional PDEBench family Darcy .

Backbone	System-core	D4-Shortlist	ASHA	SAKE	Oracle knee	Oracle best
ConvLSTM	4	4	4	4	4	5
FNO	4	3	3	3	3	4
Transformer	4	6	5	5	5	6
U-Net	4	4	4	3	3	5

Appendix

Table 13: Case-level selected windows on the additional PDEBench family Darcy .

Method	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Mean $\mathrm{Regret}_{\mathrm{best}}$	Mean unique $L$ eval.	Cost ratio
System-core	8.3%	33.3%	2.17	32.7%	33.4%	0.0	0.000
Direct-4-Shortlist	50.0%	83.3%	0.83	10.8%	11.5%	7.4	0.073
ASHA	58.3%	83.3%	0.67	9.3%	10.1%	9.7	0.094
SAKE	66.7%	91.7%	0.50	7.1%	7.9%	5.4	0.052

Appendix

Table 14: Aggregate effectiveness on the additional PDEBench family IncompNS .

Backbone	System-core	D4-Shortlist	ASHA	SAKE	Oracle knee	Oracle best
ConvLSTM	4	6	6	5	5	6
FNO	4	6	6	6	6	7
Transformer	4	5	4	4	4	6
U-Net	4	4	4	3	3	5

Appendix

Table 15: Case-level selected windows on the additional PDEBench family IncompNS .

Method	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Mean $\mathrm{Regret}_{\mathrm{best}}$	Mean unique $L$ eval.	Cost ratio
System-core	8.3%	33.3%	2.33	36.8%	37.5%	0.0	0.000
Direct-4-Shortlist	41.7%	75.0%	1.00	12.1%	12.9%	7.6	0.074
ASHA	50.0%	75.0%	0.83	10.6%	11.4%	9.9	0.096
SAKE	58.3%	83.3%	0.67	8.4%	9.2%	5.6	0.054

Appendix

Table 16: Aggregate effectiveness on the additional PDEBench family CompNS .

Backbone	System-core	D4-Shortlist	ASHA	SAKE	Oracle knee	Oracle best
ConvLSTM	4	7	6	6	5	7
FNO	4	6	6	6	6	7
Transformer	4	5	5	4	4	6
U-Net	4	4	4	3	3	5

Appendix

Table 17: Case-level selected windows on the additional PDEBench family CompNS .

Backbone	Input/context encoding	Core spatiotemporal block	Output head	Key hyperparameters	Trainable parameters (DiffReact / RDB)
U-Net	causal temporal encoder over the full history	TemporalEncoder $+$ 2D U-Net	C-channel next-step field decoder	temporal hidden $32$ , temporal layers $2$ , kernel $3$ , max context $8$ ; U-Net base $32$ , depth $3$	1,946,212 / 1,945,826
FNO	causal temporal encoder over the full history	TemporalEncoder $+$ 2D FNO	C-channel spectral decoder	temporal hidden $32$ , temporal layers $2$ , kernel $3$ , max context $8$ ; width $48$ , modes $16$ , depth $4$	4,751,460 / 4,751,298
ConvLSTM	raw frame sequence warm-up over the full history	1-layer ConvLSTM with hidden-state recurrence	$1\times 1$ conv to next-step field	hidden channels $64$ , layers $1$ , kernel $3$	152,450 / 150,081
Transformer	per-patch causal tokenization of the full history	patchwise Transformer encoder	linear patch reconstruction to field space	patch size $8$ , $d_{\mathrm{model}}=96$ , heads $4$ , depth $2$	939,936 / 927,584

Appendix

Table 18: Backbone-specific architecture fields for the four one-step autoregressive simulators used in the main experiments. Trainable-parameter counts are exact for the default code instantiations at resolution

128\times 128

, reported as DiffReact / RDB because the two datasets have different channel counts. The purpose of this table is to make the backbone implementations explicit while keeping the selector comparison itself on a shared footing.

Item	Setting used in the main experiments
Datasets	DiffReact, RDB
Candidate windows	shared candidate grid $\mathcal{L}=\{1,\dots,16\}$ for the main-text and appendix evaluations
Training seeds	$\{0,1,2\}$
Split	trajectory-level 0.8/0.1/0.1 train/validation/test split
Optimizer	Adam
Learning rate	$3\times 10^{-4}$

Appendix

Table 19: Shared training and evaluation protocol for all four backbones. These settings are held fixed so that differences in the main results reflect selector behavior rather than incompatible model pipelines.

Stage-1 component	Symbol / object	Role in the method	Setting used in the main experiments
System representation	PCA projector	clean-system state summary	randomized PCA on flattened normalized coarsened fields; $800$ samples, $99\%$ variance target, max $64$ components
Lightweight dynamics model	ridge VAR( $L$ )	low-cost system-risk estimation	one-step linear autoregressive model with ridge $10^{-3}$ fit for each $L\in\mathcal{L}$ on the shared $\{1,\dots,16\}$ grid used throughout the paper
Long-history reference	$L_{\max}$	tail-risk reference window	$16$ under the shared main protocol
Tail tolerance fraction	$\rho$	defines $L_{\mathrm{core}}$	$0.05$
Plateau threshold	$\tau_{\mathrm{pl}}$	defines $L_{\mathrm{plateau}}$	relative-gain threshold $0.05$
Uncertainty estimate	bootstrap UCB	confidence control for anchor extraction	$300$ bootstrap resamples, confidence level $1-\alpha$ with $\alpha=0.05$

Appendix

Table 20: Stage-1 settings for system-anchor construction. The reported fields specify how the clean-system estimator is instantiated before any downstream simulator pilots are run.

Stage-2 component	Symbol / object	Role in the method	Setting used in the main experiments
Coarse pilot budget	stage-1 pilots	first-pass ranking over $\mathcal{S}_{0}$	$2$ epochs; max $1024$ train pairs; $8$ train trajectories, $4$ validation trajectories; rollout horizons $8/4$ (train/val) with max $4$ validation rollout trajectories
Top-candidate count	top- $k$	number of refined seeds	$k=2$
Neighborhood radius	$h$	local grid expansion around promising candidates	$h=1$ hop on the ordered window grid
Refined candidate set	$\mathcal{S}_{1}$	stage-two evaluation set	union of top- $2$ neighborhoods, capped at $6$ candidates, with smallest and largest anchors retained
Stronger pilot budget	stage-2 pilots	second-pass evaluation budget	$6$ epochs; max $4096$ train pairs; $24$ train trajectories, $16$ validation trajectories; rollout horizons $32/16$ (train/val) with max $8$ validation rollout trajectories
Rollout anchors	$A_{\mathrm{eval}}$	multi-anchor validation used in stage two	$4$ rollout anchors per candidate

Appendix

Table 21: Stage-2 settings for knee-aware downstream selection. The key point is that these settings are fixed globally and used to compare SAKE against the direct-search baselines under matched downstream budgets.

Method	Initial candidate construction	Uses system anchors?	Downstream budget / final rule
System-core	clean-system estimate only	yes	none / direct return of $L_{\mathrm{core}}$
Direct-3-Shortlist	uniformly spaced 3-point shortlist	no	initial shortlist $\{L_{\min},\operatorname{round}(0.5(L_{\min}+L_{\max})),L_{\max}\}$ on the active discrete grid
Direct-4-Shortlist	uniformly spaced 4-point shortlist	no	initial shortlist formed from the two internal equally spaced grid quantiles on the active discrete grid
SAKE	$\mathcal{S}_{0}=\{1,L_{\mathrm{core}},L_{\mathrm{plateau}}\}$	yes	same pilot family with knee-aware final rule

Appendix

Table 22: Selector-side comparison setup. The direct baselines share the same downstream pilot family and local decision rule, while SAKE differs by replacing the generic shortlist with a system-anchored one.

Condition	Anchor-source definition
Clean full-state	reference anchor input
Gaussian noise $\sigma=0.01$	same trajectories with additive mild observation noise
Gaussian noise $\sigma=0.05$	same trajectories with moderate observation noise
Gaussian noise $\sigma=0.10$	same trajectories with heavy observation noise
Downsample $\times 2$	anchor extraction from half-resolution trajectories
Downsample $\times 4$	anchor extraction from quarter-resolution trajectories

Appendix

Table 23: Anchor-source conditions used to test robustness of the clean-system stage. The downstream selector is unchanged across rows.

Anchor condition	Mean $\|\Delta L_{\mathrm{core}}\|$	Mean $\|\Delta L_{\mathrm{plateau}}\|$	Knee-in-band	Knee in $\mathcal{S}_{0}$	Knee in $\mathcal{S}_{1}$	Mean active-set size
Clean full-state	0.00	0.00	50.0%	25.0%	95.8%	4.0
Gaussian noise $\sigma=0.01$	0.04	0.13	50.0%	25.0%	95.8%	4.1
Gaussian noise $\sigma=0.05$	0.08	0.50	45.8%	25.0%	91.7%	4.3
Gaussian noise $\sigma=0.10$	0.46	1.58	37.5%	20.8%	79.2%	4.7
Downsample $\times 2$	0.13	0.38	47.9%	25.0%	95.8%	4.1
Downsample $\times 4$	0.38	1.50	41.7%	20.8%	83.3%	4.6

Appendix

Table 24: Aggregate anchor diagnostics relative to the clean full-state reference. These results show how quickly anchor quality degrades and whether the downstream refinement still recovers a useful candidate band.

Anchor condition	Method	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Cost ratio
Clean full-state	System-core	0.0%	37.5%	1.88	49.7%	0.000
Clean full-state	SAKE	75.0%	100.0%	0.25	6.7%	0.043
Gaussian noise $\sigma=0.01$	System-core	0.0%	29.2%	1.96	53.8%	0.000
Gaussian noise $\sigma=0.01$	SAKE	70.8%	95.8%	0.29	7.5%	0.043
Gaussian noise $\sigma=0.05$	System-core	0.0%	25.0%	2.04	58.6%	0.000
Gaussian noise $\sigma=0.05$	SAKE	66.7%	95.8%	0.38	8.9%	0.044

Appendix

Table 25: Aggregate downstream-selection quality under noisy, partial, and same-data anchor estimation.

Dataset	Backbone	Clean		Noise 0.05		Downsample x4		Mask 25%		Same-data noisy
		$L_{\mathrm{core}}$	$L_{\mathrm{plateau}}$	$L_{\mathrm{core}}$	$L_{\mathrm{plateau}}$	$L_{\mathrm{core}}$	$L_{\mathrm{plateau}}$	$L_{\mathrm{core}}$	$L_{\mathrm{plateau}}$	$L_{\mathrm{core}}$	$L_{\mathrm{plateau}}$
DiffReact	ConvLSTM	3	6	3	6	3	7	3	7	3	7
DiffReact	FNO	3	6	3	6	3	7	3	7	3	6
DiffReact	Transformer	3	6	3	6	3	7	3	7	3	7
DiffReact	U-Net	3	6	3	7	4	8	3	8	3	7
RDB	ConvLSTM	2	6	2	7	2	8	2	8	2	7

Appendix

Table 26: Case-level anchor pairs under representative robustness conditions. Each entry reports the modal anchor pair across the three training seeds for that dataset–backbone–condition tuple.

Dataset	Backbone	Clean	Noise 0.05	Downsample x4	Mask 25%	Same-data noisy
DiffReact	ConvLSTM	2	2	3	3	2
DiffReact	FNO	2	2	3	2	2
DiffReact	Transformer	1	1	2	2	1
DiffReact	U-Net	4	4	5	5	5
RDB	ConvLSTM	4	4	5	5	4
RDB	FNO	4	4	5	5	4

Appendix

Table 27: Case-level SAKE selections under representative robustness conditions. Each entry reports the modal selected window across the three training seeds for that dataset–backbone–condition tuple.

Component	Setting used in this appendix
Anchor-source conditions	same-data full-state, same-data noisy, spatial downsampled observations, sparse probe observations, masked observations, and learned-latent summaries
Representation families	randomized PCA, randomized SVD, random Gaussian projection, and convolutional autoencoder latent summaries
Primary selector comparison	System-core and SAKE under the same matched downstream budgets as the main paper
Reference direct baselines	Direct-4-Shortlist and ASHA, reported once as clean-independent search references
Reported anchor metrics	mean $\|\Delta L_{\mathrm{core}}\|$ , mean $\|\Delta L_{\mathrm{plateau}}\|$ , knee-in-band coverage, knee-in-shortlist coverage, and final active-set coverage
Reported selection metrics	Exact, Within-1, mean $\|\Delta L\|$ , $\mathrm{Regret}_{\mathrm{knee}}$ , mean unique $L$ evaluations, and cost ratio

Appendix

Table 28: Protocol for the proxy-anchor appendix. The goal is to quantify how stage one behaves when clean-system trajectories or a single preferred state summary are unavailable.

Anchor source	Mean $\|\Delta L_{\mathrm{core}}\|$	Mean $\|\Delta L_{\mathrm{plateau}}\|$	Knee-in-band coverage	Knee-in-shortlist coverage	Final active-set coverage
Same-data full-state	0.00	0.13	50.0%	25.0%	95.8%
Same-data noisy	0.08	0.75	41.7%	20.8%	83.3%
Downsample $\times 2$	0.13	0.38	47.9%	25.0%	95.8%
Sparse probes ( $16\times 16$ )	0.42	1.04	41.7%	20.8%	87.5%
Random mask 25%	0.04	1.50	37.5%	20.8%	79.2%
Learned latent summary	0.17	0.54	45.8%	25.0%	91.7%

Appendix

Table 29: Aggregate anchor-quality diagnostics when stage one is instantiated without a separate clean-system pool.

Anchor source	Method	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Mean unique $L$ eval.	Cost ratio
Same-data full-state	System-core	0.0%	37.5%	1.88	50.8%	0.0	0.000
Same-data full-state	SAKE	70.8%	95.8%	0.29	7.1%	4.0	0.043
Same-data noisy	System-core	0.0%	20.8%	2.17	63.1%	0.0	0.000
Same-data noisy	SAKE	62.5%	87.5%	0.50	11.9%	4.5	0.045
Downsample $\times 2$	System-core	0.0%	33.3%	1.92	52.2%	0.0	0.000
Downsample $\times 2$	SAKE	70.8%	95.8%	0.33	7.9%	4.1	0.043

Appendix

Table 30: Aggregate selector quality under proxy-anchor extraction. Direct-4-Shortlist and ASHA are unchanged by these anchor-source choices and are therefore used only as fixed clean-independent references rather than repeated in every row.

Representation	Mean $\|\Delta L_{\mathrm{core}}\|$	Mean $\|\Delta L_{\mathrm{plateau}}\|$	Exact	Within-1	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Cost ratio
Randomized PCA	0.00	0.13	70.8%	95.8%	7.1%	0.043
Randomized SVD	0.04	0.17	70.8%	95.8%	7.3%	0.043
Random projection	0.33	0.83	54.2%	83.3%	13.9%	0.045
Autoencoder latent	0.13	0.29	66.7%	91.7%	8.0%	0.044

Appendix

Table 31: Representation sensitivity under full-state proxy extraction.

Representation	Mean $\|\Delta L_{\mathrm{core}}\|$	Mean $\|\Delta L_{\mathrm{plateau}}\|$	Exact	Within-1	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Cost ratio
Randomized PCA	0.08	0.75	62.5%	87.5%	11.9%	0.045
Randomized SVD	0.13	0.79	62.5%	87.5%	12.1%	0.045
Random projection	0.46	1.25	45.8%	75.0%	17.2%	0.046
Autoencoder latent	0.17	0.67	58.3%	87.5%	12.0%	0.045

Appendix

Table 32: Representation sensitivity under the same-data noisy proxy condition.

Dataset	Backbone	Same-data full-state	Same-data noisy	Sparse probes	Learned latent	Oracle knee
DiffReact	ConvLSTM	2	2	3	2	2
DiffReact	FNO	2	2	2	2	2
DiffReact	Transformer	1	1	2	1	1
DiffReact	U-Net	4	5	5	4	5
RDB	ConvLSTM	4	4	5	4	5
RDB	FNO	4	4	5	4	4

Appendix

Table 33: Case-level SAKE selections under representative proxy-anchor conditions.

Quantity	Symbol	Values	Interpretation
Tail tolerance fraction	$\rho$	{0.02, 0.05, 0.10}	controls how early the long-history tail is treated as effectively closed
Plateau threshold	$\tau_{\mathrm{pl}}$	{0.02, 0.05, 0.10}	controls how early diminishing returns are declared
One-standard-error multiplier	$\kappa$	{1.0, 1.5, 2.0}	controls the smallest-reliable-window rule in stage two
Bootstrap resamples	$B$	{100, 300, 500}	controls the stability of the UCB estimates used by stage one
Knee tolerance	$\epsilon$	{2%, 5%, 10%, 15%}	changes the oracle target itself rather than only the selector rule

Appendix

Table 34: Parameter sweeps used for the sensitivity analysis.

Quantity	Value	Mean $L_{\mathrm{core}}$	Mean $L_{\mathrm{plateau}}$	Knee-in-band	Mean unique $L$ eval.	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Cost ratio
$\rho$	0.02	2.8	6.1	58.3%	4.8	75.0%	100.0%	0.25	6.5%	0.046
$\rho$	0.05	2.5	6.0	50.0%	4.4	75.0%	100.0%	0.25	6.7%	0.043
$\rho$	0.10	2.1	5.8	41.7%	4.0	70.8%	95.8%	0.33	7.4%	0.039
$\tau_{\mathrm{pl}}$	0.02	2.5	6.4	54.2%	4.9	75.0%	100.0%	0.25	6.4%	0.047
$\tau_{\mathrm{pl}}$	0.05	2.5	6.0	50.0%	4.4	75.0%	100.0%	0.25	6.7%	0.043
$\tau_{\mathrm{pl}}$	0.10	2.5	5.4	41.7%	4.0	70.8%	95.8%	0.33	7.6%	0.039

Appendix

Table 35: Aggregate sensitivity of SAKE to the fixed selector constants.

Dataset	Backbone	$\epsilon=2\%$	$\epsilon=5\%$	$\epsilon=10\%$	$\epsilon=15\%$
DiffReact	ConvLSTM	2	2	2	2
DiffReact	FNO	2	2	1	1
DiffReact	Transformer	1	1	1	1
DiffReact	U-Net	6	5	4	3
RDB	ConvLSTM	6	5	4	4
RDB	FNO	4	4	3	3

Appendix

Table 36: Case-level oracle-knee locations under alternative near-optimality tolerances.

$\epsilon$	Method	Exact	Within-1	Mean $\|\Delta L\|$	Mean $\mathrm{Regret}_{\mathrm{knee}}$	Mean $\mathrm{Regret}_{\mathrm{best}}$	Cost ratio
2%	System-core	0.0%	29.2%	2.04	58.2%	58.6%	0.000
2%	Direct-4-Shortlist	50.0%	79.2%	0.71	13.4%	13.9%	0.058
2%	SAKE	62.5%	91.7%	0.42	8.6%	9.1%	0.043
5%	System-core	0.0%	37.5%	1.88	49.7%	49.9%	0.000
5%	Direct-4-Shortlist	62.5%	87.5%	0.50	10.5%	10.8%	0.058
5%	SAKE	75.0%	100.0%	0.25	6.7%	7.0%	0.043

Appendix

Table 37: Selector performance under alternative oracle-knee tolerances.

Quantity	Definition in this appendix
Selector wall-clock	End-to-end elapsed time for one selector run, including all pilot training and validation passes
Selector GPU-hours	Sum over all GPUs used by a selector run, measured over the same pilot and validation work
Full-sweep wall-clock ratio	Selector wall-clock divided by the elapsed time of the corresponding exhaustive sweep
Queueing / scheduling	Excluded; all measurements assume dedicated local execution on the shared A100 server

Appendix

Table 38: Timing definitions used in this appendix.

Method	Mean selector wall-clock (h)	Mean selector GPU-hours	Full-sweep wall-clock ratio	Mean unique $L$ eval.	Cost ratio
Direct-3-Shortlist	1.04	4.16	0.069	5.8	0.057
Direct-4-Shortlist	1.08	4.31	0.071	5.9	0.058
ASHA	1.23	4.94	0.081	6.0	0.065
BO-MF	1.28	5.12	0.084	6.0	0.068
BOHB-style	1.25	5.00	0.082	6.0	0.067
SAKE	0.86	3.43	0.057	4.4	0.043

Appendix

Table 39: Overall timing summary across the shared A100 environment.

Dataset	Backbone	Method	Wall-clock (h)	GPU-hours	Cost ratio
DiffReact	ConvLSTM	Direct-4-Shortlist	0.62	2.48	0.054
DiffReact	ConvLSTM	ASHA	0.70	2.78	0.061
DiffReact	ConvLSTM	BO-MF	0.72	2.90	0.064
DiffReact	ConvLSTM	BOHB-style	0.70	2.82	0.063
DiffReact	ConvLSTM	SAKE	0.52	2.07	0.041
DiffReact	ConvLSTM	Full sweep	9.40	37.60	1.000

Appendix

Table 40: Case-level realized timing for the principal search baselines and the exhaustive reference.

Method	Stage-1 wall-clock	Stage-2 wall-clock	Validation overhead	Total wall-clock	Total GPU-hours
Direct-3-Shortlist	0.10	0.79	0.15	1.04	4.16
Direct-4-Shortlist	0.12	0.80	0.16	1.08	4.31
ASHA	0.18	0.89	0.16	1.23	4.94
BO-MF	0.21	0.91	0.16	1.28	5.12
BOHB-style	0.20	0.88	0.17	1.25	5.00
SAKE	0.16	0.54	0.16	0.86	3.43

Appendix

Table 41: Stage-wise timing breakdown for the low-cost selectors.

Neural PDE Solvers

Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.

See all 269 papers

Posterior-First Neural PDE Simulation: Inferring What Downstream Tasks Need from a Single Field
May 5, 2026Wenshuo Wang, Fan ZhangNeural PDE Solvers PDE Surrogate Modeling
School of Future Technology, South China University of Technology, China · State Key Laboratory of Ocean Sensing & Ocean College, Zhejiang University, China · Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, USA
The Simulacrum: Decision-Theoretic Pretraining for Near-Optimal Time-Series Forecasting and Inference
Jun 26, 2026Pablo Montero-Manso, Marcel ScharthSimulation-Based Inference Zero-Shot Time Series Forecasting
University of Sydney Business School, Australia
PDE-PFN: Prior-Data Fitted Neural PDE Solver
Oct 9, 2026Jaehyeon Park, Mingu Kang, Dongseok Lee +5Neural PDE Solvers PDE Solving
SolverX · KAIST · Seoul National University +3

May 5, 2026J/K move · Enter open · S save

Posterior-First Neural PDE Simulation: Inferring What Downstream Tasks Need from a Single Field

Wenshuo Wang, Fan Zhang

School of Future Technology, South China University of Technology, China · State Key Laboratory of Ocean Sensing & Ocean College, Zhejiang University, China · Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, USA

Neural PDE Solvers PDE Surrogate Modeling

Open paper AskPDF arXiv

Neural PDE simulators often receive only a single observed field at deployment. In this setting, a field-to-future predictor can collapse distinct latent problem states into the same deterministic interface, losing the ambiguity needed for reliable rollout and downstream decisions. We propose posterior-first neural PDE simulation, a task-to-target-to-interface paradigm: the declared downstream tasks determine the minimal task-sufficient problem state to infer, and the posterior over that state conditions the rollout and downstream queries. This posterior can be supplied by amortized posterior estimators, including simulation-based inference with learned summaries; our contribution is the target, the interface, and their theory. We show that Bayes downstream values factor through this posterior, proper scoring rules on refinement labels identify it as their population-risk minimizer, and deterministic collapse incurs an ambiguity barrier whenever the true posterior is non-Dirac. Synthetic exact-ambiguity experiments show that point-versus-posterior gaps track the predicted barrier. On metadata-hidden PDEBench tasks, posterior recovery reduces pooled rollout nRMSE from 0.175 to 0.132, closing 59.4% of the direct-to-oracle gap. Matched controls attribute the gains to task-aligned targeting and posterior deployment, and the gains persist with a latent-diffusion backbone. These results suggest that single-observation neural PDE simulation should be posterior-first rather than monolithic field-to-future prediction.

Abstract

Neural PDE Solvers

Explore similar work

Posterior-First Neural PDE Simulation: Inferring What Downstream Tasks Need from a Single Field

The Simulacrum: Decision-Theoretic Pretraining for Near-Optimal Time-Series Forecasting and Inference

PDE-PFN: Prior-Data Fitted Neural PDE Solver