Efficient Neural Field Learning via Adaptive Coverage and Focused Sampling

Guang Zhao; Xihaier Luo; Huan-Hsin Tseng; Seungjun Lee; Shinjae Yoo; Yihui Ren; Wei Xu

cs.LGOct 1, 2026

Efficient Neural Field Learning via Adaptive Coverage and Focused Sampling

Authors: Guang Zhao, Xihaier Luo, Huan-Hsin Tseng, Seungjun Lee, Shinjae Yoo, Yihui Ren, Wei Xu

Organizations: Computing and Data Sciences, Brookhaven National Laboratory Upton, NY 11973, USA

Abstract

Implicit neural representations (INRs) provide a flexible framework for modeling high-dimensional continuous fields, but their training is often inefficient due to uniform subsampling that ignores spatial heterogeneity. Existing adaptive sampling methods partially address this issue by prioritizing high-error samples, but typically operate at the point level, often leading to redundant sampling in localized regions and insufficient coverage of the domain. We propose ACES (Adaptive Coverage-aware Efficient Sampling), a structured sampling framework that improves training efficiency by decoupling coverage and importance. ACES constructs adaptive spatial partitions to ensure domain coverage and reduce redundancy, and applies region-level importance weighting to prioritize informative regions during training. We provide a theoretical analysis showing that adaptive partitioning reduces gradient variance by increasing within-region homogeneity, and that controlled bias in region-level weighting may improve optimization efficiency relative to standard unbiased estimators. Experiments on scientific field learning tasks demonstrate that ACES achieves faster convergence and lower error than uniform and pointwise adaptive sampling baselines, with the largest gains in fields with highly localized complexity.

Figures & tables

Figure 1: Qualitative comparison of sampling strategies on a Navier–Stokes vorticity field at a training iteration. Uniform sampling provides broad coverage but allocates many samples to low-error regions, while pointwise importance sampling over-concentrates in localized high-error areas. ACES performs adaptive region-level sampling, balancing domain coverage and focused refinement. Point size indicates the region-level sampling weight.

Figure 2: Overview of ACES. (a) The loss field is highly heterogeneous, with localized high-error regions. (b) ACES builds an adaptive partition that allocates finer resolution to complex regions and coarser resolution elsewhere. (c) Region-based sampling ensures structured coverage of the domain while avoiding redundant samples. (d) Bias-controlled weighting emphasizes informative regions by assigning larger weights to high-error areas, leading to more effective optimization.

Panel of Figure 3 — Figure 3: Training efficiency comparison. Left: one representative snapshot per dataset. Middle: full-grid evaluation relative MSE versus iteration. Right: full-grid evaluation relative MSE versus runtime. ACES achieves substantially faster convergence per iteration than Uniform, INT, and EVOS on both datasets. Under equal runtime budgets, ACES remains competitive on NS2D and achieves the largest gains on PoolBoiling2D, where informative regions are highly localized.

PoolBoiling3D
Method	200 epochs	1,200 epochs	2,400 epochs	5,000 epochs
INT	$0.8378\pm 0.1320$	$0.2600\pm 0.0026$	—	—
Uniform	$0.2948\pm 0.0012$	$0.1396\pm 0.0014$	$0.1133\pm 0.0004$	$0.0914\pm 0.0016$
Fixed Stratified	$0.2760\pm 0.0042$	$0.1359\pm 0.0085$	$0.1231\pm 0.0099$	$0.0914\pm 0.0043$
ACES (partition)	$\mathbf{0.2484\pm 0.0073}$	$\mathbf{0.1181\pm 0.0010}$	$\mathbf{0.0941\pm 0.0011}$	$\mathbf{0.0718\pm 0.0007}$
NS3D

Table 1: Joint spatiotemporal reconstruction on PoolBoiling3D and NS3D. All methods are evaluated at matched training checkpoints over 64 consecutive frames. Relative MSE (lower is better) is reported as mean

\pm

standard deviation over three random seeds. Within each dataset, bold denotes the lowest mean error at each checkpoint. INT results are reported through 1,200 epochs.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Empirical verification of the sufficient condition in Proposition 1 on PoolBoiling2D and NS2D using a budget-matched ACES partition (

\alpha=0

, i.e.

w=\mathbf{1}

). The curves show the mean with the min–max range over five time-frame runs per dataset. (a) The ascent slope

\left.dF/d\alpha\right|_{\alpha=0}=\frac{2\|\mathbf{g}\|^{2}}{(\|\mathbf{g}\|^{2}+V)^{2}}\sum_{k}t_{k}\log|\bar{r}_{k}|

stays strictly positive at every training checkpoint on both datasets (

60/60

configurations). (b) The underlying Spearman rank correlation

\rho(t_{k},\log|\bar{r}_{k}|)

also stays positive throughout training, confirming that regions with larger residual magnitude tend to have a more favorable alignment-to-variance ratio

t_{k}

, not only near initialization.

Figure 7: Original one-step improvement score across training steps, corresponding to

S_{w}^{2}/G_{w}

before normalization/sign transformation.

Figure 8: Validation of residual variance as a proxy for gradient variance on NS2D. Left: Spearman rank correlation between

\sigma_{g,k}^{2}

and

\sigma_{r,k}^{2}

across checkpoints for random

16\times 16

and

32\times 32

region sets. Right: top-

K

overlap between regions selected by

\sigma_{g,k}^{2}

and

\sigma_{r,k}^{2}

. Both curves stay well above the random baseline, showing that residual variance preserves region ranking and high-importance region retrieval.

Method	200	400	800	2,000	5,000
Uniform random	$\mathbf{0.9537\pm 0.0047}$	$0.9435\pm 0.0141$	$0.8656\pm 0.0730$	$0.5759\pm 0.1898$	$0.2866\pm 0.0411$
INT	$1.1370\pm 0.0818$	$\mathbf{0.9045\pm 0.0573}$	$0.7092\pm 0.0425$	$0.5009\pm 0.0324$	$0.3359\pm 0.0231$
EVOS	$2.6540\pm 0.2279$	$1.1808\pm 0.0780$	$0.7930\pm 0.0544$	$0.4755\pm 0.0292$	$0.3363\pm 0.0189$
Fixed stratified	$0.9564\pm 0.0014$	$0.9483\pm 0.0093$	$0.9050\pm 0.0366$	$0.6387\pm 0.1542$	$0.3262\pm 0.0590$
ACES	$1.4906\pm 0.1301$	$1.1236\pm 0.4569$	$\mathbf{0.6154\pm 0.3184}$	$\mathbf{0.2453\pm 0.0469}$	$\mathbf{0.1536\pm 0.0263}$

Appendix

Table 2: Relative MSE on PoolBoiling2D using a Fourier Feature Network backbone (mean

\pm

standard deviation over three seeds). Lower is better; bold denotes the lowest mean at each checkpoint.

Method	Training	Sampling	Partition	Total	Per update
Uniform	$1.2474\pm 0.0002$	$0.4140\pm 0.0008$	—	$1.6615\pm 0.0010$	—
ACES	$1.4688\pm 0.0051$	$1.3547\pm 0.0051$	$0.1609\pm 0.0007$	$2.9844\pm 0.0037$	$32.184\pm 0.144$

Appendix

Table 3: Runtime breakdown on NS2D in milliseconds per training iteration (mean

\pm

standard deviation over three seeds). Partition construction is excluded from the sampling column and reported separately as an amortized cost. The final column gives the cost of one partition update.

$\alpha$	0.1	0.2	0.5	1.0	2.0	3.0	4.0	5.0
PoolBoiling2D	0.1425	0.1363	0.1258	0.1097	0.1084	0.1096	0.1123	0.1117
NS2D	0.2816	0.2744	0.2605	0.2367	0.2196	0.2163	0.2156	0.2163

Appendix

Table 4: Sensitivity to the weighting exponent

\alpha

. Lower relative MSE is better; bold denotes the lowest value in each row.

Update interval	PoolBoiling2D	NS2D
50	$0.1228\pm 0.0041$	$0.2368\pm 0.0015$
100	$0.1242\pm 0.0046$	$0.2374\pm 0.0025$
200	$0.1235\pm 0.0071$	$0.2373\pm 0.0019$
500	$0.1235\pm 0.0039$	$0.2380\pm 0.0021$
1,000	$0.1243\pm 0.0037$	$0.2387\pm 0.0019$
Initial only	$0.1396\pm 0.0077$	$0.2457\pm 0.0021$

Appendix

Table 5: Sensitivity to the partition-update interval. Relative MSE is reported as mean

\pm

standard deviation; lower is better. “Initial only” constructs the partition at initialization and does not refresh it during the 5,000-iteration run.

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Implicit neural representations (INRs) offer a powerful framework for modeling high-dimensional physical fields, but their training remains inefficient because evaluating dense coordinate grids is computationally expensive. Existing coordinate-pruning strategies rely on local error functions that can oversample correlated points and incur substantial overhead from full-grid error evaluation. We introduce Refined Grid Sampling (RGS), which combines stratified grid sampling, error-based refinement, and dynamic grid-resolution adjustment. This design reduces spatial redundancy without evaluating errors over the full grid. On high-resolution Navier--Stokes fields, RGS achieves higher reconstruction accuracy within a fixed wall-clock budget and reaches the accuracy of full-grid training up to

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