VisionLaw: Inferring Interpretable Intrinsic Dynamics from Visual Observations via Bilevel Optimization

## Abstract

Abstract The intrinsic dynamics of an object governs its physical behavior in the real world, playing a critical role in enabling physically plausible interactive simulation with 3D assets. Existing methods have attempted to infer the intrinsic dynamics of objects from visual observations, but generally face two major challenges: one line of work relies on manually defined constitutive priors, making it difficult to generalize to complex scenarios; the other models intrinsic dynamics using neural networks, resulting in limited interpretability and poor generalization. To address these challenges, we propose VisionLaw, a bilevel optimization framework that infers interpretable expressions of intrinsic dynamics from visual observations.
At the upper level, we introduce an LLMs-driven decoupled constitutive evolution strategy, where LLMs are prompted as a knowledgeable physics expert to generate and revise constitutive laws, with a built-in decoupling mechanism that substantially reduces the search complexity of LLMs.
At the lower level, we introduce a vision-guided constitutive evaluation mechanism, which utilizes visual simulation to evaluate the consistency between the generated constitutive law and the underlying intrinsic dynamics, thereby guiding the upper-level evolution.
Experiments on both synthetic and real-world datasets demonstrate that VisionLaw can effectively infer interpretable intrinsic dynamics from visual observations. It significantly outperforms existing state-of-the-art methods and exhibits strong generalization for interactive simulation in novel scenarios.

## Introduction

With the advancement of 4D generation (Zhao et al. 2023; Bahmani et al. 2024; Jiang et al. 2024b; Ren et al. 2023), realistic interaction with 3D assets has become increasingly feasible, facilitating broad applications in areas like virtual reality, embodied intelligence, and animation (Shi et al. 2023; Lu et al. 2024a; Jiang et al. 2024a). Among these advances (Xie et al. 2024; Lin et al. 2024a, b), incorporating physical simulation (Stomakhin et al. 2013; Müller et al. 2007) stands out as a particularly prominent method, as it enables the generation of interactive dynamics that closely mirror real-world physical behavior. To ensure simulation realism, it is essential to accurately capture the intrinsic dynamics of objects, including material properties (e.g., stiffness) and constitutive laws (Chaves 2013), which describe the response behaviors of materials under applied forces.

Humans can infer the intrinsic dynamics of objects merely by observing their motion, and are even capable of predicting how these objects would interact in new scenarios. A fundamental question arises: can we enable machines to infer the intrinsic dynamics directly from visual observations, as humans do? Recent methods (Xie et al. 2024; Li et al. 2023) have attempted to bridge the gap between visual dynamics and simulation by incorporating physical simulators (e.g., Material Point Method, MPM) (Stomakhin et al. 2013) into 3D representations such as NeRF and 3D Gaussian Splatting (3DGS) (Mildenhall et al. 2021; Kerbl et al. 2023).
This integration has led to a promising paradigm for inferring the intrinsic dynamics from visual observations.
Depending on the type of intrinsic dynamics being inferred, existing methods can be categorized into two groups: material parameter estimation and constitutive law inference.

For material parameter estimation, PAC-NeRF and GIC (Li et al. 2023; Cai et al. 2024) estimate material parameters by the supervision of multi-view videos. PhyDreamer, DreamPhysics, and Physics3D (Zhang et al. 2024; Huang et al. 2025; Liu et al. 2024) distill visual dynamics priors from video diffusion models to guide the estimation process.
However, these approaches typically rely on manually defined constitutive laws, which often fail to align with the complex physical behaviors observed in practice, thereby compromising the accuracy and reliability of material parameter estimation.

For constitutive law inference,
OmniPhysGS (Lin et al. 2025) introduces constitutive Gaussians, which assign a suitable constitutive law to each Gaussian kernel from an expert-designed constitutive set. However, such a predefined set often fails to capture the full diversity of real-world physical behaviors.
NeuMA (Cao et al. 2024) learns neural constitutive laws to align with visual observations. Despite their effectiveness, these laws are represented as a black-box formulation, lacking interpretability, and thus making them difficult for humans to understand.

To overcome the aforementioned challenges, we introduce VisionLaw, an interpretable intrinsic dynamics inference framework based on bilevel optimization, which can jointly infer discrete constitutive law expressions and their corresponding continuous material properties solely from visual observations.
At the upper level, we propose an LLMs-driven decoupled constitutive evolution strategy, which: 1) unleashes the capabilities of LLMs in physical understanding and mathematical reasoning to generate and refine constitutive hypotheses; 2) introduces a decoupling mechanism to effectively alleviate the search space explosion caused by jointly evolving elastic and plastic components.
At the lower level, we construct a vision-guided constitutive evaluation mechanism. Supervised by visual observations, it optimizes the continuous material parameters of a given constitutive law using a differentiable simulator and renderer. The goal is to generate evaluation and feedback that reflect the consistency between the generated laws and ground-truth intrinsic dynamics, which in turn guides the evolution at the upper level.
Through collaborative optimization between the upper and lower levels, VisionLaw effectively captures the interpretable intrinsic dynamics underlying visual observations, which can be effectively applied to generate 4D interactions in novel scenarios. Our contributions are summarized as follows:

- •
We propose a bilevel optimization framework that can automatically infer discrete constitutive laws and continuous material properties solely from visual observations.
- •
We distill physics priors from LLMs to enable the evolution of constitutive laws. In addition, a decoupled evolution strategy is introduced to significantly improve both search efficiency and solution quality.
- •
We introduce a Vision-Guided Constitutive Evaluation mechanism to provide evaluation and feedback of a given constitutive law for the upper-level evolution.
- •
Extensive qualitative and quantitative comparisons on both synthetic and real-world datasets demonstrate that our method effectively captures the interpretable intrinsic dynamics underlying visual observations and applies them to 4D interaction generation in novel scenarios.

## Related Work

### Physics-Based 4D Interaction

Advances in 3D representation methods (Mildenhall et al. 2021; Müller et al. 2022; Barron et al. 2021; Kerbl et al. 2023) (e.g., NeRF and 3DGS) have greatly facilitated the creation of 3D assets (Poole et al. 2023; Tang et al. 2024b, a), consequently drawing significant attention to the pursuit of realistic interaction with these assets.
To enable physically plausible 4D interaction, recent works have attempted to incorporate various physical simulators (Stomakhin et al. 2013; Macklin, Müller, and Chentanez 2016) with 3D representation.
PIE-NeRF (Feng et al. 2024) enables meshless nonlinear elastodynamic simulation directly in NeRF via augmented Poisson disk sampling and quadratic generalized moving least squares (Q-GMLS) (Martin et al. 2010).
Inspired by the Lagrangian nature of 3DGS, PhysGaussian (Xie et al. 2024) pioneered the integration of MPM simulator into 3DGS.
Phys4DGen (Lin et al. 2024b) effectively perceives multiple materials within a single object and automatically assigns material properties by distilling physical priors from MLLMs (Achiam et al. 2023), enabling more accurate and user-friendly interactive dynamic generation.
VR-GS (Jiang et al. 2024a) conducts tessellation via TetGen (Hang 2015) to convert 3DGS representations into tetrahedral meshes, enabling fast XPBD simulation and physically plausible interaction in VR.

Figure: Figure 2: Given a constitutive individual—either predefined during initialization or generated by LLMs—it is embedded into a differentiable MPM simulator for forward simulation. The resulting visual dynamics are rendered and compared with real observations to compute a loss, which is backpropagated to optimize material parameters. This process produces both a fitness score and feedback for the individual. Based on fitness, the top-k individuals are selected and, along with their feedback, encoded into prompts for the LLMs. Guided by the decoupled evolution strategy, the LLMs analyzes and refines these constitutive law expressions to generate offspring for the next optimization iteration.
Refer to caption: x2.png

### Intrinsic Dynamics Learning

Understanding the intrinsic dynamics underlying observational data is highly valuable for interactive simulation (Müller and Gross 2004) and scientific discovery (Wang et al. 2023).
Deep learning (Zeng et al. 2024; Lu et al. 2024b) has advanced rapidly and is increasingly being applied to physical simulation (Sanchez-Gonzalez et al. 2020), with some methods (Pfaff et al. 2020; Ummenhofer et al. 2019) using end-to-end networks to model physical laws.
However, purely neural approaches often lack physical consistency. NCLaw (Ma et al. 2023) integrates known laws with a learnable constitutive model for refinement. SGA (Ma et al. 2024) uses LLMs to infer constitutive laws from particle trajectories. However, they rely on labeled data or high-quality motion, which are difficult to acquire.
The integration of 3D representation and physical simulation makes it possible to infer intrinsic dynamics from visual observations (Xie et al. 2024; Zhong et al. 2024). PAC-NeRF (Li et al. 2023) jointly learns NeRF representations and material parameters from multi-view videos. To avoid texture distortion, GIC (Cai et al. 2024) presents a geometry supervision framework. PhysDreamer, DreamPhysics, Physics3D, PhysFlow (Zhang et al. 2024; Huang et al. 2025; Liu et al. 2024, 2025) guide the estimation process by distilling visual dynamic priors from video diffusion models.
However, the parameters estimation process in these methods relies on expert-defined constitutive laws.
While NeuMA (Cao et al. 2024) can learn neural constitutive models from visual observations, it lacks interpretability and exhibits weak generalization ability.
In this paper, we aim to infer constitutive law expressions from visual observations that are both interpretable and highly generalizable.

## Preliminaries

### Constitutive laws

In continuum mechanics (Chaves 2013), constitutive laws define how materials respond under applied forces.
The essential reason why materials like rubber, sand, and water exhibit entirely different physical behaviors lies in the differences in the constitutive laws they follow.
To simulate the motion and deformation of materials, we need to solve a system of partial differential equations derived from the conservation of mass and momentum:

$$ $\frac{D\rho}{Dt}+\rho\nabla\cdot\mathbf{v}=0,\quad\rho\frac{D\mathbf{v}}{Dt}=\nabla\cdot\boldsymbol{P}+\rho\mathbf{g},$ (1) $$

where $\rho$ denotes the density, $\mathbf{v}$ the velocity field, $\mathbf{g}$ the gravitational acceleration, and $\mathbf{P}$ the stress tensor, which is defined by the constitutive law.
In this paper, we employ the MPM simulator to solve the above system of governing equations for simulation. Please refer to the appendix for further details about MPM. Within the MPM framework, two types of constitutive laws must be specified: (1) an elastic constitutive law that describes reversible elastic responses, and (2) a plastic constitutive law that captures irreversible plastic evolution. Their formulations are given as:

$$ $\varphi_{E}\left(\mathbf{F};\theta_{E}\right)\mapsto\boldsymbol{\tau},\quad\varphi_{P}\left(\mathbf{F};{\theta}_{P}\right)\mapsto\mathbf{F}^{\text{corrected}},$ (2) $$

where $\varphi_{E}$ and $\varphi_{P}$ denote the elastic and plastic constitutive laws, respectively. $\mathbf{F}$ is the deformation gradient, $\boldsymbol{\tau}$ is the Kirchhoff stress tensor, ${\mathbf{F}^{\text{corrected}}}$ is the corrected deformation gradient after plastic return mapping. The continuous material parameters in the elastic and plastic laws are denoted by $\theta_{E}$ and $\theta_{P}$, respectively. Several classical constitutive laws are listed in the appendix.
Despite the availability of many classical constitutive laws, they remain inadequate in capturing the diversity and nonlinear behavior of complex materials. To this end, we propose VisionLaw, which infers constitutive laws directly from visual observations.

### Physics-Integrated 3D Gaussians

3D Gaussians Splatting (3DGS) (Kerbl et al. 2023) represents the scene using a set of anisotropic Gaussian kernels $\mathcal{G}=\{\mathbf{x}_{i},\mathbf{A}_{i},\alpha_{i},\mathbf{\mathcal{C}}_{i}\}_{i\in\mathcal{K}}$, where $\mathbf{x}_{i}$, $\mathbf{A}_{i}$, $\alpha_{i}$, and $\mathbf{\mathcal{C}}_{i}$ represent the center position, covariance matrix, opacity, and spherical harmonic coefficients of the Gaussian kernel $\mathcal{G}_{i}$, respectively. To render 3D Gaussians into a 2D image from a given view, the color of each pixel can be formulated as:

$$ $\mathbf{C}=\sum_{i\in\mathcal{N}}\sigma_{i}\mathbf{SH}(d_{i},\mathcal{C}_{i})\prod_{j=1}^{i}(1-\sigma_{j}),$ (3) $$

where $\mathcal{N}$ denotes a set of sorted Gaussian kernels related to the pixel and view. $\sigma_{i}$ is the effective opacity, defined as the product of the projected 2D Gaussian weight and opacity $\alpha_{i}$. $\mathbf{SH}$ computes RGB values based on the view direction $d_{i}$ and spherical harmonic coefficients $\mathbf{\mathcal{C}}_{i}$.
Unlike NeRF’s implicit form, 3DGS offers an explicit representation that exhibits a Lagrangian nature, facilitating seamless integration with simulation algorithms.
Thus, PhyGaussians (Xie et al. 2024) pioneers the integration of MPM simulator (Stomakhin et al. 2013) into 3DGS, combining physical simulation with visual rendering.
Specifically, this method treats Gaussian kernels as particles representing the continuum, and assigns each a time property $t$, material properties $\theta$ (e.g., stiffness).
Therefore, given the constitutive law and simulation conditions (e.g., external forces and boundary), MPM can be applied to predict the displacement and deformation of Gaussian kernels at the next time step:

$$ $\displaystyle\mathbf{x}^{t+1},\mathbf{F}^{t+1}$ $\displaystyle=\mathbf{\Phi}(\mathcal{G}^{t}),$ (4) $\displaystyle\mathbf{A}^{t+1}=(\mathbf{F}$ ${}^{t+1})\mathbf{A}^{t}(\mathbf{F}^{t+1})^{T}.$ (5) $$

Here, $\mathbf{\Phi}$ is a differentiable MPM simulator, $\mathbf{F}^{t+1}$ denotes the deformation gradient at time step $t+1$, which describes the local deformation of particles (the subscript $i$ is omitted for simplicity).
Gaussian covariance $\mathbf{A}^{t+1}$ can be updated by applying $\mathbf{F}^{t+1}$, which approximates the deformation of the Gaussian kernel. After the MPM simulation is completed, a 4DGS representation is constructed, which enables rendering of visual dynamics using Eq. [3](https://arxiv.org/html/2508.13792v1#Sx3.E3).

## Methodology

In this work, we aim to infer interpretable intrinsic dynamics from a series of visual observations.
Formally, given multi-view video observations $V=\{V_{1},V_{2},...,V_{N}\}$ of moving objects along with corresponding camera extrinsic and intrinsic parameters, the goal is to infer the discrete constitutive law expressions and optimize the continuous material parameters in a unified manner.
To this end, we propose VisionLaw, a novel bilevel optimization framework:

$$ $\min_{\varphi,\Theta}\;\mathcal{L}\left(\mathcal{R}\left(\varphi,\Theta,\theta^{*};\Phi,\mathcal{G}\right),V\right),$ (6) $$

$$ $\text{s.t.}\quad h\left(\varphi,\theta;\Phi\right)\leq 0,$ (7) $$

$$ $\theta^{*}\in\arg\min_{\theta\in\Theta}\mathcal{L}\left(\mathcal{R}\left(\theta;\Phi,\varphi\right),V\right),$ (8) $$

where, $\mathcal{R}$ is a differentiable renderer defined by Eq. [3](https://arxiv.org/html/2508.13792v1#Sx3.E3).
The constitutive law $\varphi$ consists of an elastic law $\varphi_{E}$ and a plastic law $\varphi_{P}$.
$\Theta$ defines the continuous parameter space for inner-level optimization $\theta\in\Theta$.
$h(\cdot)\leq 0$ refers to the validity of the simulation (e.g. whether a constitutive law $\varphi$ is simulatable). The material parameter $\theta$ includes the elastic parameters $\theta_{E}$ and the plastic parameters $\theta_{P}$.
For the upper level, based on feedback from the lower-level, LLMs is employed to generate and refine discrete constitutive expressions $(\varphi,\Theta)$.
At the lower level, given the output $(\varphi,\Theta)$ from the upper level, the optimal continuous material parameters $\theta^{*}$ are estimated under visual observation supervision, using differentiable rendering and simulation. Meanwhile, evaluation and feedback are generated during this process to guide the upper-level evolution.
The pipeline of the proposed VisionLaw is illustrated in Fig.[2](https://arxiv.org/html/2508.13792v1#Sx2.F2).

### Upper-Level Constitutive Evolution

#### LLMs-Driven Constitutive Laws Evolution.

Recently, LLMs have shown tremendous potential in scientific discovery (Yang et al. 2023; Romera-Paredes et al. 2024; Ma et al. 2024), owing to their strong symbolic reasoning abilities and extensive physical priors.
Inspired by this, in the upper-level search, we prompt LLMs to evolve constitutive law expressions.
Specifically, we consider LLMs as an intelligent operator and construct an evolutionary search paradigm to iteratively optimize the constitutive law expressions. Each law is represented as a Python code snippet with a clear physical meaning and strong interpretability.
The optimization procedure consists of five stages, which are as follows:
i) Initialization: Several classical constitutive laws (e.g., purely elastic material models) are introduced as initial individuals. This serves as a physically plausible starting point for the evolutionary process.
ii) Fitness Evaluation: Each candidate constitutive law is passed to the lower level for simulation testing. Its fitness is evaluated based on visual observation, and feedback, such as the loss curve, is collected.
iii) Selection: to enhance population diversity and avoid local optima, we first remove duplicate constitutive individuals with fitness differences below a threshold $\epsilon$. Then, we select the top-k constitutive individuals with the highest fitness from the remaining population as ”parents” for the next round of evolution.
iv) Expression Correction: we prompt LLMs to (1) analyze the parent expression and identify any shortcomings based on its feedback; (2) design an improvement plan and determine how to modify the expression to increase fitness; (3) generate a set of physically plausible constitutive law expressions as candidate individuals.
This process is formalized as:

$$ $\{\varphi^{m},\Theta^{m}\}_{m\in|M|}=\text{LLM}\left(\left\{\mathcal{\varphi}^{k},\Theta^{k},o^{k}\right\}_{k\in|K|},\mathcal{P}\right),$ (9) $$

where, $K$ denotes parent size, $M$ denotes offspring size, $o$ represents the feedback obtained from the lower level and $\mathcal{P}$ denotes the prompt provided to LLMs.
v) Stages ii) to iv) constitute a complete evolutionary iteration. Multiple evolutionary iterations are executed until the predefined number of iterations is reached.
Eventually, the algorithm evolves constitutive laws that not only simulate dynamic behaviors consistent with visual observations but also exhibit strong physical interpretability.

**Table 1: Quantitative Comparison via Chamfer Distance on Synthetic Datasets. Lower values indicate better alignment with true intrinsic dynamics.**
| Method | BouncyBall | ClayCat | HoneyBottle | JellyDuck | RubberPawn | SandFish | Average |
| --- | --- | --- | --- | --- | --- | --- | --- |
| PAC-NeRF (Li et al. 2023) | 516.30 | 15.38 | 2.21 | 137.73 | 15.47 | 1.71 | 114.80 |
| NeuMA (Cao et al. 2024) | 1.78 | 1.24 | 1.09 | 10.96 | 1.01 | 1.07 | 2.86 |
| VisionLaw (Ours) | 1.08 | 0.77 | 0.79 | 5.19 | 0.94 | 1.10 | 1.65 |

#### Decouple Evolution Strategy.

In the MPM simulation framework, a complete constitutive law $\varphi$ consists of an elastic part $\varphi_{E}$ and a plastic part $\varphi_{P}$, which together determine the simulation output of the system.
However, optimizing both types of constitutive laws simultaneously greatly enlarges the search space, increases the difficulty of LLMs search, and makes convergence to high-quality solutions more challenging.
To address the above issue, we propose a decoupled evolution strategy that splits the coupled constitutive optimization task into two independently solvable sub-tasks, thereby effectively reducing the search space. This strategy consists of two phases: 1) Alternating Evolution: In each evolutionary iteration, we prompt the LLM to optimize only one part of the constitutive law expression (either elastic or plastic), while keeping the other fixed to be updated in the next iteration. The two types of constitutive law expressions are optimized in an alternating manner over multiple iterations.
2) Joint Evolution: After the alternating optimization phase, we prompt the LLM to jointly optimize both elastic and plastic components to further enhance the overall performance. This phase serves as a fine-grained refinement of the existing high-quality expressions from a global perspective. Through the proposed decoupled evolution strategy, we effectively reduce the search space, enhance the stability and efficiency of LLM-based search, and substantially improve the quality of the final generated constitutive laws.

### Lower-Level Constitutive Evaluation

To effectively evaluate whether a candidate constitutive expression can accurately capture the intrinsic dynamics of motion observed in visual data, and to provide high-quality feedback to the upper-level evolution, we design a vision-guided constitutive evaluation mechanism.
First, a static 3DGS representation is reconstructed from the first frame of multi-view video inputs.
Then, the candidate constitutive law expression $\varphi(\theta)$, with continuous material parameters, is seamlessly embedded into a differentiable MPM simulator.
We integrate the MPM simulator with 3DGS to drive the simulation and render the predicted visual dynamics $V$ from given views. The supervised loss between the predicted and observed visual dynamics can be formulated as:

$$ $\mathcal{L}=\frac{1}{N}\sum_{n=1}^{N}[\lambda\mathcal{L}_{2}(\hat{V}_{n},V_{n})+(1-\lambda)\mathcal{L}_{\text{D--SSIM}}(\hat{V}_{n},V_{n})],$ (10) $$

where, $\hat{V}_{n}$ denotes the rendered video from the $n$-th viewpoint, and $\mathcal{L}_{2}$ is the L2 norm loss.
Since both the renderer $\mathcal{R}$ and the MPM simulator $\Phi$ are differentiable, the evaluation loss can be backpropagated to optimize the continuous material parameters. During this process, we collect the loss curve and the material parameter update trajectory as feedback to construct the LLMs’ prompts. Meanwhile, the minimum loss achieved during optimization is used as the fitness score of the constitutive candidate to guide the selection process at the upper level.

Figure: Figure 3: Quantitative Comparison of Visual Fidelity on Synthetic Datasets. Average PSNR, SSIM, and LPIPS metrics computed across multiple viewpoints between simulated visual dynamics and ground-truth observations. Higher PSNR and SSIM, and lower LPIPS, correspond to better visual fidelity.
Refer to caption: x3.png

Figure: Figure 4: Comparison on Real-World Datasets. Quantitative metrics (i.e., PSNR) between the predicted and observed frames are reported in the bottom row.
Refer to caption: x4.png

Figure: Figure 5: Generalization Analysis. We infer the intrinsic dynamics using only the first 200 frames of observation and simulate the subsequent 200 frames. Quantitative metrics (i.e., Chamfer distance) are reported in the bottom row.
Refer to caption: x5.png

Figure: Figure 6: Ablation study on decouple evolution strategy. The figure shows the loss of the best solution averaged across seeds at different iterations. The shaded area indicates the range between the minimum and maximum values.
Refer to caption: x6.png

## Experiments

### Experimental Setup

#### Implementation Details

Given a series of multi-view videos of a scene, we first perform 3D reconstruction using the multi-view images from the initial time step.
To infer intrinsic dynamics from visual observations, we utilize only single-view videos as ground-truth observations across all datasets.
For the upper-level evolution, we employ GPT-4.1-mini to generate constitutive hypotheses.
Details of the prompt design are provided in the appendix.
For all scenarios, the initial constitutive individual is only defined as a purely elastic model that combines fixed corotated elasticity with identity plasticity.
For lower-level optimization, we conduct MPM simulations under standard gravitational acceleration ($9.8m/s^{2}$) within a unit cube domain $[0,1]^{3}$.
We employ the Adam optimizer with a learning rate of $1\times 10^{-3}$ to tune the material parameters in the constitutive law.
For each scene, we perform five independent runs using different random seeds.
All experiments are conducted on an NVIDIA A40 (48GB) GPU. For more detailed information on the experimental settings, please refer to the appendix.

#### Baselines

We evaluated our proposed method on both synthetic and real-world datasets by comparing it with three state-of-the-art intrinsic dynamics inference methods. PAC-NeRF (Li et al. 2023), Spring-Gaus (Zhong et al. 2024), and NeuMA (Cao et al. 2024). PAC-NeRF is capable of inverting material parameters of MPM simulator from video input. Spring-Gaus models elastic objects using a spring-mass system with Gaussian kernels. It achieves intrinsic dynamics by learning material parameters such as spring stiffness.
We compare it with our method in real-world data.
NeuMA is the most relevant work to ours, as it learns neural constitutive models directly from visual observations.

#### Datasets and Metrics

To thoroughly evaluate the effectiveness of our method, we conduct experiments on both synthetic and real-world datasets.
For synthetic data, we adopt six dynamic scenes from NeuMA, each with varying initial conditions (including object shapes, velocities, and positions), material properties, and simulation time intervals.
In addition, this dataset provides ground-truth particle trajectories for each time step.
For real-world evaluation, we conduct experiments on two scenes-Bun and Burger-provided by Spring-Gaus. Following prior works (Guan et al. 2022; Cao et al. 2024), we adopt the L2-Chamfer distance (Erler et al. 2020) between the simulated and ground-truth particles to quantify the accuracy of the inferred intrinsic dynamics. To assess the visual fidelity of 3D asset interaction simulations, we follow 3DGS (Kerbl et al. 2023) and employ PSNR, SSIM, and LPIPS as quantitative metrics.

### Performance on Intrinsic Dynamics Inference

#### Synthetic Dataset.

To evaluate the effectiveness of VisionLaw, we conducted comparative experiments against PAC-NeRF (Li et al. 2023) and NeuMA (Cao et al. 2024) on synthetic dataset.
As shown in Tab. [1](https://arxiv.org/html/2508.13792v1#Sx4.T1), we report the Chamfer distance between the predicted and ground-truth particle positions.
PAC-NeRF, while capable of estimating material parameters from visual observations, relies heavily on manually designed constitutive laws.
This limits its ability to capture the true underlying dynamics, resulting in poor performance, particularly in complex scenarios such as BouncyBall and JellyDuck where the Chamfer distance reaches 516.30 and 137.73, respectively.
NeuMA improves flexibility by learning neural constitutive laws from visual inputs. However, its black-box nature limits interpretability and often leads to suboptimal fitting.
In contrast, our VisionLaw approach achieves the best average performance across all six benchmarks, with an average Chamfer distance of 1.65, significantly outperforming NeuMA (2.86) and PAC-NeRF (114.80).
These results demonstrate the superior ability of VisionLaw to recover intrinsic dynamics directly from visual observations, while retaining interpretability. To further evaluate visual fidelity, we report the average PSNR, SSIM, and LPIPS metrics from multiple viewpoints in Fig. [3](https://arxiv.org/html/2508.13792v1#Sx4.F3). VisionLaw significantly surpasses NeuMA, demonstrating its strong capacity to generate visual dynamics that closely match the ground-truth visual observations. This suggests that VisionLaw holds promise for generalizing to novel scenes and enabling interactive 4D simulations.

#### Real-world Dataset.

We conducted experiments on a real-world dataset and compared our method with Spring-Gaus (Zhong et al. 2024) and NeuMA (Cao et al. 2024). The visual results and PSNR metrics are presented in Fig. [4](https://arxiv.org/html/2508.13792v1#Sx4.F4). Spring-Gaus models elastic deformation using a spring–mass system. While effective for simulating simple linear behaviors, it struggles to capture the complex nonlinear elasticity often present in real deformable objects. As a result, it predictions deviate significantly from ground-truth observations.
NeuMA uses neural networks to approximate nonlinear dynamics and can represent a wide range of material behaviors.
However, it is sensitive to observation noise and limited data, and the lack of explicit physical constraints result in physically implausible predictions.
These shortcomings hinder NeuMA’s ability to reproduce the subtle deformations observed in real-world objects.
In contrast, VisionLaw integrates a broad range of physical priors through LLMs, providing a stronger inductive bias toward physically plausible dynamics. This enhances both generalization and learning stability.
As shown in Figure 4, VisionLaw generates results that are more consistent with real observations, both visually and quantitatively. These results demonstrate that VisionLaw can accurately capture the intrinsic dynamics of deformable objects and highlight its practical effectiveness in real-world scenarios.

### Analysis and Ablations

#### Generalization to Unseen Observations.

We conducted a generalization analysis on four examples, comparing our method with NeuMA (Cao et al. 2024).
In each scene, the first 200 frames of visual observations were used to infer the intrinsic dynamics, which were then used to predict the future 200 frames.
As shown in Fig. [5](https://arxiv.org/html/2508.13792v1#Sx4.F5),
NeuMA struggles to generalize beyond the observed frames. Its predictions diverge significantly from the ground truth, likely due to overfitting and the lack of strong inductive bias in neural networks.
In contrast, VisionLaw achieves consistently high predictive accuracy across both visual appearance and Chamfer distance metrics, even with limited observation data.
We attribute this superior performance to the inherent regularization effect introduced by the interpretable symbolic structure, which not only improves physical plausibility but also constrains the solution space in a meaningful way.
These results highlight that VisionLaw combines strong generalization with interpretability,
making it practical for forward simulation in previously unseen temporal regimes.

#### Ablation study on Decoupled Evolution Strategy

To evaluate the effectiveness of our proposed decoupled evolution strategy, we perform an ablation study comparing two settings over five iterations: 1) With Decouple: the elastic and plastic components are optimized alternately for four iterations, followed by a final joint refinement step; 2) Without Decouple: all five iterations are performed with joint optimization. As illustrated in Fig. [6](https://arxiv.org/html/2508.13792v1#Sx4.F6), the decoupled strategy consistently achieves lower RGB losses across diverse scenes, indicating it leads to better constitutive law discovery. By decomposing the search into simpler sub-tasks, this approach reduces the search space, making optimization more focused and efficient. Moreover, the shaded regions are noticeably larger under the decoupled setting, reflecting greater diversity in the solutions. This helps avoid early convergence to poor local minima. Overall, the decoupled evolution strategy better unleashes the potential of LLMs by not only sharpening exploitation but also broadening exploration.

## Conclusion

In this paper, we introduce VisionLaw, a bilevel optimization framework that infers interpretable intrinsic dynamics directly from visual observations by jointly optimizing the constitutive law and its material parameters.
At the upper level, LLMs are prompted to propose and refine constitutive hypotheses. Meanwhile, a decoupled evolution strategy is introduced to reduce the complexity of jointly searching over elastic and plastic components and to improve the quality of the solutions.
At the lower level, material parameters are optimized based on visual supervision, while evaluation and feedback on the consistency of intrinsic dynamics are provided to guide the upper-level evolution.
This closed-loop design effectively bridges the gap between visual data and physical nature, achieving a balance between interpretability, physical plausibility, and generalization.
Experimental results show that our method accurately captures intrinsic dynamics from visual observations and generalizes effectively to novel scenarios for 4D interaction.

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## Peer Reviews

confidence: 3
contribution: 4
initial_rating: 8
presentation: 2
questions: - **Q.1:** More of a comment: Your contributions are weirdly written. I think contibution 1 is actually made up of of contibutions 2 and 3. So that should be one bullet point. And running experiments that validate your method is not a contribution, so contribution 4 should be removed.
- **Q.2:** Another suggestion: since MPM simulation is so important to your method, maybe spend a short paragraph explaining it to everyone in the main body of the paper.
soundness: 4
strengths: - **S.1:** Great idea. I think this is a really beautiful and elegant idea. Having the physics defined in the outer loop and the parameters tuned in the inner loop is great.
- **S.2:** Clear Results. As far as I can tell, the results look pretty impressive and the paper compares its method to a variety of contemporary baselines, which is great.
- **S.3:** Reproducibility. I appreciate that the authors released their source code, additional videos, and all LLM prompts.
summary: The paper proposes a bilevel optimization framework for estimating the physics of a scene. This is done with an LLm at the high level iteratively refining the physical formulas governing the particles in the scene (through an evolutionary algorith) and a low-level differentiable particle-based simulator refining the individual particle properties.
weaknesses: - **W.1:** The writing is incredibly dense. I've worked in parameter identification for simulator tuning and I've written my own physics engines and I was barely able to follow all of this. A bit simpler writing would greatly benefit this paper. For example, it took me a while to understand why you're not just using the gradient that you get from the differentiable simulator rollout to also tune the high-level physics.

---

confidence: 4
contribution: 2
initial_rating: 4
presentation: 3
questions: I believe the author should clarify the difference between [1,2] and their work. If this concern is addressed, I will consider changing my score.
soundness: 3
strengths: - The presentation is straightforward and easy to understand. 

- The discussed topic—inferring interpretable intrinsic physics—is fascinating.

- The experiments conducted are thorough and comprehensive.
summary: This work, titled VisionLaw, proposes a bilevel framework for the visual grounding of physical laws. At the upper level, the Vision-Language Model (VLM) generates a hypothesis about the physical materials. At the lower level, a differentiable simulation-and-rendering pipeline verifies this hypothesis. The main contribution claimed is the bilevel framework for symbolic estimation. The authors evaluate VisionLaw on real and synthetic datasets.
weaknesses: Please take the time to read the following two papers:  
[1] "LLM and Simulation as Bilevel Optimizers: A New Paradigm to Advance Physical Scientific Discovery."  
[2] "Neuma: Neural Material Adaptor for Visual Grounding of Intrinsic Dynamics."  

In comparison to these papers, one concern I have is that the contribution of the current work seems weak. The bilevel framework closely resembles that of Paper [1] (which also uses a bilevel framework to search optimal material parameters), and the vision-guided constitutive evaluation mechanism appears to be based on concepts from Paper [2].

---

confidence: 5
contribution: 2
initial_rating: 2
presentation: 2
questions: - The plastic corrections are derived from plastic flows applied on a specific elastic model. That is, even with the same plastic flow, different elastic model can result in different discrete deformation corrections (called plastic return mappings). For example, the von-Mises plasticity and Drucker-Prager plasticity used in the paper assume stVK elasticity model. The combination of fixed-corotated elasticity and von-Mises plasticity in H.1 BOUNCYBALL is fundamentally wrong.

- How many lower optimizations are expected to run? It seems that each upper-level optimization are followed with multiple independent lower optimizations. Considering the above combination constraints, there are not many available constraints, for example, fewer than 10. I think it is doable to run brute-force lower-level optimizations to find the most fit, i.e., PAC-NeRF + brute-force sweep. This could be a strong baseline to compare running time.
soundness: 2
strengths: - LLM-based code generation is a promising way to handle discrete constitutive models, as it can reduce the need for mechanics experts to manually set up these models.
summary: The paper propose a framework for evolving symbolic constitutive models and optimizing material parameters from multi-view videos. The task setting follows PAC-NeRF and use PhysGaussian-based simulator for better reconstruction quality. The main contribution compared to PAC-NeRF is its LLM-based constitutive law evolution framework: the non-differentiable discrete constitutive class optimization is done by LLM code generation based on feedbacks from the lower-level continuous physical parameter optimization. The lower-level optimization uses differentiable simulation and differentiable rendering, similar to PAC-NeRF.
weaknesses: - Since the upper-level optimization is the main contribution, a more thorough examination is needed:
    - How many upper-level evolutions are need to converge to the correct one?
    - How are different elasticity models distinguished by LLM? Since many models behave similarly at small deformation magnitudes. I don't think LLM can differentiate StVK, neo-Hookean, fixed-corotated when the object is pure elastic.
    - What if the LLM outputs python code with bugs that can crush the program?
    - Why does direct joint evolution not perform well? It appears that the difference lies primarily in prompting and the available information and feedbacks are the same. Ablation studies are needed to motivate decoupled evolution.


- Although the high-level ideas are clear, some details are lacking for reproducibility:
    - What (modalities) are included in the feedback from the lower-level optimization?
    - How is the first batch of constitutive model candidates are initialized? Are they generated by LLM based solely on multiview videos?
    - How are the initial physical parameters set when the predicted constitutive models change?

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: Do the learned laws transfer across different objects with similar physical properties in real scenes without re-search?
soundness: 3
strengths: 1. The idea of combining LLM-based hypothesis search with differentiable, vision-guided evaluation to discover interpretable constitutive laws directly from videos is very compelling.
2. Implementation details are transparent and the source code is provided, largely promoting reproducibility.
3. The paper is clearly written and easy to understand.
4. The comprehensive experimental results are solid, proving the effectiveness very well.
summary: This paper proposes VisionLaw, a bilevel framework that uses LLM-driven search to generate/edit symbolic elastic and plastic constitutive laws (upper level) and a differentiable MPM+renderer loop to fit continuous material parameters from video supervision (lower level). The experiments report the best average Chamfer distance on synthetic data, and improved visual fidelity/generalization vs baselines.
weaknesses: 1. While the decoupled strategy is motivated, the paper could quantify search efficiency (e.g., wall-clock, simulator calls, accepted offspring) and compare against naive joint search under equal budgets.
2. The pipeline depends on accurate camera parameters and a 3DGS reconstruction from the first frame. Its sensitivity to calibration errors, occlusions, and reconstruction artifacts is not analyzed.


Decision: Accept (Poster)