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3D Field Data Reduction with Adaptive Sample-Based Gaussian-Encoded Reconstruction

Michael R. Martin, Joseph Insley, Victor A. Mateevitsi, Silvio Rizzi, Kwan-Liu Ma

arXiv:2609.16024v1cs.GRcs.CEcs.CVcs.LG

TL;DR

Scientific field data are stored as grids, particles, or unstructured meshes, but reduction methods often remain tied to particular data forms or representation choices. This paper introduces a sample-based fixed-budget Gaussian encoder for these inputs and reports improved reconstruction comparisons, including up to 44× fewer primitives in one grid comparison. Its temporal experiments also evaluate warm starts against independently trained models within a dataset-specific operating regime.

  • Problem

    Data-reduction methods remain tied to particular data forms or representation choices, while resampling particle and mesh data to grids can create dense products whose storage and movement grow with resolution.

    Source: §1 · p. 1

  • Method

    The encoder initializes scalar-carrying Gaussian primitives from particle, cell, or grid samples and refines them against a field-space reference while preserving a prescribed primitive count.

    Source: §1 · p. 1, §3.2 · pp. 2–3

  • Results

    Across the reported comparisons, sample-based encoding improves reconstruction over grid-seeded alternatives; on Vortex, 32K primitives reach 49.67 dB versus 44.90 dB for a 1.4M-primitive published model [23].

    Source: §5.2 · p. 6, §5.3 · p. 6

  • Takeaways & Limitations

    The framework provides a scalar-field representation for particle, unstructured-cell, and regular-grid data with a prescribed capacity and reported exports of approximately 12 bytes per primitive on disk.

    Source: §1 · p. 1, §7 · p. 9

    The unstructured path assumes axis-aligned hexahedral cells and uses a materialized reference grid during optimization; temporal warm-start quality reaches independent-model levels only at tested dense strides of 501–591 cycles in this series.

    Source: §6.2 · p. 8

Abstract

from arXiv · show

In scientific simulation, regular grids, unstructured meshes, and particle-based formats are chosen to represent field data for computational efficiency, geometry/adaptive flexibility, and following motion/deformation, respectively. Each of these field data formats is often handled through separate data-specific processing pipelines. We present a unified sample-based Gaussian encoding method that represents these data forms under a single fixed-budget formulation. The method initializes and refines Gaussian primitives directly from the input samples while preserving a prescribed primitive count and encoded size to achieve a desired level of data reduction. Across structured, unstructured, and particle data, the sample-based formulation improves reconstruction accuracy with measurably fewer primitives in comparison to prior formulations, achieving up to 4.8 dB higher PSNR with an approximate 44x reduction in primitive count. For time-varying data, warm-starting from the previous timestep reduces the optimization required to reach independently trained reconstruction quality. Together, these results demonstrate a unified fixed-budget Gaussian encoding framework for structured, particle, unstructured, and time-varying scientific data with predictable storage, higher reconstruction accuracy, and improved temporal encoding efficiency.

1 INTRODUCTION

The paper targets a unified scalar-field encoding with predictable memory across grids, particles, and unstructured data, rather than separate data-specific pipelines. It introduces fixed-budget sample-based Gaussian encoding and evaluates its accuracy, storage, and temporal reuse.

1 INTRODUCTION

  • A sample-based fixed-budget Gaussian encoder allocates a prescribed set of scalar-carrying anisotropic primitives directly from particles, unstructured cells, or grid samples.The primitive budget fixes the full-precision footprint before training and retains scalar attributes for post hoc rendering changes.Source: §1 · p. 1
  • The method adds budget-preserving refinement and temporal warm starts, evaluating when warm starts approach independently trained reconstruction quality.The evaluation spans particle, structured-grid, unstructured-cell, and time-varying data, with matched-budget comparisons against the grid-seeded predecessor.Source: §1 · p. 1
  • The paper compares its encoder with a learned-Gaussian encoder at near-matched primitive counts and with implicit neural representations on size-fidelity reference points.The learned-Gaussian comparison follows that method’s public release schedule.Source: §1 · p. 1

2 RELATED WORK

Prior scalar-volume encoders provide flexible Gaussian rendering or compact neural fields, but their allocation, supervision, or input assumptions constrain a unified fixed-budget treatment of grids, particles, and unstructured cells. The paper positions its sample-based formulation against these representation-specific approaches.

2.1 Gaussian representations for scalar volumes.

  • The sample-based formulation extends fixed-budget scalar-carrying Gaussians beyond regular-grid inputs to native particles and unstructured cells, with budget-preserving relocation and temporal reuse.Earlier approaches include image-supervised VEG, field-supervised W-VEG with error-guided densification and pruning, and a fixed-budget grid encoder [5], [4], [23].Source: §2.1 · p. 1

2.2 Neural volume representations.

  • Neural volume methods can provide compact, high-fidelity approximations, but storage depends on network architecture and feature-table choices, and evaluation requires network inference.The passage cites coordinate networks [20] [34], feature-grid and multiresolution hash encodings [25] [40], NeuralVDB, adaptive multi-grid networks [42], and UGINR.Source: §2.2 · p. 2

2.3 Time-varying fields and conventional compression.

  • Time-varying neural models and conventional voxel compressors address different storage settings; conventional compressors reconstruct voxel arrays, complementing learned field models.The passage identifies MoE-INR, ANTIC [2], lossless prediction-based compression, zfp, and SZ.Source: §2.3 · p. 2

2.4 Particle and unstructured grid data.

  • Mesh-preserving renderers and particle-specific Gaussian splatting serve individual data forms, but do not provide one fixed-budget scalar-field encoding across particles, cells, and grid samples.ParticleGS optimizes image fidelity for selected visualization settings [12].Source: §2.4 · p. 2

3 METHOD

The method uses one fixed-count scalar-carrying Gaussian representation for samples from particles, unstructured cells, and grids. It refines those primitives against a field-space reference, reallocates underused capacity without changing model size, and reuses models across stored time steps.

3 METHOD

  • The encoder retains the grid-seeded method’s scalar-carrying Gaussian representation, fixed primitive budget, and export format while replacing grid-specific allocation and refinement with a common sample formulation.Particles, unstructured cells, and grid voxels initialize primitives; field-space references supervise refinement.Source: §3 · p. 2

3.1 Representation and Export

  • A model stores a fixed set of anisotropic Gaussian primitives, each with center, log-scales, rotation, scalar logit, and weight logit.Field evaluation clamps standard deviations to 0.4–8.0 voxel units and uses a truncated Gaussian kernel.Source: §3.1 · p. 2

3.2 Sample Interface

  • Input samples encode position, initial local scale, normalized scalar value, and importance weight in the reference-grid index frame.Particles and cells initialize the model directly, while a reference grid supplies field-space supervision and evaluation; unstructured-cell references use volume-weighted deposition.Source: §3.2 · pp. 2–3

3.3 Seeding from Samples

  • The prescribed budget is split among importance, uniform, and peak sample streams, which initialize one Gaussian per selected sample.The streams receive 45%, 40%, and 15% of the budget, respectively; importance and uniform seeds use widened initial support, while peak seeds retain local sample scale.Source: §3.3 · p. 3

3.4 Fidelity Objective and Relocation

  • Refinement samples stratified field locations, emphasizes high-valued structure, and promotes mixture support at occupied locations; scheduled relocation moves the least-supported primitives to high-error points.Relocated primitives receive the error-point center, an isotropic 0.8-voxel scale, the local reference value, and weight 0.5, preserving primitive count and model footprint.Source: §3.4 · p. 3

3.5 Warm Start Across Timesteps

  • Each stored timestep initializes the next model with its primitive arrays and fixed count; dense-stride warm starts retain relocation, whereas coarse-stride warm starts omit it.Each timestep remains a complete model rather than a delta, and relocation does not preserve one-to-one primitive correspondence across steps.Source: §3.5 · p. 3

4 EXPERIMENTAL SETUP

The experiments evaluate six inputs spanning particle, unstructured-cell, and regular-grid data, with training protocols, baselines, and evaluation measures specified for the comparisons.

4.1 Datasets

  • Six inputs span HACC particles, adaptive-mesh cells, temporal water-fraction fields, and regular-grid Vortex, Bubble Plume, and Miranda data.The HACC and unstructured-cell experiments use cosmology and Deep Water Impact data, respectively; the temporal experiment uses 26 stored fields on a fixed lattice.Source: §4.1 · p. 3, Table 2 · p. 4

4.2 Training Configuration

  • Sample-based runs use Adam with cosine-decayed learning rates, dataset-specific iteration schedules, and stratified samples; warm-started steps load the preceding timestep’s model.Relocation is used for HACC, grid, cell, and dense-stride warm-start runs, while independently encoded time steps and coarse-stride warm starts omit it. The matched training-budget protocol uses 1,500 iterations and 3,072 points per iteration.Source: §4.2 · pp. 3–4

4.3 Baselines

  • The grid-seeded encoder is the primary baseline; HACC also uses NeurComp and a multiresolution hash-grid, while W-VEG is evaluated through its mesh-based preprocessing path.W-VEG is evaluated on regular-grid inputs through that path; evaluating HACC would require converting its native particles to a mesh or deposited grid.Source: §4.3 · p. 4

4.4 Evaluation

  • Evaluation compares reconstructed scalar fields with voxel PSNR, rendered images with PSNR and SSIM, and reports on-disk sizes for Gaussian encodings.Figure 1 shows HACC reference and model columns, with labels for size, reduction factor, supervision lattice, rendered PSNR, and SSIM; its bottom panels show absolute luminance differences. Float32 Gaussian files use 48 bytes per primitive plus metadata, while 8-bit exports use mixed-precision quantization and lossless archive compression.Source: §4.4 · pp. 4–5, Figure 1 · p. 4

5 RESULTS

The results evaluate sample-based Gaussian encoding across particle, regular-grid, unstructured-cell, and time-varying data, including comparisons at matched budgets and storage sizes. Across these settings, the method shows improved reconstruction quality, while temporal warm-start outcomes depend on stride and field evolution.

5.1 Particle Data: HACC

  • 13.8 dB higher rendered PSNR and 4.1× less storage distinguish the matched 300K, 256^3-supervision HACC sample-based export from the grid-seeded model.For gas temperature and metal mass fraction, 8-bit exports reduce storage by 20× and 19× relative to their deposits while retaining rendered fidelity and visible structures.Source: Figure 2 · p. 5, §5.1 · pp. 5–6, Table 3 · p. 5

5.2 Regular Grids as Samples

  • 49.67 dB voxel PSNR at 32K primitives exceeds the largest published grid-seeded model’s 44.90 dB at 1.4M primitives, using approximately 44× fewer primitives.At matched budgets, voxel PSNR improves by 21.35 dB on Vortex and 6.18 dB on Bubble Plume.Source: Table 4 · p. 6, §5.2 · p. 6

5.3 Evaluation at Matched Primitive Counts

  • Full-schedule models exceed W-VEG by 12.45 dB on Vortex and 2.28 dB on Miranda; matched schedules differ by +7.87 and −0.74 dB, using 41% and 22% of its optimization time.These are voxel-PSNR comparisons at 33K Vortex and approximately 200K Miranda primitives.Source: §5.3 · p. 6, Table 5 · p. 6

5.4 Unstructured Cells: Deep Water Impact

  • With 400K Gaussians initialized from 44,851,632 native cells, the unstructured-cell model reaches 29.25 dB voxel PSNR and 31.4 dB rendered PSNR, preserving the sea sheet and localized splash.The visible residual is concentrated near the seam between coarse and fine cells.Source: §5.4 · p. 6

5.5 Temporal Reuse and Primitive Relocation

  • Dense-stride warm starts with relocation reach their independently trained final near iteration 800 and finish within 0.17 dB; coarse-stride warm starts without relocation finish 1.69–3.75 dB below.At cycle 17,260, warm starts improve water-fraction voxel PSNR from 31.6 to 34.0 dB, while temperature and asteroid fraction remain below their independent encodes.Source: §5.5 · pp. 6–7

5.6 Encoding Cost and Short Schedules

  • At matched training budgets, sample-based optimization takes 128 s versus 223 s on Vortex and 91 s versus 387 s on Bubble Plume, while achieving higher reported PSNR.Short schedules and smaller primitive budgets provide additional operating points: a 10K Vortex model reaches 42.59 dB voxel PSNR, 14.27 dB above the 32K grid-seeded model, at about one-third its file size.Source: §5.6 · pp. 7–8

6 DISCUSSION

The discussion reports that sample-based initialization and refinement perform well in controlled grid comparisons, while identifying limits in reference-grid use, supported cell shapes, temporal reuse, and evaluation scope.

6.1 Representation Behavior

  • Matched-budget, size, and decoder grid comparisons favor sample-based initialization and refinement, but do not isolate which changed component accounts for the gains.Source: §6.1 · p. 8

6.2 Limitations and Future Work

  • Particle and cell inputs still require a materialized reference grid during refinement, and the unstructured path currently assumes axis-aligned hexahedral cells.Source: §6.2 · p. 8
  • The workflow is offline, and single-run optimization-loop timings on one laptop GPU do not measure end-to-end in-situ costs or establish a general 8-bit fidelity bound.Source: §6.2 · pp. 8–9, Table 8 · p. 8
  • At tested 501–591-cycle dense strides, water warm starts reach or exceed independent quality during refinement and finish within 0.17 dB; larger strides lag.Source: §6.2 · p. 8, Figure 8 · p. 9
  • Relocation preserves primitive count, not correspondence, so evaluated reuse improves time-to-quality rather than enabling delta-coded temporal storage.Source: §6.2 · p. 9
  • Scale and anisotropy constraints restrict the smallest representable features, while the highest-accuracy HACC hash-grid result uses the largest reported model and neural evaluation.Source: §6.2 · p. 9

7 CONCLUSION

The fixed-budget sample-based Gaussian encoder provides a shared representation across evaluated grid, particle, cell, and temporal settings. Results show strong reconstruction quality per primitive, while temporal reuse is most effective when successive stored fields are sufficiently close for relocation to adapt capacity.

7 CONCLUSION

  • 4.77 dB higher voxel PSNR: the 32K-primitive Vortex model reached 49.67 dB versus 44.90 dB for the 1.4M-primitive grid-seeded model, using approximately 44× fewer primitives.On matched-budget regular-grid comparisons, rendered PSNR rose from 22.8 to 41.1 dB on Vortex and from 23.7 to 28.9 dB on Bubble Plume.Source: §7 · p. 9
  • The encoder uses one fixed-capacity representation interface across evaluated grid, particle, adaptive-cell, and temporal settings, with predictable storage and strong reconstruction quality per primitive.Full precision requires 48 bytes per primitive; reported 8-bit exports occupy approximately 12 bytes per primitive on disk, including lossless archive compression.Source: §7 · p. 9
  • Warm starts with budget-preserving relocation reduced optimization toward independently trained water-fraction encodings for the dense stored strides evaluated.Temporal reuse was most effective when successive stored fields were sufficiently close for relocation to adapt primitive capacity to evolving structure.Source: §7 · p. 9
  • The Deep Water Impact experiment initialized primitives from 44.9 million adaptive-mesh cells and produced a 19.2 MB encoding with 31.4 dB rendered PSNR.Source: §7 · p. 9
  • Direct particle- and cell-query supervision, support for more general cell types, and correspondence-preserving temporal updates remain important next steps.Source: §7 · p. 9

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