Physics-Informed Deformable Gaussian Splatting

• PIDG is a method that integrates physics-based constraints into 3D Gaussian splatting to achieve high-fidelity, artifact-free dynamic scene reconstructions.
• It employs techniques like MLS-MPM and PBD to assign material properties and simulate realistic deformations in accordance with continuum mechanics.
• The framework leverages differentiable rendering and optical flow supervision, enhancing convergence speed and enforcing physically plausible motion.

Physics-Informed Deformable Gaussian Splatting (PIDG) refers to a family of explicit 3D scene representations and modeling methodologies that integrate data-driven 3D Gaussian splatting with physically-informative constraints. These constraints draw from continuum mechanics, constitutive modeling, or differentiable simulation, yielding systems capable of high-fidelity dynamic scene reconstruction and rendering under physically plausible or empirically consistent motion. PIDG methodologies unify neural rendering with Lagrangian material modeling, often employing differentiable rendering losses, optical flow supervision, and multi-material simulation, enabling artifact-free reconstructions and generalization to diverse motion regimes (Xiao et al., 9 Jun 2025, Dinkel et al., 13 May 2025, Hong et al., 9 Nov 2025).

1. Fundamental Principles of Physics-Informed Deformable Gaussian Splatting

PIDG builds on 3D Gaussian Splatting (3DGS), in which a scene is represented as a set of spatially localized, parameterized Gaussian primitives $\{g_i\}$, each defined by position $\mu_i \in \mathbb{R}^3$, scale $s_i$, rotation $q_i$, opacity $\alpha_i$, and color coefficients $c_i$. The key innovation in PIDG is to endow each Gaussian primitive with physically meaningful attributes—such as material parameters or time-evolving field values—and to supervise its evolution with explicit physics-based constraints or simulation. This association enables the representation to respect principles such as conservation of momentum, elasticity, and collision.

Variants of PIDG can be characterized by:

• Treating Gaussians as Lagrangian material points with time-varying constitutive parameters.
• Embedding per-object or per-particle physical properties (density $\rho$, Young’s modulus $E$, Poisson’s ratio $\nu$).
• Imposing physical constraints via explicit simulation (MLS-MPM), position-based dynamics (PBD), or residual forms of the Cauchy momentum equation.
• Supervising scene dynamics by aligning motion flows (e.g., particle flow) with camera-compensated optical flow.

2. Pipeline Architectures and Workflow

The canonical PIDG workflow consists of the following core stages, as seen across major works:

Stage	Primary Operation	References
3D Gaussian Reconstruction	Optimize 3DGS model from multi-view images	(Xiao et al., 9 Jun 2025, Hong et al., 9 Nov 2025)
Object-Level Segmentation / Partition	Fast 2D$\to$3D mapping via rendered depth & 2D masks	(Xiao et al., 9 Jun 2025)
Physical Property Assignment	Assign per-object material parameters (e.g., $\rho$, $E$)	(Xiao et al., 9 Jun 2025)
Physics Simulation & Deformation Modeling	MLS-MPM, PBD, or learned field prediction of velocity/stress	(Hong et al., 9 Nov 2025, Dinkel et al., 13 May 2025)
Rendering and Observation Loss	Differentiable rendering, photometric and flow supervision	(Hong et al., 9 Nov 2025, Dinkel et al., 13 May 2025)

Notable Variants

• In (Xiao et al., 9 Jun 2025), PIDG is realized as a unified pipeline integrating 3DGS with fast object segmentation, look-up assignment of material parameters, coupled multi-material simulation in MLS-MPM, and adaptive eigen-clamp artifact removal.
• "DLO-Splatting" (Dinkel et al., 13 May 2025) specializes PIDG for rope-like objects, combining a PBD physics prediction (with explicit stretch constraint, implicit damping), differentiable Gaussian-splat rendering, and prediction–update-style filtering.
• (Hong et al., 9 Nov 2025) proposes an end-to-end differentiable architecture with 4D decomposed hash encoding (static/dynamic fields), time-evolving material fields (for velocity and stress), a Cauchy momentum residual loss, and optical flow matching for supervision.

3. Physical Modeling and Simulation within PIDG

Physical modeling in PIDG frameworks encompasses several forms:

MLS-MPM (Moving Least Squares Material Point Method)

In (Xiao et al., 9 Jun 2025), every object-segmented Gaussian is mapped to MLS-MPM particles endowed with their object's physical parameters ($\rho$, $E$, $\nu$). The MLS-MPM solver proceeds via standard P2G (Particle-to-Grid) and G2P (Grid-to-Particle) transfers, computing per-particle deformation gradients $F_p$. The simulation updates particles' positions, velocities, and applies grid-based collision/contact. Gaussian covariance matrices $\Sigma_i$ are updated via pushforward: $\Sigma_i' = F_p \Sigma_i F_p^T$ Eigen-decomposition is used for axis-aligned scaling and rotation clamping, enforcing geometric fidelity.

PBD (Position-Based Dynamics)

"DLO-Splatting" (Dinkel et al., 13 May 2025) instantiates PIDG for tracking deformable linear objects (DLOs). The prediction step advances state using unconstrained Verlet integration: $$\mathbf X_0^{t+1} = \mathbf X^t + (\mathbf X^t - \mathbf X^{t-1})\Delta t + \frac{1}{2}\mathbf F^t \Delta t^2$$ followed by projection of node pairs onto an inextensible manifold: $$\mathbf x_i^{t+1} \gets \mathbf x_i^{t+1} - \frac{1}{2} \Delta l_i^{t+1} \mathbf n_i;\quad \mathbf x_{i+1}^{t+1} \gets \mathbf x_{i+1}^{t+1} + \frac{1}{2} \Delta l_i^{t+1} \mathbf n_i$$ Here, constraint satisfaction is enforced only for stretch; shape smoothness and damping are handled via integration/timestep selection.

Learned Material Fields and Physics Constraints

(Hong et al., 9 Nov 2025) introduces neural modeling for time-evolving fields: each Gaussian's position and index embedding, combined with 4D hash encodings and Fourier features, conditions a multi-head MLP outputting

$(v(x,t), \sigma(x,t)) = f_\theta(F),$

where $v$ is velocity and $\sigma$ is Cauchy stress. The Cauchy momentum residual is calculated and penalized: $$\mathcal{L}_{\rm CMR} = \frac{1}{M} \sum_{i=1}^M \left\| \rho\left(\frac{\partial v}{\partial t} + (v \cdot \nabla) v\right) - \nabla \cdot \sigma \right\|_2^2$$ Automatic differentiation facilitates evaluation of derivatives within the network.

4. Supervision, Losses, and Observation Consistency

All PIDG implementations utilize a combination of differentiable rendering and physics-motivated losses for supervision.

• Photometric Rendering Loss: The color at each pixel is rendered by depth-sorted, alpha-blended Gaussians:

$$\hat I_k(u) = \sum_j c_j \alpha_j(u) \prod_{\ell<j}(1 - \alpha_\ell(u))$$

The observation loss is

$$\mathcal{L}_{\rm obs} = \sum_{k=1}^K \sum_u \| \mathcal{I}_k^t(u) - \hat{\mathcal I}_k^t(u) \|_2^2$$

for all camera views $k$ and pixels $u$.

• Optical Flow Flow Matching (Hong et al., 9 Nov 2025): PIDG aligns Lagrangian particle or Gaussian flow to camera-compensated optical flow. Two kinds of flow—"Gaussian flow" and "velocity-advected flow"—are rendered and matched against motion flow extracted via backward warping:

$$\mathcal{L}_{\rm LPFM} = \lambda_g \| \text{flow}_g - \text{flow}_{gt} \|_1 + \lambda_v \| \text{flow}_v - \text{flow}_{gt} \|_1$$

where $\text{flow}_{gt}$ is the backward motion flow and $\lambda_g, \lambda_v=0.5$.

• Physics Residual Losses: Penalize violations of the Cauchy momentum equation over the Lagrangian field.
• Artifact Suppression (Xiao et al., 9 Jun 2025): Adaptive eigen-clamp is utilized in transforming deformation gradients to Gaussian scale/rotation, as unconstrained updates cause "needle" or vanishing artifacts. Soft corrections with empirical weights ($\lambda_S\approx0.8$, $\lambda_R\approx1.2$) are applied to maintain geometric plausibility.

5. Object Segmentation, Multi-Material Representation, and Coupled Interactions

PIDG augments 3DGS segmentation using fast 2D$\to$3D mapping of multi-view instance masks and depth rendering. Each Gaussian is assigned the modal object ID over all views, yielding precise 3D object clusters. No feature distillation is needed—depth priors and 2D masks suffice. Performance on segmentation benchmarks is strong, with reported mean IoU $>$95% on 3D-OVS and $>$90% on LERF-Mask (Xiao et al., 9 Jun 2025).

Once segmented, each object is assigned material parameters from a lookup table—typical values for density, Young’s modulus, and Poisson ratio are documented. The MLS-MPM solver admits all particles to a single global grid, naturally enabling collision, contact, and strongly coupled multi-material behaviors, such as soft–stiff object interactions, without need for explicit collision geometry.

6. Neural Network Architectures and Optimization Strategy

In the learning-based PIDG of (Hong et al., 9 Nov 2025), network modules are specialized:

• Spatial and Temporal MLPs: A 1-layer FC + ReLU over hash-based features.
• Deformation Decoder: 2-layer MLP (width 256) outputs Gaussian rigid and scale increments.
• Time-Evolving Material Field: Multi-head 2-layer MLP (width 256) produces per-Gaussian velocity and stress.
• Embedding and Hashing: Four distinct 3D hash tables ($G_{xyz}, G_{xyt}, G_{yzt}, G_{xzt}$) enable efficient encoding of static and spatio-temporal geometry.

A two-stage schedule is employed: all Gaussians are densified and jointly optimized (static+dynamic); then static geometry is frozen, and only the dynamic regions are refined via motion masks. Adam optimizer is used, hash grids receive 10–50$\times$ MLP learning rate, with step decay every 20k iterations.

7. Experimental Results, Benchmarks, and Quantitative Impact

PIDG achieves strong quantitative results across reconstructed dynamic datasets:

• On (Xiao et al., 9 Jun 2025), segmentation mean IoU reaches 95–97% on 3D-OVS and 80–91% on LERF-Mask, outperforming baselines (e.g., PhysGaussian, Feature Splatting).
• PIDG effectively eliminates rendering artifacts such as region drifting, penetration, and needle/hole formations arising in earlier 3DGS-based physical reconstructions.
• (Hong et al., 9 Nov 2025) demonstrates best-in-class PSNR, SSIM, LPIPS and MS-SSIM across synthetic (D-NeRF, SC-GS, Grid4D, MoDec-GS, MotionGS) and real-world dynamic video benchmarks (HyperNeRF). For example, on synthetic scenes, the addition of the physics-informed loss raises PSNR by 0.1–0.4 dB over the next best method; on soft-body and fluid dynamics, PIDG exhibits the largest gains.
• The physics-informed inductive bias and flow-matching acceleration yield improved convergence speed and superior generalization to unseen motion regimes, especially in fluid and elastoplastic cases.

8. Significance, Limitations, and Development Directions

PIDG constitutes the first suite of methods to jointly deliver high-precision 3D object-level segmentation, materials-aware simulation, and physically plausible novel-view dynamic synthesis in explicit Gaussian-splatting representations (Xiao et al., 9 Jun 2025, Hong et al., 9 Nov 2025, Dinkel et al., 13 May 2025).

• A plausible implication is that these methods, by narrowing the hypothesis space with physical constraints, systematically improve convergence rates and prevent overfitting to visual cues—particularly in challenging scenes with motion ambiguity, transparency, or underconstrained kinematics.
• These results suggest expanded applicability to robotics, visual digital twins of dynamic environments, and physically-grounded inverse graphics.

There remain limitations:

• Some implementations rely on explicit physics (MLS-MPM, PBD) with hand-assigned material parameters, constraining generalization to complex constitutive behaviors unless learned (as in time-evolving material fields).
• The computational cost of differentiable simulation and high-resolution Gaussian fields imposes nontrivial resource requirements—training iterations and memory scale with scene dynamic complexity.
• In certain variants, regularization for bend/smoothness is implicit, leaving higher-order behaviors to be enforced only by frictional/contact coefficients or timestep tuning.

Ongoing research investigates tighter integration of data-driven and physically-informed losses, efficient multi-material learning with overlapping deformation fields, and robust handling of fluid–structure interactions. The landscape of PIDG continues to evolve, encompassing both direct simulation-based approaches and neural field-based methods leveraging physics-informed surrogate loss terms.