A differentiable Lagrangian-coupled 3D Gaussian Splatting-SPH model for forward simulation and inverse analysis in solid mechanics

Tian Xu; Soroush Atashi; Tianju Xue

cs.CVOct 3, 2026

A differentiable Lagrangian-coupled 3D Gaussian Splatting-SPH model for forward simulation and inverse analysis in solid mechanics

Authors: Tian Xu, Soroush Atashi, Tianju Xue

Organizations: Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology

Abstract

Recent advances in generative world models have increased interest in digital models that reproduce both the appearance of real objects and their response to physical interaction. Three-dimensional reconstruction techniques, including 3D Gaussian Splatting, capture detailed surface geometry and appearance from images and videos. However, extending these representations beyond plausible animation to mechanically interpretable models for constitutive behavior, boundary conditions, and inverse parameter identification remains less explored. In this work, a differentiable Lagrangian-coupled 3DGS-smoothed particle hydrodynamics (SPH) model is proposed for forward simulation and inverse analysis of deformable solids. The observed object is first reconstructed from multi-view calibrated visual dataset as a 3DGS rendering model. An envelope-based procedure then generates an independent SPH support for the solid-mechanics model, avoiding the direct use of rendering primitives as mechanical particles. A reference-configuration Lagrangian transfer maps SPH deformation to Gaussian positions and covariances, thereby coupling the physical model and the image observation model while preserving a differentiable computational path. The SPH formulation supports linear elastic, hyperelastic, and Kelvin--Voigt viscoelastic responses, together with fixed, free, and Robin-type boundary conditions. Numerical studies validate the SPH response against finite-element results, assess accuracy and efficiency against a conventional model using Gaussian centers as surface SPH particles, and demonstrate forward simulations on beam, bridge, and liver-shaped examples. Inverse analyses further estimate constitutive and boundary parameters from rendered deformation observations, including noisy cases, demonstrating the feasibility of the proposed model for mechanics-based parameter identification from image data.

Figures & tables

Figure 1: Workflow of the proposed Lagrangian-coupled 3DGS-SPH model for forward simulation and inverse analysis. (a) Reference multiview images; (b) prescribed scene configuration; (c) constitutive and boundary parameters; (d) 3DGS rendering model and SPH physical model coupled through the proposed Lagrangian transfer; (e) simulated deformation frames; (f) inverse analysis using the discrepancy between simulated and observed frames; and (g) observed deformation frames. Brown arrows indicate model construction, red arrows indicate forward simulation, and dashed blue arrows indicate inverse analysis, including gradient propagation through the coupled model for parameter estimation.

Figure 2: Envelope-based generation of the SPH support from a reconstructed Gaussian model. Expanded Gaussian ellipsoids first define a finite-thickness shell. After gap closing and interior filling, occupied grid-cell centers are retained as surface and interior SPH supports.

Figure 3: Schematic support relation in the total-Lagrangian SPH discretization. The approximation at a reference support uses neighboring supports within the compact kernel radius.

Figure 4: Reference-configuration support relation for the Lagrangian SPH-GS transfer. Each Gaussian center selects neighboring SPH supports within the transfer kernel radius, forming a fixed bipartite graph for deformation transfer.

Figure 5: Schematic setting for inverse analysis. The observed image sequence is used to identify selected physical parameters, including elastic modulus, Robin boundary stiffness, and, for dynamic viscoelastic cases, Kelvin–Voigt viscosity parameters.

Geometry	Sim. type	Material model	B.C. setting
Beam	Quasi-static	Corotated linear	Robin: weak, strong, heterogeneous
Beam	Quasi-static	neo-Hookean	Robin: weak, heterogeneous
Beam	Dynamic	Corotated linear + Kelvin–Voigt	Weak Robin
Beam	Dynamic	neo-Hookean + Kelvin–Voigt	Weak Robin
Bridge	Quasi-static	Corotated linear	Fixed lower band and central top load
Liver-shaped	Dynamic	Corotated linear + Kelvin–Voigt	Fixed lower region, Robin region, and end load

Table 1: Design of the forward mechanics experiments.

Figure 6: Multiview reconstruction of the undeformed speckle-textured beam used in the forward simulations. The upper row shows four reference images from $\mathfrak{D}_{\mathrm{r}}$ , and the lower row shows the corresponding renderings generated by the reconstructed 10,000-kernel 3DGS model.

Figure 7: Quasi-static corotated-linear beam response under different Robin support fields. Subfigure (a) show the boundary condition settings of the beam case. Subfigures (b)–(d) compare weak, strong, and heterogeneous Robin support, respectively. The upper row shows the rendering images of final deformation, and the lower row shows the corresponding SPH particle deformation colored by downward displacement $w=-u_{z}$ .

Elastic model	Robin support	$k_{R}$ ( $\mathrm{s^{-2}}$ )	Final $w_{\mathrm{load}}$ (m)	Final $w_{R}$ (m)
corotated linear	weak	$10^{-2}$	0.5244	0.1478
corotated linear	strong	$10^{4}$	0.04265	$3.21\times 10^{-5}$
corotated linear	heterogeneous	$10^{4},10^{2}$	0.08104	$5.03\times 10^{-3}$
neo-Hookean	heterogeneous	$10^{4},10^{2}$	0.08101	$5.03\times 10^{-3}$
neo-Hookean	weak	$10^{-2}$	0.5245	0.1475

Table 2: Quasi-static forward response for the beam. Downward displacement is

w=-u_{z}

Figure 8: Dynamic bending response of the beam for different viscous coefficients. The gray curve is the numerically damped quasi-static reference; CL denotes the corotated-linear model and KV denotes Kelvin–Voigt viscosity. Increasing $\eta_{s}=\eta_{b}$ reduces the oscillation amplitude.

Elastic model	Viscous model	Final $w_{\mathrm{load}}$ (m)	Peak $w_{\mathrm{load}}$ (m)	Final $\max\\|\bm{u}\\|$ (m)	time (s)
corotated linear	none	0.9188	1.0369	0.9916	11.15
corotated linear	KV, $\eta_{s}=\eta_{b}=1000\,\mathrm{Pa}\cdot\mathrm{s}$	0.7701	0.9677	0.8186	37.83
corotated linear	KV, $\eta_{s}=\eta_{b}=5000\,\mathrm{Pa}\cdot\mathrm{s}$	0.5540	0.7602	0.5794	71.11
corotated linear	KV, $\eta_{s}=\eta_{b}=10000\,\mathrm{Pa}\cdot\mathrm{s}$	0.5267	0.6173	0.5501	92.68
neo-Hookean	KV, $\eta_{s}=\eta_{b}=5000\,\mathrm{Pa}\cdot\mathrm{s}$	0.5548	0.7628	0.5804	72.55

Table 3: Dynamic forward response with weak Robin support.

Figure 9: Multiview reconstruction of the undeformed bridge (a–d) and liver-shaped object (e–h). For each object, the upper row shows reference images from its dataset $\mathfrak{D}_{\mathrm{r}}$ , and the lower row shows the corresponding renderings generated by the reconstructed 3DGS model.

Geometry	$E$ (Pa)	$\nu$	$\rho_{0}$ ( $\mathrm{kg}\cdot\mathrm{m^{-3}}$ )	$\eta_{s}=\eta_{b}$ ( $\mathrm{Pa}\cdot\mathrm{s}$ )	Load (N)
Bridge	$10^{5}$	0.20	2400	0	500
Liver-shaped	$10^{4}$	0.45	1060	3000	8

Table 4: Material parameters and loading for the bridge and liver-shaped forward simulations.

Figure 10: Boundary configurations for the bridge and liver-shaped forward simulations. The bridge case uses a lower fixed band and a central loaded patch on the top layer. The liver-shaped case uses surface supports for the lower fixed region, the adjacent Robin region, and the loaded end region.

Figure 11: Forward deformation responses for the bridge and liver-shaped examples. Initial and deformed Gaussian renderings are shown together with the corresponding mean downward displacement histories. (a) The bridge case illustrates a quasi-static elastic response on a multi-member structure. (b) The liver-shaped case illustrates a transient viscoelastic response with both loaded-region and Robin-region displacement curves.

Geometry	SPH particles	Gaussians	Final $w_{\mathrm{load}}$ (m)	Peak $w_{\mathrm{load}}$ (m)	time (s)
Bridge	2381	11310	0.05116	0.06392	32.85
Liver-shaped	2677	10000	0.11434	0.19272	123.28

Table 5: Forward simulation results for the bridge and liver-shaped cases. Final and peak

w_{\mathrm{load}}

denote the mean loaded-region displacement at

t=20\,\mathrm{s}

and its maximum over the simulated interval.

Geometry	Unknown parameters	Material model
Beam	$E$	corotated linear
Bridge	$E$	neo-Hookean
Beam	$E,\eta_{s},\eta_{b}$	corotated linear + Kelvin–Voigt
Beam	$k_{R}$	corotated linear
Beam	$E,k_{R}$	corotated linear / neo-Hookean
Liver-shaped	$E,k_{R}$ , then $\eta_{s},\eta_{b}$	neo-Hookean + Kelvin–Voigt

Table 6: Design of the inverse-identification experiments.

Figure 12: Iteration histories for constitutive identification under prescribed boundary conditions. (a) Image-data mismatch objective $\mathcal{J}_{\mathrm{o}}$ normalized by its initial value. (b) Elastic modulus relative to its reference. (c) Recovered Kelvin–Voigt coefficients in the five-frame beam test. Horizontal dotted lines denote the reference values.

Figure 13: Forward-evaluated states for constitutive identification. Columns show the initial estimate, recovered state, and synthetic reference. Rows correspond to the corotated linear beam, neo-Hookean bridge, and the last observed frame of the Kelvin–Voigt beam.

Figure 14: Joint identification of material parameters and boundary conditions for the beam. (a,c) Normalized image-data mismatch for the corotated-linear and neo-Hookean models. (b,d) Corresponding Young’s modulus estimates normalized by the reference value. Particlewise and RBF boundary representations use the same scene configuration. The corotated-linear particlewise history reports best-so-far evaluations.

Figure 15: Final rendered states for joint identification of material parameters and boundary conditions for the beam. Columns show initialization, recovery, and reference. Rows compare particlewise and RBF Robin fields under corotated linear and neo-Hookean elasticity.

Figure 16: Joint identification of material parameters and boundary conditions for the beam using an image with 1% salt-and-pepper noise. (a,b) Clean reference and noisy observation. (c,d) Simulated images with the initial and recovered parameters. (e) Image MSE between the simulated image and noisy observation during optimization; the dashed line denotes the MSE between the clean reference and noisy observation. (f) Elastic modulus estimates and reference value. (g,h) Reference, initial, and recovered Robin stiffness and final vertical displacement over the Robin region. Particles in (g,h) are ordered by increasing reference $x$ -coordinate from the fixed-end side to the loaded-end side.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 1: Reference validation of the solid SPH discretization. Panels (a)–(d) show the final deformed configurations of the FEM reference, the corrected SPH model (ours), the corrected surface-dense SPH model, and the uncorrected SPH model, respectively. Panel (e) compares the section-averaged displacement magnitude along the normalized axial coordinate $\xi=(X_{x}-X_{\min})/(X_{\max}-X_{\min})$ .

Model	Supports/nodes	Maximum $\\|\bm{u}\\|_{2}$ (m)	$\\|\bm{u}\\|_{2}$ NRMSE
FEM reference	4961 nodes	0.7183	–
Corrected SPH	4000 supports	0.7343	$4.27\%$
Surface-enriched corrected SPH	5000 supports	0.7900	$8.16\%$
Uncorrected SPH	4000 supports	0.9543	$18.9\%$

Appendix

Table 1: Comparison of the FEM reference and SPH models for the linear beam benchmark.

Figure 2: Physical discretizations and B.C. assignments used in the transfer experiment. (a) Co-located model, in which every surface support coincides with a Gaussian center and is therefore not overplotted. (b) Proposed Lagrangian-coupled model with an independently discretized SPH domain; Gaussian centers are omitted to expose its support distribution. Panels (c) and (d) show the fixed, weak Robin-supported, loaded, and free supports selected in the two models.

Figure 3: Rendering and Gaussian-displacement agreement between the co-located construction and the proposed 3DGS–SPH model with Lagrangian SPH–GS transfer. The comparison uses matched deformation amplitudes. (a) Co-located model. (b) Proposed model. (c) Pixelwise mean absolute RGB difference between (a) and (b), with the color scale truncated at the 99.5th percentile. (d) Global relative $L_{2}$ Gaussian-displacement error $\varepsilon_{2}(t)$ of the proposed model over simulation time. (e) Pointwise normalized displacement errors $e_{g}$ at five representative states. Both displacement-error measures use the co-located response as the baseline; boxes denote the interquartile range, center lines the median, and whiskers the 5th and 95th percentiles.

Parameter (unit)	Material/B.C. field	Evaluation	Reference	Dim.	Relative error
$E$ (Pa)	corotated linear	5000	10000	1	$4.6788\times 10^{-2}$
$E$ (Pa)	neo-Hookean	5000	10000	1	$6.3272\times 10^{-3}$
$k_{R}$ ( $\mathrm{s^{-2}}$ )	Robin stiffness field	1000	100	105	$7.4073\times 10^{-3}$
$(\eta_{s},\eta_{b})$ ( $\mathrm{Pa}\cdot\mathrm{s}$ )	Kelvin–Voigt	1000	5000	2	$5.0778\times 10^{-3}$

Appendix

Table 2: AD–FD comparison for the image-mismatch term

\mathcal{J}_{\mathrm{o}}

in Eq. ( 24 ). Evaluation values specify where derivatives are computed; reference values generate the reference images. Dim. denotes the parameter dimension.

Figure 4: Automatic differentiation validation for the beam 3DGS–SPH coupled pipeline. Relative errors between AD and FD directional derivatives of $\mathcal{J}_{\mathrm{o}}$ are evaluated through the complete SPH rollout, SPH–GS transfer, covariance update, and differentiable rendering path. Here $E_{\mathrm{C\text{-}L}}$ and $E_{\mathrm{N\text{-}H}}$ denote Young’s-modulus checks for the corotated-linear and neo-Hookean material models, respectively.

Symbol	Definition
$g=1,\ldots,G$	Gaussian index and number of rendered Gaussians.
$i,j=1,\ldots,N$	SPH support indices and number of physical supports.
$\bm{\mu}_{g}^{0},\bm{\mu}_{g}^{n}$	Reference and current Gaussian centers.
$\bm{\Sigma}_{g}^{0},\bm{\Sigma}_{g}^{n}$	Reference and current Gaussian covariances.
$\bm{X}_{i},\bm{x}_{i}^{n},\bm{v}_{i}^{n}$	Reference position, current position, and velocity of SPH support $i$ .
$m_{i},\Delta V_{i},\rho_{0}$	Support mass, reference volume, and reference density.

Appendix

Table 3: Principal notation used in the coupled formulation.

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LagrangeGS: Non-Conservative Lagrangian System on Dynamic 3D Gaussian Splatting

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3D Gaussian Splatting Dynamic Scene Reconstruction Dynamical Systems

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Dynamic 3D Gaussian Splatting (3DGS) achieves photorealistic reconstruction of time-varying scenes, and recent physics-aware extensions improve extrapolation by explicitly predicting velocity fields. However, these extensions merely fit vector fields to visual deformations without satisfying Lagrangian mechanics, leading to three major issues: (i) physically inconsistent trajectories, (ii) lack of time-reversibility, and (iii) geometric collapse during long-term extrapolation. In this paper, we propose LagrangeGS, which formulates dynamic 3DGS as a non-conservative Lagrangian system. While this Lagrangian formulation fundamentally solves (i), a direct application of general LNNs to dynamic 3DGS requires a large velocity-Hessian inversion for millions of Gaussian particles. To overcome this computational bottleneck, we approximate the velocity-Hessian as an identity matrix, decoupling particle dynamics for computational tractability. For (ii), we restrict the non-conservative forces to be explicitly time independent, enabling consistent backward integration. Finally, to address (iii), we introduce local rigid alignment that regularizes particle trajectories. Extensive evaluations on dynamic scene benchmarks demonstrate that LagrangeGS enables stable long-term extrapolation, consistent time reversal, and counterfactual physics-based editing without retraining.

Abstract

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