AdaVoMP: Adaptive Volumetric Material Fields

• The paper introduces AdaVoMP, an adaptive method predicting dense spatially varying mechanical properties (Young’s modulus, Poisson’s ratio, density) from static 3D assets for realistic simulations.
• It leverages learned Sparse Adaptive Voxels and an autoregressive transformer encoder-decoder to generate an adaptive material tree, achieving up to a 16³ resolution increase over fixed grids.
• The method reduces memory and compute demands by refining only critical regions where material properties change, resulting in efficient, high-fidelity simulation-ready assets.

Searching arXiv for the specified AdaVoMP paper and closely related prior work to ground the article. AdaVoMP, short for Adaptive Volumetric Mechanical Property Fields, is a feed-forward method for predicting dense spatially varying mechanical properties inside 3D objects from a static 3D asset. It estimates Young’s modulus $E$, Poisson’s ratio $\nu$, and density $\rho$ as volumetric fields, with the stated goal of converting ordinary 3D shapes into simulation-ready deformable assets without manual material annotation. Its central departure from prior fixed-grid approaches, especially VoMP, is the replacement of a dense voxel lattice by a learned Sparse Adaptive Voxels (SAV) structure and a sparse transformer encoder-decoder that autoregressively generates an adaptive material tree for each input shape (Dagli et al., 16 Jun 2026).

1. Problem setting and target representation

AdaVoMP addresses the absence of physical parameters in most visually detailed 3D assets. For realistic elastic or deformable simulation, the object interior must be endowed with spatially varying material properties, rather than a single global label. The paper formulates this as prediction of a field

$$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$

for points $\mathbf{x}$ inside the object volume $\Omega$ (Dagli et al., 16 Jun 2026).

The method explicitly targets spatially varying volumetric properties. Instead of assigning one material to an entire shape, it predicts a hierarchical field in which each voxel stores a constant material vector

$\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$

with components corresponding to $E$, $\nu$, and $\rho$. This design is motivated by the observation that fine internal structures and material boundaries are poorly represented by coarse dense grids.

A central claim is that the representation is invariant to resolution in the sense of being able to scale to much finer effective resolution without a dense cubic memory blowup. The paper states that AdaVoMP can produce material fields at an effective resolution of $\nu$0, compared to the $\nu$1 fixed grid used by VoMP and Pixie, achieving a $\nu$2 increase relative to prior art (Dagli et al., 16 Jun 2026). This is presented as the main mechanism by which the method improves both fidelity and efficiency.

2. Sparse Adaptive Voxels and multiscale geometry-material encoding

The core representation is SAV, or Sparse Adaptive Voxels. A bounded domain

$\nu$3

is partitioned by an adaptive voxel tree $\nu$4. Voxels may occur at multiple levels $\nu$5, where $\nu$6 is the finest level, $\nu$7, and the finest grid resolution is set to $\nu$8 (Dagli et al., 16 Jun 2026).

To support joint transformer processing across scales, each voxel is mapped into the finest-grid coordinate system by

$\nu$9

The method also uses an octant identifier

$\rho$0

which specifies the child location of a voxel inside its parent. These definitions give the hierarchy an explicit spatial semantics suitable for sparse attention.

The same adaptive formalism is used for both input geometry and output materials. For the input shape, the object is voxelized at $\rho$1, and multi-view visual features are aggregated into a sparse adaptive tree $\rho$2. The paper uses DINOv3 ViT-H+/16 patch features lifted from multiple views into the volume. Feature aggregation is depth-attenuated: $\rho$3

$\rho$4

This weighting reduces feature dilution from distant or weakly observed views.

For the ground-truth material field, AdaVoMP constructs a second tree $\rho$5. The paper uses a value-range refinement rule: materials are aggregated bottom-up, a voxel is subdivided only if the within-voxel range exceeds tolerance $\rho$6, and otherwise the descendant mean is stored. This ensures that coarse nodes remain meaningful averages of their descendants, so truncated trees still correspond to valid lower-resolution fields.

The claimed advantage of SAV over dense voxel grids is threefold: it uses few coarse voxels in homogeneous regions, subdivides only where the field changes rapidly, and stores shape and materials in the same sparse hierarchical structure. This directly underpins the paper’s claims about higher effective resolution, better accuracy near part boundaries, and lower memory cost (Dagli et al., 16 Jun 2026).

3. Transformer architecture: AGT and AMG

AdaVoMP is organized around two sparse transformer modules: the Adaptive Geometry Transformer (AGT) and the Adaptive Material Generator (AMG) (Dagli et al., 16 Jun 2026).

AGT is the geometry encoder. Its input tokens are the leaf voxels of $\rho$7, embedded as

$\rho$8

where $\rho$9 is the feature for voxel $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$0, $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$1 is a linear projection, and $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$2 is a learned level embedding. The encoder uses RoPE on unified coordinates $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$3, sparse 3D shifted-window self-attention, and feed-forward layers, yielding contextual latents $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$4.

AMG is the autoregressive generator for the output material tree. It proceeds coarse-to-fine, from $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$5 down to $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$6, and only evaluates a sparse frontier $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$7 of candidate voxels rather than all voxels at a given scale. For each candidate voxel $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$8, AMG predicts:

1. a structure action over Empty, Keep, or Subdivide;
2. a 2D latent $$M(\mathbf{x}) = (E(\mathbf{x}), \nu(\mathbf{x}), \rho(\mathbf{x}))$$9 used for material decoding.

The query embedding is

$\mathbf{x}$0

This combines level information, octant information, and the parent hidden state. The parent term is particularly important because the frontier contains only nodes selected for refinement, so coherence across levels must be carried by the parent representation.

AMG then performs cross-attention to AGT latents, sparse windowed self-attention among candidates, and prediction through separate heads for structure logits $\mathbf{x}$1 and material latent $\mathbf{x}$2. The material latent is decoded with the frozen MatVAE decoder from VoMP to obtain $\mathbf{x}$3. This use of a learned latent material decoder constrains outputs to the material manifold encoded by the prior model, while allowing the new adaptive structure to be generated independently.

4. Autoregressive tree generation, training losses, and inference scaling

Generation is breadth-first and coarse-to-fine. The initial frontier is

$\mathbf{x}$4

At each level, AMG predicts one of three actions. If a voxel is Subdivide, its eight children are added to the next frontier: $\mathbf{x}$5 If the prediction is Keep, the voxel becomes a leaf; if Empty, it is discarded (Dagli et al., 16 Jun 2026).

A notable consequence is that lower-resolution outputs are valid truncated versions of the same tree, while additional test-time compute enables more refinement and thus finer material fields. The paper explicitly emphasizes that fewer levels mean less compute, so output resolution can be traded against runtime during inference.

Training is end-to-end with teacher forcing. During training, candidate expansion follows the ground-truth tree rather than the model prediction: $\mathbf{x}$6 This also introduces explicit empty-space negative examples, because all eight children of a subdivided voxel are instantiated.

The overall loss is

$\mathbf{x}$7

For the structure term, the ground-truth label is

$\mathbf{x}$8

The structure loss is a weighted NLL: $\mathbf{x}$9 with

$\Omega$0

Material supervision is applied only to non-empty candidates

$\Omega$1

using

$\Omega$2

where

$\Omega$3

The targets are normalized before regression: $\Omega$4 and $\Omega$5 by log-minmax, and $\Omega$6 by min-max. The paper uses

$\Omega$7

The implementation is correspondingly elaborate. The paper reports BF16 mixed precision, AdamW, 60k steps, linear warmup and decay, gradient clipping at global norm $\Omega$8, EMA of model weights, and HSDP training: ZeRO-3/FSDP-2 + DDP. Larger models are trained on 32× A100-80GB for 5 days. SAV is implemented with sparse tensors, hashed lookup, batched coordinate flattening, cached coordinate-dependent computations, and chunked attention and MatVAE decoding. Because frontier sizes can still become large, candidate counts per level are capped by selecting a contiguous spatial window, which the paper states preserves locality while controlling memory (Dagli et al., 16 Jun 2026).

5. Empirical behavior, scaling laws, and comparison to prior methods

AdaVoMP is evaluated against NeRF2Physics, PUGS, Phys4DGen, Pixie, and VoMP. The paper reports that it is consistently best or near-best across Young’s modulus error, Poisson ratio error, density error, mass estimation, and material validity, with particularly large gains at high resolution and on difficult shapes (Dagli et al., 16 Jun 2026).

The most prominent empirical claim concerns high-resolution prediction. Prior fixed-grid methods are described as effectively limited to $\Omega$9, whereas AdaVoMP produces fields at $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$0 effective resolution. On GVT-Hard, which contains objects with fine details missed by coarse grids, the paper states that AdaVoMP’s advantage becomes even larger. This suggests that the adaptive tree is especially beneficial when material boundaries or small geometric structures cannot be faithfully represented on a coarse uniform lattice.

The paper also reports explicit structure-compactness statistics. On GVT-Test, ground-truth SAV leaves at $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$1 are only 7.24% of the occupied voxels in a dense $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$2 voxelization, with median compactness even lower. Generated trees recover 79.28% as many leaves as the ground-truth trees while still occupying only 9.14% of the occupied voxels of a dense $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$3 representation. These figures are used to support the claim that adaptive trees match the intrinsic sparsity of many material fields.

Scalability is quantified by reported empirical laws: $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$4 where $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$5 is memory, $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$6 is compute, $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$7 is model parameters, and $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$8 is resolution. The paper presents these exponents as evidence that adaptivity avoids the cubic blowup of dense volumetric processing. It further states that AdaVoMP can achieve better accuracy with less test-time compute than all prior art in some regimes, because refinement is concentrated where the material field is complex (Dagli et al., 16 Jun 2026).

6. Relation to VoMP, simulation pipeline implications, and scope

VoMP is the most directly relevant predecessor in the paper’s framing. The contrast is explicit in three respects. First, VoMP uses a fixed $\mathbf{m}_{\ell,i} \in \mathbb{R}^3,$9 voxel grid, whereas AdaVoMP uses a learned sparse adaptive tree. Second, AdaVoMP represents homogeneous regions with coarse voxels and refines only at material boundaries. Third, VoMP processes all active voxels at the finest level, while AdaVoMP processes only the refinement frontier. These differences are presented as the reason AdaVoMP is both more accurate and more memory efficient (Dagli et al., 16 Jun 2026).

The practical implication is automatic conversion of high-resolution 3D content into physically meaningful, simulation-ready assets. The paper states that meshes, Gaussian splats, and other 3D objects can be annotated with volumetric materials and then passed into FEM, Simplicits, Isaac Sim, PhysX, and related simulators. Example outcomes described in the paper include furniture stable under gravity, robots interacting with fruits and other objects, Gaussian splats converted into physically simulated assets, and realistic end-to-end deformable simulations.

The scope of the method is specific. It predicts $E$0, $E$1, and $E$2 from static 3D geometry and associated visual features; it is not a general simulator, nor a direct dynamics predictor. A plausible implication is that its main contribution lies at the asset-generation layer of the simulation stack: it enriches geometric assets with spatially varying constitutive information so that downstream simulators can operate on them. Within that role, the paper positions AdaVoMP as a step toward automated physical annotation of 3D content at a scale and resolution not attainable with fixed dense voxel fields (Dagli et al., 16 Jun 2026).