Efficient Neural Field Learning via Adaptive Coverage and Focused Sampling

1 Introduction

• Figure 1 contrasts broad but inefficient uniform coverage, localized pointwise concentration, and ACES’s region-level balance of coverage and refinement.Point size represents region-level sampling weight.Source: Figure 1 · p. 2, §1 · p. 2
• Uniform sampling can waste computation on simple regions while under-resolving complex regions that dominate approximation error.INR training may have millions to billions of candidate points, but computational and memory constraints limit how many are used per iteration.Source: §1 · pp. 1–2
• Pointwise adaptive methods can over-concentrate on nearby high-error samples, creating redundancy at the expense of global coverage.The paper also identifies their non-uniform sampling bias as insufficiently analyzed and systematically exploited.Source: §1 · p. 2
• ACES adaptively partitions the domain and applies bias-controlled region-level importance weights to balance coverage, variance reduction, and convergence.Partitioning respects spatial organization; within regions, weighting prioritizes informative samples.Source: §1 · p. 2

Adaptive sampling for implicit neural representations

• INR adaptive-sampling methods prioritize high-residual, uncertainty, or other locally important points, while ACES adds adaptive partitioning and region-level weighting (Zhang et al., 2024, 2025; Kheradmand et al., 2024; Tack et al., 2023).The cited approaches primarily operate at individual points and largely ignore higher-level spatial structure.Source: §2 · p. 3

Variance reduction through structured mini-batches

• Importance sampling and stratified mini-batches address variance or informative-example selection, but are primarily designed for fixed datasets rather than spatially structured continuous fields (Needell et al., 2014; Katharopoulos and Fleuret, 2018; Zhao and Zhang, 2014).ACES adaptively partitions continuous domains to jointly improve coverage and variance reduction.Source: §2 · p. 3

Weighted sampling and controlled bias

• Optimization performance can depend on the relationship between gradient variance and update quality, motivating ACES’s study of controlled bias through region-level weighting (Faghri et al., 2020).ACES also uses gradient-alignment analysis.Source: §2 · p. 3

3.1 Problem Setting

• Large datasets and heterogeneous learning signals make uniform minibatch sampling inefficient, while pointwise adaptation can over-concentrate and reduce domain coverage.ACES instead samples and weights at the region level, with periodic partition updates and per-iteration weight updates.Source: §3.1 · pp. 3–4

3.2 Variance Reduction via Partitioning

• Adaptive refinement of spatially coherent regions is expected to reduce within-region gradient variance by making regions more homogeneous.The refinement score favors regions that are both large and internally heterogeneous.Source: §3.2 · pp. 5–6

3.3 Bias–Variance Tradeoff and Acceleration

• A biased region-weighted estimator can improve the local optimization proxy when reduced relative variance outweighs lost directional alignment.The proxy compares directional alignment with relative variance; the analysis motivates weights emphasizing informative regions.Source: §3.3 · pp. 6–7

3.4 Practical Implementation

• ACES approximates gradient variability with residual statistics to avoid the cost of estimating gradient statistics directly for every region.For scalar-output squared loss, the variance approximation neglects a cross term; the simplified form further assumes within-region Jacobian smoothness.Source: §3.4 · pp. 7–8
• The practical weighting uses mean residual as a lightweight informativeness proxy, and its local improvement proxy increases with small positive bias at evaluated checkpoints.Estimating residual variance at every iteration would add overhead and can be unreliable when regions receive few samples.Source: §3.4 · p. 8

4 Experiments

• The experiments evaluate spatial snapshots and 64-frame joint spatiotemporal fields using relative MSE, with results averaged over three random seeds.Spatial tasks use 2 × 10^-3 of candidate points per iteration and joint tasks use 7 × 10^-4; there is no conventional train/test split.Source: §4 · pp. 8–9

4.1 Reconstruction Efficiency

• ACES achieves the lowest final relative MSE on both spatial datasets, while adaptive partitioning lowers joint-task error versus uniform sampling by 21.4% on PoolBoiling3D and 3.3% on NS3D at 5,000 epochs.Spatial comparisons cover iteration and runtime budgets; the joint-task result evaluates adaptive partitioning without residual-powered loss weighting.Source: §4.1 · pp. 9, 11, Figure 3 · p. 10, Table 1 · p. 11

4.2 Ablation Study

• On PoolBoiling2D, adaptive partitioning improves convergence over fixed-partition weighting, and adding residual-aware weighting further speeds convergence.The comparison indicates complementary benefits from structured coverage and focused weighting.Source: §4.2 · p. 11, Figure 4 · p. 12

4.3 Empirical Correlation with Optimization Behavior

• ACES has higher one-step improvement proxy scores than unbiased partitioning and uniform sampling, and proxy ordering generally matches faster convergence and lower final error.The comparison uses the bound-derived S2 w/Gw quantity, normalized against uniform sampling and a full-gradient oracle.Source: §4.3 · pp. 11–12, Figure 5 · p. 12

• ACES combines adaptive spatial partitioning with residual-aware weighting to balance domain coverage and focused refinement.Source: §5 · p. 12
• The analysis attributes reduced gradient variance to increased within-region homogeneity, while experiments report faster convergence and improved sample efficiency, especially for highly localized complexity.Source: §5 · p. 12
• The current implementation adds sampling overhead, and the analysis characterizes local one-step behavior rather than long-term nonconvex dynamics.Source: §5 · p. 12
• The source code and reproduction scripts are publicly available.Source: §5 · p. 13

A Proof of Variance Reduction

• Refining a cell into congruent subcells reduces its sampling-variance contribution by a fixed multiplicative factor.The analysis assumes a local variogram model for the gradient field.Source: A Proof of Variance Reduction · pp. 14–15
• The total variance after refinement is the sum of the variance contributions from the resulting subcells.Source: A Proof of Variance Reduction · pp. 14–15
• The expected gain from refinement is proportional to the cell’s current variance contribution, so the largest-contribution cell is the most beneficial split.Source: A Proof of Variance Reduction · p. 15

B Derivation of the Gradient Variance Approximation

• The residual is treated as scalar and the Jacobian as vector-valued; the derivation approximates regional gradient variance using a first-order expansion that neglects second-order and cross terms.Source: B Derivation of the Gradient Variance… · p. 16

C.1 A Sufficient Condition for Local Improvement with Residual-Power Weighting

• When region improvement scores and log-residual magnitudes are positively correlated, a sufficiently small residual-power bias strictly improves the one-step proxy over unbiased stratified estimation.Source: §C.1 · p. 17

C.2 Empirical Verification of the Ascent Condition

• The sufficient condition holds in all 60 tested configurations across PoolBoiling2D and NS2D, with positive rank correlations throughout training on both datasets.The correlations range from 0.225 to 0.606 on PoolBoiling2D and from 0.312 to 0.774 on NS2D.Source: §C.2 · p. 18

D Original One-Step Improvement Score

• The appendix presents the original one-step improvement score corresponding to the proxy, while the main text affinely normalizes that proxy to assign uniform sampling 0 and the full-gradient oracle 1.Source: D Original One-Step Improvement Score · p. 18

E Validity of Residual Variance as a Proxy

• The proxy comparison evaluates Spearman rank correlation and top-K overlap between regional residual-variance and gradient-variance scores on NS2D at 16 × 16 and 32 × 32 granularities.Source: E Validity of Residual Variance as a… · p. 18
• 0.719 ± 0.059 (16 × 16) and 0.660 ± 0.070 (32 × 32) top-20% overlap exceed the random baseline of 0.20.Across training, residual variance shows strong rank agreement with gradient variance: ρ = 0.807 ± 0.031 (16 × 16) and ρ = 0.820 ± 0.054 (32 × 32).Source: E Validity of Residual Variance as a… · pp. 18–19

F Generalization to a Fourier Feature Network

• 0.1536 ± 0.0263 relative MSE for ACES versus 0.2866 ± 0.0411 for uniform sampling at 5,000 iterations.Relative MSE is averaged over three random seeds.Source: F Generalization to a Fourier Feature… · p. 19
• Uniform sampling and INT have lower mean error at the earliest checkpoints, while ACES has the lowest mean error from 800 iterations onward.Source: F Generalization to a Fourier Feature… · p. 19
• The later-stage advantage of adaptive region-level sampling is observed with the Fourier Feature Network, though ACES can have a slower initial phase with this architecture.Source: F Generalization to a Fourier Feature… · p. 19

G Experimental Settings and Hardware

• Unless otherwise stated, experiments use a SIREN backbone, draw 2 × 10−3 of candidate points per spatial iteration and 7 × 10−4 per joint spatiotemporal iteration, and report relative MSE averaged over three seeds.Source: G Experimental Settings and Hardware · p. 20
• Spatial experiments use depth 3, hidden width 155, and w0 = 30, with learning rates of 10−5 for NS2D and 10−3 for PoolBoiling2D.The experiments use SGD with momentum 0.9, Nesterov acceleration, zero weight decay, and zero dampening.Source: G Experimental Settings and Hardware · p. 20
• Table 4 evaluates weighting-exponent sensitivity using relative MSE, where lower values are better and bold marks the lowest value in each row.Source: Table 4 · p. 21
• Joint spatiotemporal experiments use depth 10, hidden width 384, and w0 = 60, with dataset-specific learning rates and adaptive partitioning without residual-powered loss weighting.The learning rates are 2 × 10−3 for NS3D and 5 × 10−3 for PoolBoiling3D for Uniform, Fixed Stratified, and ACES, and 10−4 for INT.Source: G Experimental Settings and Hardware · pp. 20–21
• All reported runs use one NVIDIA A100-SXM4-80GB GPU, the specified runtime environment, and hardware used for runtime and wall-clock comparisons.Table 3 describes an NS2D runtime breakdown in milliseconds per iteration, including mean ± standard deviation over three seeds and a separate amortized partition-construction cost.Source: G Experimental Settings and Hardware · p. 21, Table 3 · p. 21

H.1 Runtime and Overhead Analysis

• ACES costs approximately 1.80× as much per iteration as uniform sampling, with partition construction contributing only 0.161 ms per iteration after amortization.Most of the difference comes from region-wise sample allocation and weighted-minibatch construction; higher training-stage cost reflects weighted-loss operations.Source: §H.1 · p. 21

H.2 Sensitivity to Weighting and Partition Updates

• Increasing residual-weighting exponent α from a weak regime improves reconstruction on both datasets, but gains saturate at moderate values and stronger concentration gives no consistent further gain.Source: §H.2 · p. 21
• Performance changes little for partition-update intervals of 50–1,000 iterations, while refreshing only at initialization increases error on both datasets.The initial-only setting constructs the partition at initialization and does not refresh it during the 5,000-iteration run.Source: §H.2 · p. 22, Table 5 · p. 22