We present GCA (Gaussian Constitutive Alignment), a framework for learning implicit constitutive laws from monocular dynamic video of deformable objects represented by 3D Gaussians. Given a static multi-view scan for geometric initialization, our method learns intrinsic physical dynamics solely from a single fixed-viewpoint video of the moving object. Existing implicit methods often suffer from local minima under noisy supervision and lack physical interpretability, while explicit approaches rely on predefined constitutive equations, limiting generalizability and becoming unstable in monocular settings. To address these challenges, our framework unifies LoRA-based adaptation with two key alignment modules. First, we propose Rank-based Depth-Geometric Anchors (RDGA) to establish robust geometric constraints from monocular dynamic observations via scale-invariant rank-based depth alignment, reducing the reliance on unreliable pixel-level color supervision. Second, a Constitutive Prior Regularizer (CPR) integrates classical constitutive models as soft differentiable priors, regularizing the optimization while preserving the flexibility of implicit modeling---even when the actual material is absent from the hypotheses. Extensive experiments on synthetic, real-to-sim, and real-world datasets demonstrate that GCA outperforms existing methods, achieving 48% lower Chamfer Distance than the strongest baseline on synthetic benchmarks while remaining robust under monocular supervision.
Figures & tables
Figure 1: The core idea of GCA. Learning implicit constitutive laws from dynamic video faces two key challenges: geometric ambiguity from sparse visual observations (left), and material ambiguity from unknown constitutive models and parameters (right). GCA bridges both sides through visual–physical bidirectional alignment: Rank-based Depth-Geometric Anchors (RDGA) establish robust geometric constraints via scale-invariant depth consensus, while a Constitutive Prior Regularizer (CPR) softly guides optimization using classical constitutive hypotheses. Together, they enable reliable learning of implicit constitutive laws (bottom) that generalize across diverse material types.
Figure 2: Overview of GCA . Our framework builds on Low-Rank Adaptation (LoRA) to fine-tune neural material laws while maintaining integration with PDE-based simulation, and introduces two coordinated alignment modules: (i) Rank-based Depth-Geometric Anchors (RDGA) that resolve geometric ambiguities via scale-invariant rank-based depth alignment, and (ii) a Constitutive Prior Regularizer (CPR) that provides soft physical priors while preserving generalization.
Material
Elastomer
Gel
Rubber
Plasticine
Granular
Non-Newt.
Object
Ball
Duck
Pawn
Cat
Fish
Bottle
Average
NCLaw [ 37 ]
4.085
2.934
2.031
1.909
0.536
1.631
2.188
NeuMA [ 7 ]
1.123
1.863
0.517
0.844
0.322
1.056
0.954
Ours
0.922
0.702
0.200
0.318
0.077
0.757
0.496
Table 1: Quantitative comparison on synthetic dataset (Chamfer Distance ↓ ). Our method achieves 48% lower average CD than NeuMA across diverse materials.
Figure 3: Chamfer Distance during physical simulation. Our method consistently maintains lower CD throughout the simulation.
Figure 4: Rendering metrics on synthetic dataset. Our method achieves superior PSNR, SSIM, and LPIPS.
Figure 5: Qualitative comparison on real-to-sim dataset. Our method accurately captures complex dynamics for real-world geometries.
Object
Plast. Dragon
Sand Wolf
Gel Pudding
NCLaw [ 37 ]
25.021
49.420
38.149
NeuMA [ 7 ]
3.527
9.803
13.804
Ours
2.081
5.842
0.906
Table 2: Chamfer Distance on real-to-sim dataset ( ↓ ). Our method achieves the lowest error across all objects.
Figure 6: Qualitative comparison on real-world dataset. Our method achieves better visual consistency using only monocular supervision. The PSNR and LPIPS values are computed after filtering the background.
Configuration
Elast.
Gel
Rubber
Plast.
Gran.
Non-Newt.
Note
NeuMA (Baseline)
1.123
1.863
0.517
0.844
0.322
1.056
Prior best
w/o Global Align.
>100 (Diverged)
Divergence
w/o Anchor Superv.
1.024
0.963
0.637
0.510
0.164
1.166
Loss of detail
Replace Rank w/ L1
1.064
1.098
0.846
0.919
0.147
0.989
Scale ambiguity
w/o RDGA
3.842
3.045
1.975
1.821
0.559
1.531
No geom. anchor
w/o CPR
1.024
0.694
0.243
0.510
0.084
0.769
No physics prior
Table 3: Detailed ablation study (Chamfer Distance ↓ ). Each component contributes meaningfully. Replacing rank-based depth alignment with L1 loss degrades performance below the baseline , validating our scale-invariant design.
Method
Mem. (GB)
Train (min)
Infer. (s) ↓
CD ↓
NeuMA [ 7 ]
28.9
73.3
32.4
1.31
Ours (Full F.T.)
76.9
408.6
32.5
1.35
Ours (LoRA)
30.5
51.2
31.5
0.50
Table 4: Computational efficiency analysis. LoRA fine-tuning achieves superior physical accuracy with lower cost.
Method
Supervision
Ball
Cat
Status
PAC-NeRF [ 30 ]
Multi-view
0.85
0.29
Reference
NeuMA [ 7 ]
Multi-view
0.98
0.35
Reference
PAC-NeRF + Depth
Monocular
> 100
> 100
Diverged
NeuMA [ 7 ]
Monocular
1.12
0.84
Suboptimal
Ours
Monocular
0.92
0.32
Robust
Table 5: Comparison with explicit methods under monocular setting (CD ↓ ). Explicit methods diverge even with depth supervision; GCA remains robust.
Figure 7: Direct law-level validation on held-out deformation gradients. Left: normalized constitutive response under analytic JellyDuck ground truth. Right: quantitative response error. GCA more accurately recovers the underlying constitutive response than NeuMA, reducing the average error from 39.6% to 8.0%.
Configuration
Elastomer
Plasticine
Composite
(target excl.)
(target excl.)
(undefined)
NeuMA [ 7 ]
1.123
0.844
3.628
Ours (Target Removed)
0.976
0.397
–
Ours (Full Library)
0.922
0.318
1.275
Table 6: Robustness to out-of-distribution materials (CD ↓ ). GCA outperforms baselines even when the correct hypothesis is absent or for composite materials.