VisionLaw: Inferring Interpretable Intrinsic Dynamics from Visual Observations via Bilevel Optimization
Authors: Jiajing Lin
Shu Jiang
Qingyuan Zeng
Zhenzhong Wang
Min Jiang
Venue: ICLR 2026 Poster (2026)
Keywords: 3D Gaussian Splatting
Physics-Driven 4D Interaction
Scientific Discovery
Intrinsic Dynamics
Primary area: applications to computer vision, audio, language, and other modalities

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 align with actual intrinsic dynamics; 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 to act as physics experts 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.

Published as a conference paper at ICLR 2026
VISIONLAW: INFERRING INTERPRETABLE INTRIN- SIC DYNAMICS FROM VISUAL OBSERVATIONS VIA BILEVEL OPTIMIZATION
Jiajing Lin', Shu Jiang", Qingyuan Zeng2.3, Zhenzhong Wang', Min Jiang" School of Informatics, Xiamen University.
Institute of Artificial Intelligence, Xiamen University.
'The Hong Kong University of Science and Technology (Guangzhou) {jiajinglin, shujiang, 069202211531 0 xmu, edu. cn (zhenzhonwwan,, minjiang edu..uuu.commmmmmmmmmmmmmc..
ABSTRACT
The intrinsic dynamics of an object governs its physical behavior in the real world, playing a critical role 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 manually defined constitutive priors, making it difficult align with actual intrinsic dynamics; 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 frame- work that infers interpretable expressions of intrinsic dynamics from visual obser- vations. At the upper level, we introduce an LLMs-driven decoupled constitutive evolution strategy, where LLMs are prompted act as physics experts to generate and revise constitutive laws, with built-in decoupling mechanism that substan- tially reduces search complexity LLMs.. At the lower level, we introduce vision-gu lision-guided constitutive evaluation mechanism, which utilizes visual simula- tion to evaluate the consistency between the generated constitutive law and the underlying intrinsic dynamics, thereby guiding the upper-level evolution. Experi- ments on both synthetic and real-world datasets demonstrate that VisionLaw can effectively infer interpretable intrinsic dynamics from visual observations. It sig- nificantly outperforms existing state-of-the-ard methods and exhibits strong gen- eralization for interactive simulation in novel scenarios. Our implementation is available at gighttt.bb..m. com/ Jia iotg://Jaaaiiiiiiii/iiioooooooo
INTRODUCTION
With the advancement of 4D generation Zhao et al. (2023): Bahmani et al. (2024); Jiang et al.
(2024)); Ren al. (2023), realistic interaction with 3D assets has become increasingly feasible, fa- cilitating broad applications in areas like virtual reality, embodied intelligence, and animation Jiang al. 2024b) Shi et al. (2023; L (2024a); [Sun et al. (2024) (2024)2025) Among these advances Xie et al. 22024) Lin et al. (2024b). incorporating physical simulation Stomakhin et al.) (2013); Müller al. (007)) stands out particularly prominent method, as it enables the generation of interac- tive 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 roughly infer the intrinsic dynamics of objects merely by observing their motion, and are even capable predicting how these objects would interact in new scenarios. A fundamental question arises: can we enable machines infer the intrinsic dynamics directly from visual obser- vations, as humans do? Recent methods Xie et al. 2004 Li et al. (2023) have attempted to bridge
"Corresooniigg author: Min Jiang, minjiang xmm.cdu.cn

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Published as a conference paper at ICLR 2026
Method "Struggle olign with actual dynamics"
Method: "Strong Interpretabllity and generalization"
Manually Defined Constitutive Laws
You are thinker. Please compleese the geecration and correction the conuttivtivn lom.
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Method 2: "Poor Interpretability and generalization"
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Neural Elastic Law
Neural Plastic Law
Visual
MPM
Supervise
Observation
Supervise
Simulator
Figure 1: Existing works either rely manually defined constitutive laws, which struggle to align with actual intrinsic dynamics, or learn neural constitutive laws, which suffer from poor inter- pretability and generalization. In contrast, our approach can automatically infer interpretable in- trinsic dynamics solely from visual observations.
the gap between visual dynamics and physical simulation by incorporating physical simulators (e.g.. Material Point Method, MPM Stomakhin al. (2013) into 3D representations such NeRF and 3D Gaussian Splatting (3DGS) Mildenhall et al. (2021): ; Kerbl et al. (003)). This integration has led promising paradigm for inferring the intrinsic dynamics from visual observations. Depend-
ing 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. mate material parameters by the supervision of multi-view videos. Phy Premmrr,er,
2023 ; Cai et al. (2024) esti- TreamPhhysics,
and Physics3D Zhang et al. (2024); Huang et al.] (2025); Liu et al. (2024) distill visual dynamics priors from video diftusion models to guide the estimation process. However, these approaches typ- ically rely manually defined constitutive laws, which often fail to align with the complex physical behaviors observed in practice, thereby compromising the accuracy of 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 constitu- tive set. However, such predefined set often fails to capture the full diversity of real-world physical behaviors. NeuMA Cao et al.) (2024) learns neural constitutive laws from visual observations. De- spite its effectiveness, it has notable limitations: 1) The learned laws black-box representations, which lack interpretability and are difficult for humans and LLMs to understand; 2) Due to the lack of physical inductive biases, neural networks tend to mechanically reconstruct visual observations instead of modeling the underlying dynamics, resulting in overfitting and poor generalization.
To overcome the aforementioned challenges, we introduce VisionLaw, an interpretable intrinsic dy- namics inference framework based on bilevel optimization, which can jointly infer symbolic consti- tutive laws 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 physical understanding and mathematical reasoning to gener- ate and refine symbolic 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 rision-guided constitutive evaluation mechanism. Supervised by vi- sual observations, it optimizes the continuous material parameters 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-trui intrinsic dynamics, which turn guides the evolution at the upper level. Our contributions summarized as follows:
We propose bilevel optimization framework that unifies constitutive evolution and vision- guided constitutive evaluation, achieving the inference of symbolic constitutive laws and material properties from visual observations.
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Published as a conference paper at ICLR 2026
We introduce physical inductive biases through LLMs to guide the evolution of constitutive laws. In addition, a decoupled evolution strategy is proposed to markedly improve both search efficiency and solution quality.
Extensive experiments on both synthetic and real-world datasets demonstrate that our method effec- tively captures the interpretable intrinsic dynamics underlying visual observations and generalizes them to novel scenarios for 4D interaction.
2
PRELIMINARIES
2.1 CONSTITUTIVE LAWS AND MATERIAL POINT METHOD
In continuum mechanics Chaves 2013), constitutive laws define how materials respond under ap- plied forces. The essential reason why materials like rubber, sand, and water exhibit entirely dif- ferent physical behaviors lies in the differences in the constitutive laws they follow. To simulate the motion and deformation of materials, we need to solve system of partial differential equations derived from the conservation of mass and momentum:
Dp - P DE DE
Dv
V Pg.
where p denotes the density, v the velocity field, g the gravitational acceleration, and P the stress tensor, which is defined by the constitutive law. The system becomes closed and solvable only after a specific constitutive relation for P is prescribed.
In this paper, we employ an MPM simulator Sttoakhiin al. 2013) solve the above governing equations because of its versatility in handling various materials. Intuitively, the MPM discretizes the continuum into set of material points that carry physical quantities such as mass, velocity, and deformation gradient, and updates system states through particle-gri transfers and time integration. At each time step, the particles first project their quantities onto background grid; the momentum conservation equation is then solved on the grid to compute nodal velocities and accelerations; finally, these updated grid quantities are interpolated back to the particles, advancing their positions and deformation gradients F, which describes the local deformation. For more details about MPM, please refer to Appendix D
Within the MPM framework, two types of constitutive laws must be specified: (1) an elastic consti- tutive law that describes reversible clastic responses, and (2) a plastic constitutive law that captures irreversible plastic evolution. Their formulations are given as:
Pr (F;0F) + pp (F;0p) + peotrected
where pg and pp denote the elastic and plastic constitutive laws, respectively. F is the deformation gradient, т is the Kirchhoff stress tensor, Feomrected is the corrected deformation gradient after plastic return mapping. The continuous material parameters in the elastic and plastic laws are denoted by 8g and 6p, respectively. Several classical constitutive laws are listed in Appendix E 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.
2.2 PHYSICS-INTEGR.AA INTEGRATED 3D GAUSSIANS
3D Gaussians Splatting (3DGS) Kerbl et al. (0023) represents the scene using set of anisotropic Gaussian kernels = [xi, A,, a; C. Chex where x;, A;, (i and C: represent the center position, covariance matrix, opacity, and spherical harmonic coefficients of the Gaussian kernel G,, respec- tively. To render 3D Gaussians into a 2D image from a given view, the color cach pixel can be formulated as:
C= C=EoSH C,) (.)
o,).
(3)
j=1
where N denotes a set of sorted Gaussian kernels related to the pixel and view. σ; is the effective opacity, defined as the product of the projected 2D Gaussian weight and opacity dg SH computes
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Candidate Constitutive Individuals
Decoupled Evolution Strategy
2steisigers
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Given constitutive individual-either predefined at initialization or generated by LLMs-it is embedded into a differentiable MPM simulator for forward simulation. The resulting dynamics are rendered and compared with observations to compute 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, en- coded into prompts for the LLMss. Guided by the decoupled evolution strategy, the LLMS analyze and refine these constitutive law expressions to generate offspring for the next optimization cycle.
Figure 2:
RGB values based on the view direction d; and spherical harmonic coefficients C.. Unlike NeRF's implicit form, 3DGS offers explicit representation that exhibits Lagrangian nature, facilitating seamless integration with simulation algorithms. Thus, PhyGaussians Xie et al. (2024) pioneers the integration of MPM simulator Stomakhin 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 0 (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 Gaussian kernels the next time step:
t1 F++1
(g')),
A 1+ F1+1 A A
(5)
Here, + is differentiable MPM simulator, Ft-1 denotes the deformation gradient at time step t + 1 (the subscript i is omitted for simplicity). Gaussian covariance A 4+1 can be updated by applying F++1, which approximates the deformation of the Gaussian kernel. After completing the MPM simulation, 4DGS representation is constructed, which enables rendering of visual dynamics using Eq. 3
METHODOLOGY
In this work, we aim to infer interpretable intrinsic dynamics from series of visual observations. Formally, given multi-view video observations V = [V1, ..., VN} 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, propose VisionLaw, novel bilevel optimization framework:
min 8.8
L (R (vp, O 0"; (0))), V),
s.t. h 0 <0 0 E arg min L (R (,
(7)
(8)
6EB
where R is a differentiable renderer defined by Eq. 3 The constitutive law & consists of an elastic law bg and a plastic law ppp. O defines the continuous parameter space for inner-level optimization
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0 +. h(-) I 0 refers the validity of the simulation (e.g. whether constitutive law yp is simulatable). The material parameter includes the elastic parameters 0g and the plastic parameters 8p. For the upper level, based on evaluation and feedback from the lower level, LLMs are employed to generate and refine constitutive law expressions (sp, θ). At the lower level, given the output (p,0) from the upper level, the optimal continuous material parameters 0" estimated under visual observation supervision, using differentiable rendering and simulation. During this process. evaluation and feedback are provided. The pipeline of the proposed VisionLaw is illustrated in Fig.
3.1 UPPER-LEVER CONSTITUTIVE LAWs EVOLUTION
3.1.1 L.MS-DRIVEN CONSTITUTIVE EVOLUTION
Recently, LLMs have shown tremendous potential in scientific discovery Yang et al. (2023); Romera-Paredes et al. 2024); Ma et al. (2024): : Wang et al. (2026a]b), owing to their strong sym- bolic reasoning abilities and extensive physical priors. Inspired by this, in the upper-level search, we prompt LLMs to evolve discrete constitutive law expressions. Specifically, we consider LLMs intelligent operator and construct evolutionary search paradigm to iteratively optimize the constitutive law expressions. Each law is represented as Python code snippet with a clear physical meaning and strong interpretability.
The optimization procedure consists of five stages, which are as follows: i) Initialization: Clas- sical 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 Eval- uation: Each candidate constitutive individual is passed the lower level for simulation testing, where its fitness is evaluated from visual observations and the corresponding feedback information collected. iii) Selection: To improve population diversity and avoid local optima, we first remove duplicate constitutive individuals with fitness differences below a threshold 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 modify the expression to increase fitness; 3) generate set of physically plausible constitutive law expressions as candidate individuals. This process is formalized as:
{ 9Th IMEIM = LLM (IF* (t O* JREEK" PP.
where K denotes parent size, M denotes offspring size, O represents the feedback obtained from the lower level and P denotes the prompt provided to LLMs. v) Iteration: Repeat steps ii) to iv) until a predefined number of iterations reached. Eventually, the algorithm discovers constitutive laws that not only simulate dynamic behaviors consistent with visual observations but also exhibit strong physical interpretability.
3.1.2 DECOUPLED EVOLUTION STRATEGY
In the MPM simulation framework, a complete constitutive law &p consists of an elastic part e and a plastic part ypp, which together govern the system's simulation behavior. However, simultancous optimization of these components significantly enlarges the search space, increases the difficulty of LLMs search, and hinders convergence high-quality solutions. 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 iteration, we prompt the LLM to optimize only one component of the constitutive law expression (elastic or plastic), while the other remains fixed and is updated in the subsequent iteration. The two components of con- stitutive laws are optimized alternately across multiple iterations. 2) Joint Evolution: After the alternating optimization phase, we prompt the LLM to jointly optimize both elastic and plastic com- ponents to further enhance overall performance. This phase serves as a fine-grained refinement of the existing high-quality expressions from global perspective. Through the proposed decoupled evolution strategy, we effectively reduce the search space, enhance the stability and efficiency of L.LM-based search, and substantially improve the quality of the final constitutive laws.
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Method
BouncyBall Ball
ClayCat
HoneyBottle
JellyDuck
RubberPawn
SandFish
Average
PAC-NeRF et aL 2023
516.30
15.38
2.21
137.73
15.47
1.71
114,80
NCLaw Ma cl 2023
56.69
2.35
0.92
11.97
3,91
1.30
12.86
NeuMA 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
Table 1: Quantitative Comparison of Intrinsic Dynamics Consistency Synthetic Datasets. The Chamfer distance was employed to quantify the similarity between simulated and ground-truth particle trajectories. Lower values indicate better alignment with ground-truth intrinsic dynamics.
3.2 LOWER-LEVEL CONSTITUTIVE LAWS EVALUATION
To effectively evaluate whether a candidate constitutive expression can accurately capture the intrin- sie 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 (00), with continuous material parameters, is seamlessly embedded into differentiable MPM simulator. We integrate the MPM simulator with 3DGS to drive the simula- tion and render the predicted visual dynamics V from given views. The supervised loss between the predicted and observed visual dynamics can be formulated
1 Σ (AC V., +1 SSIIMMS V,))], N
L
(10)
where Vn, denotes the rendered video from the n-th viewpoint, and L2 is the L2 norm loss. Since both the renderer R and the MPM simulator & are differentiable, the evaluation loss can be back- propagated to optimize the continuous material parameters & as described in Eq. 8 During this process, we collect the loss curve and the material parameter update trajectory as feedback to construct the L.MMs' prompts. Meanwhile, the minimum loss achieved during optimization is used as the fitness score of the constitutive candidate guide the selection process at the upper level.
EXPERIMENTS
4.1
EXPERIMENTAL. SETUP
4.1.1
IMPLEMENTATION DETAILS
Given multi-view videos of a scene, we follow NeuMA Cao et al. (2024) to perform 3D recon- struction and Particle-GS GS binding using multi-view images from the iitiial time step. We use only single-view videos as ground-truth observations to infer intrinsic dynamics across all experiments. For all scenarios, the initial constitutive individual is only defined as purely elastic model that com- bines linear isotropic elasticity with identity plasticity. For the upper-level evolution, we employ GPT-4. 1 1-minn to generate constitutive hypotheses. Details of the prompt design are provided in Appendix G The decoupled evolution strategy is executed through four iterations of alternating optimization, followed by three iterations of joint optimization. For lower-level optimization, we conduct MPM simulation Xie et al. 2024 under gravitational acceleration (9.8m/s² /²²). We employ the Adam optimizer with a learning rate of x 10-3 to tune the material parameters. For each scene, perform five independent runs using different random seeds. All experiments are conducted on an NVIDIA A40 (48 GB) GPU. Detailed experimental settings are provided in Appendix A.I
4.1.2 BASELINES
We compare our method with state-of-the-art intrinsic dynamics inference methods: PAC-NeRF Li et al. (2023), NCLaw Ma et al. (2023). NeuMA Cao et al. (2024), and Spring-Gaus Zhong et al. (2024). PAC-NeRF is capable of inverting material parameters from video input. NCLaw only fits neural constitutive laws to known dynamics, whereas NeuMA extends this by introducing visual information for adaptation. NeuMA is the most relevant work to ours, as it learns neural constitutive laws directly from visual observations. mass system with Gaussian kernels. It achieves intrinsic dynamics inference through the estimation of spring stiffness. All baseline experimental settings follow the original setup.
Sering-Gaus models elastic objects by integrating spring-

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Published as a conference paper at ICLR 2026
VisionLaw
NeuMA
(a)
PSNR
(b)
40
PSWR
SouncySall
Claycet
HonoyBottle
32
30
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Ball
Cat
Bottle
Duck
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Fish
Figure 3: Quantitative Comparison of Visual Fidelity on Synthetic Datasets. (a) Average PSNR over all non-training views. Higher PSNR values reflect improved visual fidelity; (b) PSNR com- parison at different views, with View 0 denoting the training view.
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Figure 4: Comparison on Real-World Datasets. (a) Quantitative metrics (i.e., PSNR) between the predicted and observed frames reported in the bottom row; (b) The intrinsic dynamics inferred from the Bun scene, represented as Python code, exhibit strong interpretability.
4.1.3 DATASETS AND METRICS
To thoroughly evaluate the effectiveness of our method, conduct experiments both synthetic and real-world datasets. For synthetic data, we adopt six dynamic scenes from NeuMA Cao et al. (2024), each with varying initial conditions (including object shapes, velocities, and positions), intrinsic dynamics, and simulation time intervals. Each synthetic scene includes 10 videos captured from different views, each containing 400 frames, and the dataset further provides ground-truth particle trajectories. For real-world evaluation, we conduct experiments "urger') provided by Spring-Gaus Zhong et al. (2024). Each real- weal-world scene includes 3 videos captured from different views, each containing 19 Trames. More details of the datasets are provided Appendix A.2 Following prior works Guan et al. (2022); Cao et al. (2024), we use L2- Chamfer distance Erler et al. (2020) between the simulated and ground-truth particle trajectories to quantify the accuracy of intrinsic dynamics inference. To assess the visual fidelity, we follow 3DGS Kerbl al. 2023) and employ PSNR, SSIM, and L.IPS quantitative metries.
two scenes ("Bun' and
4.2
PERFORMANCE ON INTRINSIC DYNAMICS INFERENCE
4.2.1
SYNTHETIC DATASET.
Comparison of Intrinsic Dynamics Consistency. In synthetic datasets, ground-truth particle tra- jectories are generated from ground-truth intrinsic dynamics. We evaluate aligmment between in- ferred and ground- truth intrinsic dynamics by measuring the Chamfer distance between simulated and ground-truth trajectories, as summarized in Tab. T PAC-NeRF relies heavily on manually de- signed constitutive laws and is highly sensitive material parameter initialization. This restricts

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Published as a conference paper at ICLR 2026
Observation NeuMA
Ours
Observation NeuMA
Ours
Observation NeuMA
Ours
Observation
NeuMA
Ours
Time
Bouncy8all 13.6
1:17
Clayoot
7.93
0.95
HoneyBottle
1.24
0.96
RubberPown
1.39
0.93
Figure 5: Generalization to Unseen Observations. We infer the intrinsic dynamics using only the first 200 frames of visual observation and simulate the subsequent 200 frames. Quantitative metrics (i.e., Chamfer distance) are reported in the bottom row.
its ability to capture actual dynamics, leading to poor performance, especially in complex scenar- ios such BouncyBall and JellyDuck. Similarly, NCLaw learns predefined constitutive laws and suffers from the same limitations as PAC-NeRF, NeuMA improves flexibility by learning neural constitutive laws from visual inputs. However, its black-box nature limits interpretability and of- ten leads to overfitting. In contrast, our VisionLaw approach achieves the best overall performance across all six benchmarks, with an average Chamfer distance of 1.65, significantly outperforming the baselines. These results demonstrate the superior ability of VisionLaw to recover intrinsic dynamics directly from visual observations, while maintaining interpretability.
Comparison of Visual Fidelity. To further evaluate visual fidelity. we compute the PSNR between rendered dynamics and ground-truth observations. As shown in Fig. B (a), we report the aver- aged PSNR over all non-training views. The results show that VisionLaw significantly outperforms NeuMA, achieving superior visual fidelity. In Fig. (b). we further compare PSNR across different views, including the training view (View 0). NeuMA exhibits pronounced variability, with higher PSNR at the training view and its neighbors (View 1 and View 9), but considerably worse perfor- mance unseen views. This shows that NeuMA tends to overfit the training views, which limits its ability to generalize. In contrast, VisionLaw performs consistently across different views and still produces robust results unseen views, even when trained on only one. This stability arises from introducing physical inductive biases through LLMs into the evolution of constitutive laws, which effectively mitigates the overfitting commonly observed in purely neural methods. Overall, these findings confirm that our approach not only captures more faithful intrinsic dynamics but also delivers dynamic reconstructions of higher visual fidelity.
4.2.2 REAL-WORLD DATASET.
We evaluated our method on a real-world dataset against Spring-Gaus Zhong et al. (2024) and NeuMA Cao et al. (2024), with visual results and PSNR metrics shown in Fig. (). Spring-Gau Gaus models elastic deformation using a spring-man mass system, which works well for simple linear behav- iors, but fails capture the complex nonlinear elasticity of real deformable objects. Consequently, its predictions deviate markedly from the ground-truth observations. NeuMA employs neural net- works approximate nonlinear dynamics and capture diverse material behaviors. However, it is sensitive to observation noise and lacks explicit physical constraints, which limits its ability to re- produce the subtle deformations of real-world objects. In contrast, VisionLaw integrates a broad range of physical priors through LLMs, providing a strong inductive bias toward physically plau- sible dynamics. This improves both generalization and learning stability. As shown in Fig. [ (a), VisionLaw generates results that are more consistent with real observations, both visually and quan- titatively. These results demonstrate that VisionLaw can accurately capture the intrinsic dynamics of deformable objects and highlight its practical effectiveness in real-world scenarios. Meanwhile, Fig. (b) illustrates the inferred intrinsic dynamics in the Bun scenario, expressed in the form of Python code. This form offers strong interpretability. allowing humans to intuitively grasp the phys- ical meaning underlying the formulas, thereby facilitating scientific discovery. Moreover, symbolic expressions provide implicit regularization effect, which helps prevent overfitting.
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Image-to-4D
3D-to-4D
Clay
Bounch
Rubber
Jell
Honey
Sand
Time
Figure 6: Generalization to Novel Scenarios for 4D Interaction. The left text indicates the intrin- sic dynamics applied, which are learned from visual observations through VisionLaw.
With Decoupling
Without Decoupling
RGB
30
BouncyBall Ball
ClayCat
JellyDuck
RubberPawn
10
Iteration
Iteraion
Iteration
Iteration
Figure 7: Ablation Study on Decoupled Evolution Strategy. The figure shows the loss of the best solution averaged across seeds different iterations. The shaded area indicates the range between the minimum and maximum values.
4.3 GENERALIZATION ANALYSIS AND ABLATION STUDIES
4.3.1 GENERALIZATION TO UNSEEN OBSERVATIONS.
We conducted a generalization analysis on four examples, comparing our method with NeuMA Cao al. (2024). For each scene, the first 200 frames of visual observations were used to infer the intrinsic dynamics, which were then used to predict the next 200 frames. As shown in Fig. 3 NeuMA struggles generalize beyond the observed frames. Its predictions diverge significantly from the ground truth, likely due to overfitting. In contrast, Visionl achieves consistently high predictive accuracy across both visual appearance and Chamfer distance metrics, with limited observation data. We attribute this advantage to the physical inductive bias introduced by knowledge-riet LLMs, which not only improves physical plausibility but also constrains the solution space meaningful way. These results highlight that VisionLaw combines strong generalization with interpretability, making it practical for forward simulation previously unseen temporal regimes.
4.3.2 GENERALIZATION TO NOVEL SCENARIOS
To further verify the generalization and transferability of the interpretable intrinsic dynamics learned by VisionLaw from visual observations, we apply the dynamics learned from different scenarios to novel 4D generation tasks. The 3D-to-4D and image-to-41 tasks follow the paradigms of Phys- Gaussian Xie et al. (2024) and 12y Lin et al.] (024a)) respectively, and all experiments are conducted under gravitational conditions. As shown in Fig. 6 all examples generate dynamics con- sistent with the original observations, such as the slow deformation of clay, the elastic recovery of rubber, and the dispersive behavior sand. These results demonstrate that the intrinsic dynamics inferred by VisionLaw are not only interpretable but also transferable to unseen scenarios, enabling the 4D interaction aligned with real physical behaviors. This cross-scenario generalization opens new possibilities for dhysics-driven 4D interaction.

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Iteration
Iooraion
Figure 8: Comparison across Difffernt Lower-level Optimization Paradigms. We compare two types of lower-level parameter optimization: gradient- based optimization (default) and evolutionary search, where Evolution()) denotes an evolutionary strategy with the specified population size. The left axis report the RGB loss computed against observations, while the right column records the runtime.
4.3.3 ABLATION STUDY ON DECOUPLED EVOLUTION STRATEGY
To evaluate the effectiveness of our proposed decoupled evolution strategy, we perform ablation study comparing two settings over five iterations: 1) with decoupling: the elastic and plastic com- ponents are optimized alternately for four iterations. followed by a final joint refinement step; 2) without decoupling: all five iterations are performed with joint optimization. As illustrated in Fig, the decoupled strategy consistently yields lower RGB losses across diverse scenes, indicating leads to better constitutive law discovery. By decomposing the search into simpler sub-tasks, it narrows the search space, making optimization more efficient. Moreover, the shaded regions are noticeably larger under the decoupled setting, indicating greater solution diversity. This helps avoid early con- vergence to poor local minima. Overall, the decoupled evolution strategy more effectively unleashes the potential of LLMs by not only sharpening exploitation but also broadening exploration.
4.3.4 POTENTIAL ANALYSIS IN MON-DIFERENTIABLE SIMULATION ENVIRONMENTS
To demonstrate the potential of VasionLaw in non-differentiable simulation environments, we com- pare two types of lower-level parameter optimization: (i) the gradient-based optimization adopted as the default in this work, and (ii) the gradient-free evolutionary strategies implemented via the dif- ferential evolution algorithm Storn & Price (1997). For evolutionary search, we evaluate population sizes of and 10, with 10 lower-level optimization iterations matching the gradient- based baseline. As shown Fig. g even when the lower-level optimizer is replaced with evolutionary search, Vi- sionLaw consistently converges solutions comparable to those from gradient-based optimization. In certain scenarios (e.g., ClayCat and RubberPawn). evolutionary search even achieves superior results. On the other hand, evolutionary search incurs substantial computational overhead, each iteration requires multiple forward simulations to evaluate all candidate parameter individuals in the population. Nevertheless, VisionLaw's 's compatibility with gradient-frre evolutionary strategies allows it to function effectively in non-differentiable simulation environments. This highlights its broad applicability and practical significance in the real world.
CONCLUSION
In this paper, we propose VisionLaw, a bilevel optimization framework that infers interpretable in- trinsic dynamics directly from visual observations by jointly optimizing the symbolic constitutive law and its material parameters. At the upper level, knowledgeable L.MMs are prompted to generate and refine symbolic constitutive laws, thereby introducing physical inductive biases into constitu- tive evolution. Mcanwhile, a decoupled evolution strategy is introduced to reduce the complexity of jointly searching and to improve the solution quality. At the lower level, material parameters op- timized under visual supervision, while evaluation and feedback on intrinsic dynamics consistency provided to guide the upper-level evolution. This closed-loop design effectively bridges the gap between visual data and physical nature, achieving balance between interpretability, physical plau- sibility, and generalization. Experimental results show that our method accurately captures intrinsic dynamics from visual observations and generalizes well to novel scenarios for 4D interaction.
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ACKNOWLEDGMENTS
This work was supported in part by the National Natural Science Foundation of China under Grant No. 52535009. This work was also supported by the National Natural Science Foundation of China under Grant No. 62276222.
ETHICS STATEMENT
This research adheres to the ethical guidelines outlined by ICLR. We confirm that no human sub- jects involved in this study, and all datasets used have been properly sourced and are publicly available. Our methods have been designed with fairness and transparency in mind, ensuring no biases are introduced in the analysis. Privacy and security of data have been prioritized throughout the research, and we comply with all applicable legal regulations. No confficts of interest or spon- sorships have influenced the research outcomes. We are committed to upholding research integrity and have followed appropriate ethical practices throughout the study.
REPRODUCIBILITY STATEMENT
We have made efforts to ensure the reproducibility of our work. The source code for the algorithms presented in this paper is publicly available at lggitbu... www.com//eeeeeee com/JiajingLing JingLin VisionLaw Ad- ditionally, a detailed description of the experimental setup and datasets is provided in the Appendix. We encourage reviewers and readers refer these materials for complete reproducibility.
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APPENDIX
In this appendix, we will provide: i) more experimental details; ii) more experimental results; iii) re- lated work; iv) implementation details of the MPM algorithm: summary of classical constitutive laws; vi) details of the prompt design. vii) visualizations of the inferred constitutive laws.
THE USE OF LARGE LANGUAGE MODELS (LLMs)
Large language models (LLMs) were utilized in this work improve the fluency and clarity of the manuscript. Their application was specifically focused on detailed proofreading to correct spelling errors and ensure grammatical accuracy, as well as refining sentence structures to enhance the read- ability and logical flow of the paper. It is crucial to note that all scientific contributions, including the core concepts, experimental design, data analysis, and conclusions, were entirely conceived and written by the authors. The LLMs were employed solely as writing assistance tool and did not contribute to the conceptualization or analysis of the study.
A MORE EXPERIMENTAL DETAILS
A.1 IMPLEMENTATION DETAILS
Given multi-view videos of scene, first perform 3DGS reconstruction Kerbl ct al. (2023) using the multi-view images from the initial time step. Following NeuMA Cao et al. (2024), we establish relationships between simulation particles and Gaussian kernels via the Particle- GS mechanism. 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 gen- erate constitutive hypotheses. For all scenarios, the initial constitutive individual is only defined as a purely elastic model that combines linear isotropic elasticity with identity plasticity. The alternating evolution phase consists of 4 iterations. In each iteration, the top 3 individuals are se- lected, and each generates offspring independently. In the subsequent joint evolution phase, we conduct 3 iterations. In each iteration, the top five individuals are selected to jointly prompt GPT, generating offspring in one shot. For lower-level optimization, we conduct MPM simulations un- der standard gravitational acceleration (9.8 m/s²) within a unit cube domain [0, 1JS The simulation resolution is set to 32 for synthetic data and 70 for real-world data. We employ the Adam opti- mizer with learning rate of 10-3 3 and perform 10 iterations to tune the material parameters of a single constitutive law. For each scene, we conduct five independent runs using different random seeds: 0, 1, 2, 3, and 4. All experiments conducted on NVIDIA A40 (48GB) GPU.
A.2 DATASET DETAILS
The synthetic dataset is derived from NeuMA Cao et al. (2024) and consists of six scenes ( ('Bouncy- Ball', 'JellyDuck', 'RubberPawn', 'ClayCat', 'HoneyBottle" and 'SandFish'). Each scene records the motion of a single object, providing observations from 10 viewpoints with a total of 400 frames per dynamic sequence. To reduce computational resources, for the synthetic data, we select one frame every five frames from the video to create the training set. This dataset features a variety of material types, ranging from elastic bodies to granular materials, exhibiting diverse dynamic behav- iors and complex geometric shapes. Meanwhile, the synthetic dataset also provides ground-truth particle trajectories, which can be used to evaluate the consistency between the inferred and ground- truth intrinsic dynamics. The real-world dataset is taken from Spring-Gaus Zhong et al. (2024) and contains two scenes ('Bun' and 'Burger'). provides observations from 3 viewpoints, with each dynamic sequence consisting of 19 frames. In all experiments, the initial velocity Up follows the configuration provided NeuMA's dataset description. We use only a single frontal view of the object visual observation to infer its intrinsic dynamics.
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Observation
Ours
Observation
Observation
Ours
Observation
Ours
Observation
Ours
Observation
Ours
Time
BounsyBoll
ClyyCat
HoneyBottle
JellyDuck
RubberPown
Sondrish
Figure 9: Visual Results on Synthetic Dataset. We select the rendered images at frames 1, 100, 200, 300, and 400. VisionLaw exhibits dynamics similar to those observed in visual observations.
VisionLaw
NeuMA
NCLaw
Chamier Distance Chamter Distance
BouncyBall
ClayCat
12-
HoneyBottle
20-
3.-
69
2.
88
JellyDuck
RubberPawn
SandFish
175
19-
11
180
Step
Step
Step
Figure 10: Comparison of Chamfer Distance at Different Time Steps Synthetic Dataset.
B MORE EXPERIMENTAL RESULTS
B. QUALITATIVE VISUALIZATION RESULTS
We provide qualitative results six synthetic scenes assess the visual fidelity of method. As shown in Fig 9 we compare rendered outputs from our model with ground-truth observations selected time frames (1, 100, 200, 300, and 400). Our method accurately reproduces object dynamics over time, showing close alignment with the ground truth across all scenes. These results demonstrate that VisionLaw effectively captures complex deformation behaviors with visual realism.
B.2 QUANTITATIVE COMPARISON OF CHAMFER DISTANCE
As shown in Fig. 10 we compare the Chamfer distance of VisionLaw, NeuMA Cao et al. (024)
and NCLaw Ma et al. (2023) across different time steps on the synthetic dataset. NCLaw consis- tently shows the worst performance. This is because NCLaw can only fit the known dynamics, but fails to adapt to the underlying intrinsic dynamics behind the visual observations. As result, its emoor remains high across all objects. NeuMA introduces additional neural network components to capture the mapping between visual observations and intrinsic dynamics. However, due to the lack of physical inductive bias, NeuMA is mainly based memorization. leading overfitting and un- stable predictions. In contrast, VasionLaw distills physical priors from LLMs to refine constitutive laws, thereby incorporating form of physical inductive bias. This mechanism enhances its abil-
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VisionLaw
NeuMA
SSIM
LPIPS
PSNR Variance
0.995
0.10
0.985
0.08
0.975
0.570
0.00
Bell
Cet
Buttle
Duak
Pein
Fish
Cet
Bottle
Pwom
14sh
Dwl
Cat
Boftile
Duck
Pain
Bah
(a)
(b)
Figure 11: Quantitative Comparison of Visual Fidelity on Synthetic Datasets. (a) Average SSIM and LPIPS over all non-training views. Higher SSIM and lower LPIPS values reflect improved visual fidelity; (b) PSNR variance over all views, including training views.
Scener
AnuncySed ClayCat HoneySotttee
JelyOurk
SandFish
Initiil Velocity
Time
Time
Law: I Rotheer/wwn
Jelyyookk
CleyCut
Booscydull
SonaFish HoneyRottle
0.5v,
1 5.5g
0.5%g
1.5g
0.51
155
(b)
Figure 12: Generalization Analysis. (a) Generalization new scenarios. The top row shows the simulated scenes, while the bottom row presents the intrinsic dynamics inferred for the given scenarios. (b) Generalization different initial velocities. The bottom row represents the configured initial velocity, expressed as multiple of the original initial velocity.
ity to discover hidden dynamics, leading to consistently better performance across different objects and time steps. As shown in Fig. [0, VisionLaw achieves lower Chamfer distance, demonstrating stronger adaptability to complex dynamics.
B.3 QUANTITATIVE COMPARISON OF VISUAL FIDELITY
To more comprehensively evaluate visual fidelity, we report average SSIM and LPIPS across all non-training views in Fig. 11 (a). The results show that VisionLaw outperforms NeuMA Cao et al. (024)). This confirms that our method not only captures more faithful intrinsic dynamics but also produces dynamic reconstructions with higher perceptual fidelity. We further compute the PSNR variance over all views in Fig. (b), which reflects the generalization to unseen views. NeuMA exhibits high PSNR variance, indicating a tendency to overfit. In contrast, VisionLaw achieves much lower variance. This demonstrates that, even when trained from a single fixed viewpoint, our method generalizes effectively to novel views by leveraging the physical inductive bias introduced through LLMs.
B.4 GENERALIZATION ANALYSIS
generalization by applying the intrinsic dynamics inferred from scenario to simulate another. As shown in Fig [[]a), the top row presents the target scenes, while the bottom shows the intrinsic dynamics inferred from different sources. Despite the mismatch
We first evaluate
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BouncyBall
RubberPawn
JellyDuck
Obseevation
NeuMA
Ours
Figure 13: Comparison Generalization Novel Scenarios. The first row shows the visual observations used to learn the intrinsic dynamics. The rendered frames in each column correspond to the same simulation time step across all methods. Red crosses indicate numerical divergence during simulation, which terminates the rollout.
Methods
Metric
BouncyBal Ball
ClayCat
HoneyBottle
JellyDuck
RubberPawn
SandFish
Average
w/o decoupling
Accuracy
9.08
17.31
3.64
6.24
25.88
2.24
10.73
Time
11790
10997
10891
9333
1060S
14439
11343
Accuracy
7.80
8.13
3.54
5.93
15.25
2.17
7.14
w/ decoupling
Time
9319
10636
10699
11249
10839
15689
11405
Table 2: Search Efficiency Analysis of Decoupled Evolution Strategy. Comparison between naive joint optimization (w/o decoupling) and decoupled evolution strategy (w/ decoupling). Accuracy denotes RGB loss (lower is better); Time indicates time consumption in seconds (lower is better).
between the source scene and the target, method consistently produces physically plausible behaviors. This indicates that the constitutive laws discovered by VisionLaw are not merely scene- specific fits but encode transferable physical priors, demonstrating strong cross-scene generaliza- tion. We further design experiments under different initial conditions by varying the initial velocity of objects (with the baseline up specified in the NeuMA Cao al. (2024) dataset description). As shown in Fig. 12 (b), the results show that, even with varying initial velocities, the intrinsic dynam- ies inferred by VisionLaw still accurately reflect the object's behavior. This result underscores the robustness of our method in the face of variations in initial conditions, confirming that VisionLaw identifies fundamental physical laws that extend beyond the specific configurations used training.
B.5 GENERALIZATION COMPARISON
To further validate the generalization capability of VisionLaw, we compare with NeuMA on a 4D interaction generation task, with all experiments conducted under gravity only. As shown in Fig. [3 the intrinsic dynamics learned by NeuMA exhibit clear limitations when transferred to novel scenar- ios. First. NeuMA frequently suffers from numerical instabilities, causing the simulation to diverge and terminate prematurely, as observed in the BouncyBal Ball and JellyDuck cases, Even when rollouts remain stable, NeuMA produces dynamics that deviate markedly from the reference observations. For instance, the duck undergoes excessive collapse, inconsistent with the visual behavior observed in the RubberPawn reference. These failures stem from NeuMA's lack physical inductive ases, which leads it to mechanically reconstruct appearance rather than truly capturing the latent intrinsic dynamics caramicc-ultimatety ultimately resulting in poor generalization. In contrast, VisionLaw consistently generates stable 4D interactive dynamics that align with the original observations. By incorporat- ing physical inductive biases through LLMs, VisionLaw effectively captures the underlying intrinsic dynamics, enabling robust and reliable generalization unseen scenarios.
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B.6 SEARCH EFFICIENCY ANALYSIS OF DECOUPLED EVOLUTION STRATEGY
To assess the search efficiency of our decoupled evolution strategy, we conducted an ablation study under the same experimental settings described in Sec. 4.3.3 As shown in Tab. 2 the decoupled strategy significantly improves accuracy, reducing the average RGB loss from 10.73 to 7.14, while maintaining comparable computational cost (1 13443s vs. 11 11405s). This result demonstrates that the decoupled strategy effectively unleashes the LLM's capability for constitutive discovery by making the optimization more focused, producing superior solution quality without additional computational overhead.
C RELATED WORK
C.I PHYSICS-BASHH 4D INTERACTION
Advances in 3D representation methods Mildenhall et al. (2021); Müller et al. p: Kerbl et al. 2023); Zeng et al. (2025) (e.g., NeRF and 3DGS) have greatly facilitated the creation of 3D sets Poole et al. ) Tang et al. 2222b a). consequently drawing significant attention to the pur- suit of realistic interaction with these assets. To enable physically plausible 4D interaction, recent works have attempted to incorporate various physical simulators Stoomahin et al. (2013); Mack- lin et al. (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 gen- eralized moving least squares (Q-GMLS) Martin al.) (20T0). Inspired by the Lagrangian nature of 3DGS, PhysGaussian Xie al. (2224) 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 Diang al. (2024b) conducts tessellation via TetGen [Hang] (2015) to convert 3DGS representations into tetrahedral meshes, enabling fast XPBD simulation and physically plausible interaction in VR.
C.2 INTRINSIC DYNAMICS LEARNING
Understanding the intrinsic dynamics underlying observational data highly valuable for interactive
simulation Müller & Gross (2004) and scientific discovery Wang et al. (2023). ) Deep learning Gong
al. (026) 2024): 2026) Zeng et al. (0024) Lu al. (2024b) 2023) has advanced rapidly and is increas- et al. (2020); Yang et al. (2025); ) Wang cet al. (2025), with some methods PPaff et al. (2020); Ummenhofer et al. (2019) using end-to-end net- works to model physical laws. However, purely neural approaches often lack physical consistency. NCLaw Ma et al. (2023) integrates known laws with a Icarnable 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 infer intrinsic dynamics from vi- sual 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, Dream Physcss 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 parameter estimation process in these methods relies on expert-defined constitutive laws. Spring- Gaus Zhong et al. (2024) integrates spring-mant mass system BBlickhan (1989) with 3DGS to simulate elastic objects, and optimizes spring stiffness under multi-view video supervision. OmniPhysGS Lin et al. (2025) introduces learnable constitutive Gaussians that assign specific constitutive laws each Gaussian kernel. enabling interaction simulation in multi- material scenarios. While NeuMA Cao let al. (2024) can leam neural constitutive models Wong et al. 2025) from visual observations, lacks interpretability and exhibits weak generalization ability. In this paper, we aim to infer consti- tutive law expressions from visual observations that are both interpretable and highly generalizable.
ingly being applied to physical simulation Sanchez-Gonzal
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D MATERIAL POINT METHOD
Continuum mechanics studies the deformation and motion behavior of materials under forces. Mo- tion is typically represented by the deformation map x = (XX,1) 1), which maps from the undeformed material space wD to the deformed world space w' The deformation gradient scribes how the material deforms locally. MPM is a simulation method that combines Lagrangian particles with Eulerian grids and has demonstrated its ability simulate various materials. In MPM, each particle p carries various physical properties, including mass m, density p. volume V. Young's modulus E, Poisson's 's ratio v, velocity deformation gradient F and velocity gradient C. MPM operates within loop that includes particle-to- toriiid (P2G) transfer, grid operations, and grid-to- particle (G2P) transfer. In the particle-to- grid (P2G) stage, MPM transfers momentum and mass from particles to grids:
ax (X, () de-
m mi+1_
Σ
Wpp
(11)
(mv)
Σ Wip [mp v + mp Cx x
(12)
where Wip the B-spline kernel that measures the distance between particle p and grid i. After P2G stage, we perform grid operations:
v (mvv)) //m
(13)
f in - Σ T/ Vusp V
(14)
(+1 vi + At (fe in/ m + g)
(15)
where g = 9.8 m/s² denotes the gravitational acceleration. Then we transfer the results back to
particles in the grid-to-particle (G2P) stage:
++ Σ wio v
(16)
2
1+
x
Aivet!
(17)
C++1
Ax Σ
Wip v ++1 (x: x²)²;
(18)
F = (I+ AICH+1) F'.
(19)
F++1 (F
(20)
PE (F
(21)
where Φg and p denote the elastic and plastic constitutive laws, respectively. F represents the trial deformation gradient, which is subsequently corrected using the plastic constitutive law (pp. denotes the Kirchhoff stress. By following these three stages, we complete a simulation step.
E EXPERT-DESIGNER CONSTITUTIVE LAWS
Within the MPM framework, complete constitutive law consists of an elastic constitutive law and a plastic constitutive law. In our experimental setup, for all scenarios, we initialize the constitutive individual combination of fixed corotated elasticity model and identity plasticity model. Several well-known classical constitutive laws are presented in the following.
E.1 ELASTIC CONSTITUTIVE LAW
The elastic constitutive law describes reversible elastic responses of the material under deformation. Here, use the Kirchhoff stress to express the stress-strain relationship.
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E.1.1 LINEAR ISOTROPIC ELASTICITY.
The Kirchhoff stress is defined as:
[M (F+F" 21) (tr((F) 3) F
(22)
where μ and À are the Lamé parameters.
E.1.2 FIXED COROTATED ELASTICITY.
The Kirchhoff stress is defined as:
T (F (J-
(23)
where R UVT and F UΣV is the singular value decomposition of elastic deformation
gradient. is the determinant of
E.1.3 NEO-HOOKEAN ELASTICITY. The Kirchhoff stress is defined as:
14 (FF" -I) Alog(J)I.
(24)
E.1.4 STVK ELASTICITY.
The Kirchhoff stress is defined as
T= U (2ne + Atr(E))) vT,
(25)
where F UΣVT and log
E.2 PLASTIC CONSTITUTIVE LAW
The plastic constitutive law captures irreversible plastic evolution beyond the elastic limit by cor- recting the trial deformation gradient ptrial to the final deformation gradient F.
E.2.1
IDENTITY PLASTICITY.
The corrected deformation gradient is defined as:
coorected
F
(26)
The identity plasticity model does not induce any plastic effects.
E.2.2 DRUCKER-PRAGER PLASTICITY.
Given F = UEVT and ε log( E), the corrected deformation gradient defined as:
poorrected -UZ(E) VT.
(27)
I,
if ti((e)) > O,
3(E)= =
Σ,
if 8y 0nd ir(e) < 0,
(28)
exp E
87 [2T
otherwise,
sin 3-sin and of is the friction angle. = dev(e). Drucker-Prager plasticity is suitable for simulating materials like snow and sand.
Here, 8y = ||ē|| + (d)+) 2p
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E.2.3
VON MISES PLASTICITY.
The corrected deformation gradient is defined as:
Feorrected -EZ
VT,
(29)
where
Σ
8% S 0
2(2)
(30)
cxp
ppll) (
otherwise,
and
TY by = ||ē||
(31)
2p
Here ty is the yield stress. von Mises plasticity is suitable for simulating plasticity like metal and clay.
E.2.4 FLUID PLASTICITY.
The corrected deformation gradient is defined as:
peorrected
J1/31,
(32)
where is the determinant of F. Fluid plasticity is suitable simulating fluid-like materials.
F LIMITATION AND FUTURE WORK
Although our method effectively captures intrinsic dynamics from visual observations and demon- strates strong interpretability and generalization capabilities, it still has certain limitations that war- rant further research and improvement. The method relies on an evolutionary search paradigm that involves extensive evaluations. This process is time-consuming because it requires large num- ber of forward simulations and backward parameter optimization. Idcally, preliminary screening mechanism could be introduced, where only individuals with potential merit subjected to further evaluation. Such a strategy could significantly reduce evaluations and accelerate the efficiency of constitutive law discovery.
G PROMPT DESIGN DETAILS
In the following, present the prompts used to guide LLMss to enable the evolution of constitu- tive laws. To further achieve a decoupled evolution strategy, we designed distinct prompts for the alternating evolution phase and the joint evolution phase.
G.1 PROMPT DESIGN JOINT EVOLUTION
System prompt:
You are an intelligeat MI assistant for coding, physical simsistion, and scientitic diseevery, Follow the usez's requirenenta cazefully and nake suze you underatand then. strictly linited to physical simulation, materisi nathenatios,
coding.
Keep your ansvers short and to the peint. Do not provide information that is not requested.
docusenl your code your solucion.
explain the beliadd
Use Razkdown
You are very familiar with Fyshon and PyTorch.
exiwenel iibraries other Lhan Lbe Librariss used in the exanples.
User prompt for elastic and plastic constitutive law evolution:
204 Context
physical simulation built based Foint Method. The jective of problem Lo code block that result from execut ing the code natches the ground-truth result.
physicel simulation
tu.l constitutive
simulates through
reparate asses:
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gradient correction nedel. glass containa functions that divido the code into centiruocs part that defines the differentian iable paranaters and discrete part that setincs the symbolic detornation gradicat correction model. The ingut the syabolic defornation gradient correction nodel cocrested defermation gradient. 2. -=Elasticitytoooo
seplattciccooool: datines the
deformation gradient, and
datines the constitutive lav that maps corrected dotormation gradient to stress. Tais clasa containa two functions that divide the coda late concincous part that defines the differentlable paraneters
defines the symbolic constitutive The the uscrected defornet ivo gradient, the outpul
tapat symbelic conscitutive the Kizchhoff atress tenser.
the deformation ian passes this corrected deformacion gradient the "Elascicitytodd" compuse the stress.
simulation applies "iasceeilyeeell
States that caplure the physical dynanics of Lhe systen and metries that measuee the difference the greand-trath result included foodback secticn.
244 7ask
fowlingg llecet luns exarples, analyse then, generabe solution upon
then.
Coding format prompt for elastic and plastic constitutive law evolution:
*44 Pytorch Tips
when elenent-vise multiplying natrix,
dincacions their 30, you
operation, example,
muitiplying 18,1 and before the operation. Sim' larly, "Il 11 should also reshaged to 11 1iveee
Niview
1) 1. Iz you are not suze, write down every component in the expression one by one and annotate its cimension in the conrent for veritication.
then compating the cin2-2). edmn(dim)))) viewi-l, 1, 11 Avoid using traze" or "Tonsor.tracs" sinze they only rupport 20 matrix.
of tensor (8, 3), "a.diagonal Idim1-1,
Requirements
The procramiing lanquage is slxays pythor. Annotate the size the tensor
operation. For
2)".
2. The only library allowad is PyTorch. Follow the exomples provided by the user and chock the EyTorch cocumentation to learn now to use PyTorca.
the into eprimization
tuned differentiable symbali: constitutive 1as represeated by PyTorch code. Darine then init init cunction and tae 'torkard" cunction.
physicsl
respectively in the
s. always remember the only sutput of the 'fowwadd" function in *#laacciitytddee class is corvected deformation only sutaut of the 'forward' function eflaatieittWWadll class Kirchhoff stress tensor, which is defined by the narrix muitigtication between the first Fiols-Kirchhott stress tenser and the trenspase of the deformation gradient tenzor, Fernally, tea Piola-Kizehhoff direstly return any oshar type seress
Always remember
F'T', whare la the Kirchhoff deformatica gradient tensor. tensor other than Kirchaott scress senser. Compute Kirchhost strees tensor using the equation: tta P p*2",
tansoz,
propesed should
follox
fuaction signatures
syyyhho
impart torch
import
class PlasticltyMedel (na. Module))
Linit__scct, paran: float DEPAULT VALUEE
Derine trainable continuous physical paransters for difforentiable optinizationn Tentatively initialize the parmneters
default wige.
Aces: *y
(Clost): the physical neaning of (be paramelms.
super init_
sel I.prann = Pn. (Lorch, (pasan))
forl, Tensor)
Compate corsected deformation gradient from defornation gradient tensor.
(ensar) defornation gradient censor (B, 3, 3).
Belucia:
(torch.Tensone
gradient
setuen 2oooseenied
ElasticityMedi Modilee
de Init fs paren: flost DEPAUNT VATUR]:
trainable continuous physical paraneters tor ditterentiable optimizationn Tentatively initiali the paremeters wist che default arge.
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Published as a conference paper at ICLR 2026
Ares:
pazan [Ilost]; the physical meaning of the poranctor.
...
s)per And5
sell.paste Pn..aremeter (trrhhttttrrcc.cccceeeeoo (paran))
def forwardicoii, torch, Tensor) torch. Tensor:
Compute Kirchhoft acress bensor fron deformation gradient tensoz.
tenaar) dafornation gradient censor (B, 3, 3).
Beturna:
Xirchhoff_straa itorch. Tensor): Xirchhoft tenser (3, 3, 3).
Iutuen Xirelhoff_sttee
244 Solution Bequirementa
Ayne slep by step what the patent iel problen in the previous iterations based the the resulca fron previocs constitutive lave nianatched with the ground truth. not give advice about hew to optinize. Focus on the formulation
foodbask. Think about
consticutive Stack this seut ion with 4244 Analysist. iia Analyas all Hlesal ioms individuelly, and start the subsection for each iceration with 1944, Iteration. Wi, wheze x stands for the index. Benember analyze every
history.
2. Think atag-by-acop what you need to do in this iseration to improve medel perfomancee Considar bath the elast falty and plast city components.
plasticity compoeett::
Think about if the plasticity is needed improve performance. Ranenber that plasticity not necessary. If your anelysis supports plasticity, think about how to update deformat fon gradient using plastisaly. Think about how to superale your algorilli inko concinueu symbolic deformation gradient correction
physicel parameter part
part.
For the elasticity compoeent::
Think about how to separate your slgurichm late a con imsous physical parametes pact and symbolic conetitutive lax
written in great detail. Remember update the default values the trainable physical parameters based previous uptinizationa. Blacs this section wich *684 Step-by-SSep Plant.
Dexeribe your plan pseudo
Output code single block "pyshon ... oos detailed conments the block. no not add any trailing coments before or after the code block. Start this section with npe Code".
G.2 PROMPT DESIGN ALTERNATING EVOLUTION
System prompt:
You are an iacelliquat At assistant for coding, phystcal simelation, and scientific disesvery. Pal the userts requrementa carefully and make sure you under stand then. nuthenatios,
Your expectiss atzinily linited physical simiationn naturial coding.
Keep your ansvers short and to the point. Do provide
information that is not requeated,
Alvays docusent your code
explain the bahind than.
Uhe Narkdown farnat your sulution. You are very faniliar with Pyshon and PyTorch, use any external libraries other than the libraries used in the exargles.
User prompt for plastic constitutive law evolution:
1F8
Context
physical simulation environnet. The physlcal simulation is built based Naterial code block that the result from
Foint Method. objective of this promlon to execut ing the code naithes the ground Lzuth sssult.
block defires fu'l const itut behavion of simulated naterial through
separate classes:
PePlaattiiitytodee dafines the defornation for gradient ion nedel. This slass containa two Funct fona that divide the code into cont iruoua that defines the di fifrrttnt fable paraneters syabolis defornation gradient deformation gradient, and output
discrele part defines
nodel, ingut
the symbolic deCornation gradient correction model
the one corrected deformation ian gradient.
2. aellaatiiiyttoddll
defines the cottiitutive lav that corsected defosmat gradient that
stress, Tala
pare that the difrerent lable paramtters and discrete part that defines the symbolie constitat lax. The imput to the symbelic comsiitu ive Lav is the corsected defonnation ion gradient, and the outgut is the Ki Kirdhooff stress censor.
contains
code
simulation applies the "pascieltyMddd"" Clrat co correct the deformation ion gradient, passes this corrected deformstion ion gradient into the 18l ast iolty to conpute the stress.
States that cepture physicai cynamics at eysten and netrics neasure the citterence neeu are included in the feedback sect ien.
from the ground-t both truth
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1FE Task
the current task, EascieltyWede already been cinalized and ahould renain unchanced. Flease focus exclusively analyzing and inprovira the #lasticitytodcl class. book at the tollowing iterations as exampies, analyze thon, and generate a better plastic constitutive model based on than.
Coding format prompt for plastic constitutive law evolution:
aE PyTozch: Tipa
1. When element wise multiglylng ng two natrix, make sure their numbee of dimensions necch before the examlee,
miliiglynng 13" (B,1 and 'I" 13 before the operation. larly, "Id 13 shoule also be reshaged to 'id 11 H.veewtt)) 1,
opeccal
3), you should du N.vieet
13 1. 18 you are not sure, wrice down every composenc in the expression one by ane and annobate it.s dimension in the comment for verlfication.
When computing the
of tensor
S, 3h, 1A.eiagonal (diml=l,
din2=2).. (um((din)))) .vient 1, 1, 11", Avoid using "tarch-traww" or "Tansor. sindew they only s support 22 natzix.
244 Code Reguliements
1. The programming language is a xays pythan,
2. Annolate the size of Lbe Lenxus comment after each Lenwor operation. For exemple, "4 (8,
3)",
3. The only lbbrar loved is PyTorch, Pollow the exanples provided by the user and check the
Pyforch documeniation Lo Iearn hus vo use PyTurea,
Separate the code into continueus physical parameters that can be tured with dsfterentiable
the symbolie const tut lau represented by PyTorch code, Define then swapectively the "inni"" Cunduion ion and the "forward" Sunction.
iprinization
5, Always remenber the only sutput of the "torward' function in #fllacciiitytodeee class is corrected deformation gradient.
#ElantiestyMhdee class is Kirchhoff streas tersor, which detined by the necrix nultiplicet icstiiiiiiinn between the Eirst P Fiols Ki Kirchhoff stress tensor and the transpose of the deformat fon gradient tensor. Pornally, teau tirst Piola-Kizathot direstly reture any other type
Always remenser the only sutgut or the 'torward" tunction
F'TS, where tau is the Kirchhort stress is the deformation gredient tensor, tenear other than Kirchhoff serexs cenaor. Compute Kirchhoff streaa tenaor using she equation: tau P 3 FTr",
strese
tensor,
The proposed code should strictly follow the structure and fonation signatures belous
"gython
(code)
H Solution fon Reguirements
potential problen in the previcus iterations bases the tootback. Think about. say the sesulcs from previous constitutive lavs nismatched truth. Do not give sdvice about hew to optinizo. Focus on the termulstion or the constitutive lsx. Start this section with *HH Ans ysis", Analyze all iterations indivioually, and start the subsection for each itcration with 'ettr Iteratior. H", where stands for the index. Renember analyze every iteration
inalyze step-by-step
the ground
history.
2. Think stop-by-stop what you need to do in this iscration to improve medsl perforaanee Considar the clasticity For the plasticity componants:
plasticity components.
Think about i2 the plasticity is needed to inprove performanco. Romamber that plasticity is not necescary, It your analysis supports plasticity, think anout how to update decormation your aigorithe lata concinusus
aradient using plastisity. Think about how to physicel parameter past and symbolic dsformation gradient correction model part.
For the clasticity components:
v22o not analyze oz nodlfy this partes. Flease focus improving the plastls components. the -=Blssticitytoosii class must senain exactly the reproduced exactly as originslly detines.
Piease
every
steraticn,
Describe your plan in pscudo-csdo, writton out in great detail. Remenber to update the dafault values of the trzinahie physical parancters based previsus optimizations. Start this socticn with Stap-b plan".
3. Cutput the code in single code Elock "py'stton ... with detailed comments in the code block. Do not add any trailing comments before or after the code block. Seart this section with *etr code".
User prompt for elastic constitutive law evolution:
444
Context
This physical sinulation anvironmen. The physical simulation is built based the Naterial Point Mothed, The object fow of this problen is bo fill in a code block ad that the result from execul ing the code natches Che ground sround-Leuth sesult.
The code block defines the fu'l conkt ivv behavior of the vinulated naterial through two suparate classes:
that divide defines the deformation for gradient correct fon nodel, This class containe two inucus part that defines different iable peraneters and discrete past that defines the symbolic defocnation gradient correction nodel. The input the symbolic deformstion gradient correction model is the deformation gradient, and the outpot is the cocrected deformetion gradient.
code
cileeticieytooelll derines the constitutive Law that maps corrected decosnation grasient stress. Tals cless contains is two Funct lons that divide the code into cont Enpore pare that defines
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the differentiable paraneters and discrate part that defines the syabolic canstitacive lax. The input to the symbelic constitutive Law is the corracted decormation gradient, and the output is the Ricchaot: strese tensor,
The simalation applies the "gaaciiittyeede" cirst to correct the defermation gradient, then passes this corrected setormation gradient into the "Blascicttytodel to compute the strese.
Statos that
physical cynanies of the systen and netries that masure the differance fron the greand Leuth swault are included in the fewdback section.
1tt Task
current task, the Flascicttyedel already bean cinalized and ahould ranain unchanced. Flcase rocus exclusively sn analyzing and improving the slasticitytode class. Look at the following iterations exanples, analyze than, and generate better wlastic isutivee model based
Coding format prompt for elastic constitutive law evolution:
*t PyTorch
then element-xise multiplying natrix, make sure
nunber dimensions match betore the
openet fon, For example, when mulsiplying ying 'P' the and '1" (R, 3, 3), you should do M.view(-1, 1, 1)" betore the operation. Similarly, "IJ - 1) saould alse be reshaped to 'Id 11 vieweew--, 1, write dow every component in the expression cimcnsion in the comment for verizscation.
and annotate its 2. then computing the trace st a tensor A (B, 2, 3h, use '8.ciegonal isimi-1, cinz=2) Laun (dimml))))) (eddimm)...iiiieed... viexi-l, 1. 11', Avoid using "torch.trase" or "Tensoritrace" sirce they only cupport matrix.
% I5
H Code Requi rements
programinng language is s.xays pythor.
the
the tensor es comeett each teneor operation. for exarple,
2)"
3. The only library alloved is PyTorch. Follow the examples provided by the user and check the hox Separate the codo into concinucus physicel parmmeters that can ke tuned aifcerentiable
Eytorch documentation to
optimization and the symbolis constitutive 1av represented by PyTorch code, Detine then gespectively in the -nit_ function and the 'forardd' function.
Always remenber the only sutput the 'forward' function in *flastiittybodil class corrected dezormation gradient.
6. Always ranerber the only sutput of the 'forward" function in -Elsscccitttddel class is Kirchhott stress tensoz, which is detined by ite narzix multiplication between the tizst Kirchhoitthhoft
and a F'T', whare tau is the Birchhoff stress teasor, tensor, and 7 is the deforration gradient tensor. Do not directiy return any other type ot strees tensor other than. Kirchhoft scress censer. Compute Kirchhoff stzesa tenser using the equation: "tau FTT".
of deformation gradient first Hiola-Kischhoff stress
Pornallv, ttas
The propesed code should strictly tollox the structure tuaction signatures bolow:
"python (code)
44 Solution Requirementa
Analyze step-by-stop what the posential problen in the previcus itcrations based. Toudback. Think about shy the resulca from previors constitutive ive lavs niamatched with the ground truth. Do not give advice about hew to optinize. Pocus on the fornulation of the constitutive 1sw. Analysis". Analyae itesations individuelly, 1H4 Iteration R", shese stands for the index. Remembar
Stact subsection for each iceration analyza every itcration in the history.
the
Think step-by-acep what you both the clasticity and plasticity components.
this iteration to improve medel perfomance. Consider
For the plasticity compoeeti::
vilo not analyze oz modify this partes, Please focus improving the slastle components. inityModel
please ensure
must exent ly ileration, and must reprodhced exautly as originally defined.
the
every
For the elasticity componants:
Think about how to seperate your algorithm into cont incoux physicsl paraneter part and syskolic cssstitative Sas part.
Dascribo your plan pacudo-code, written our in great detail. Renenbor to update the dafauit valuws of the trainable physical parameters based on previous opt Intiinaaaion lana. Stars this sect ion with *444 Step by Step Plant.
single code block tpyshann ... block. Do not add any Leaiting comment beforw or aftes the code block, Seart this section with "ste Code",
Output the code
detailed conment a in the code
H VISUALIZATION OF INFERRED INTERPRETABLE CONSTITUTIVE LAW
In this section, we show the inferred constitutive laws under different visual scenarios, including "BouncyBall", "ClayCat", "HoneyBottle,, "JellyDuck", "RubberPawn", "SandFish", "Bun" and "Burger". Since these laws are expressed in the form of Python code snippets, these laws exhibit strong interpretability and readability, making them easily understandable humans.
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H.1 BOUNCYBALL
In the BouncyBall scenario, the constitutive law inferred by method is presented.
import torch
import Lorch.on
class Plasticil yadde KondRedal))
Lintt Iswlf, threaalde 0,51:
9
Deffne trainable physical paranecer for plast plasticity yield threshold. Tnitial uoed to 0.5 balance plastig effectx based on feedback,
10
Arges
sheld_shreehold (flost): logerichnle strain clamp threshold,
supert) init__))
self.yieeld an.Paeaneeer (torch. lensor (yield_
17
def forxardize's f, torch,Tensori Korch,"enxone
Dorrect detornation gradient by clanping logacitanio principal strains.
Args:
(corch, Teneaci: detornstion gradient tenser (B, 3, 3)
25
Returns:
F_correctes NorchhTensorij corrected deforaaiin gradient tensor 1B, 3, 31.
SVD at detormation gradient
29
U, Signo, Vh = torch,linelg. lin.llg.sg........... (vc D:. (B,3,3), Signar (B,3), Vhr 18,3,5)
Clanp singular values Signa signa_clampes corch.clerp.mme (Siginnns, 16-61 (8,3)
ronerizel problens
33
* Logarsthnic principal streine loq_aigma torch. (Siena__lanped) (B,3)
37
4 Entorde positive yield tarestold via sottplue
yleld_thrash torch.nn. fuactional...... wtw.iiinn......ssseeee... wtt..a/suuuattt..oooo...aaaaaaaa
sealer
opsiion_clampes torch.clanp sign,, nin--pieid_threen maxyyiiid (B,
41
corrected singciar
sirna_corsected torch. lepsiloo_ (B,
45
Recompose corrected defornation gradient
[_correctad torch.natmcl totch-natmal (torch, www.:/eooooiiiiinnoooooeooe (3,3)
48
seturn F ecoorrcttdd
class ElasticityM:de Kodule)
52
_init_ younga lomgaanadoleagg fluat 10.18, saiasonnral-ooaegnaeee floss =0.5)
Define trainable continaous paysical parenstere for Corotated Blasticity.
56
Aros:
youngs_nodrles__u (flost): log Young's 5 modules.
wawasooo_rttie_eo (floas): pazanstes bafore aignoid for Polsson's ratio.
60
super (nis_
hol...younga modutuus uas log an.Paranceer (torch, Lensor tyoungs mnddulss_ddd
En.Farancter teasch.tenser tenser (po.ssoos_nttisssseoso.
64
forwardisntt, Tonaoe torch.
Bampate RirchhoCC stresa Lenaus from deConnet lun gradient via Corutated Flasticlty.
68
Argss
(Locch, Demwar) deformetion gradient Lenser (B, 3, 3).
Returnes
kizdhhuff_ [Lorch, Denaus) Kisclduff slawss
B P.shage[O)
paraneters
younga modulus Eogonggs modulax log.expdl soslar poisaona_cataa
79
io wigiidg wng.d slgnoid 0.45 a scalas in (0, 0.49)
youngs youngu poisaona_ratio / (11.0 poiasoos_ratio) 11.0 2.0 poi ssons__rattin retio]] 4 scs ar
ratio soslar
detormation gradiert gigne, Vh. torch, avd (B, 3,33, Sigres (8,3), Whr (8,3,3)
27

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Published as a conference paper at ICLR 2026
Clamp singular values
85
sigme_clampce terch. ciamccccn.ii..nm.m.. 1e-61 (B,3)
Rotation maczix
R tocch.natmul 13, Vh) f (B,2,3)
92
determinant
vien 13 (3,1,1)
96
Idettaty tensor
Losch.eye13, device diypesP diypenP.ddyge)) .dlyge) uungueeee expand 3,,3,
Rashape ocalars for broadcast
100
nu.vies (-1, 1, if ne.din =* 0 elaw la.dim() else
101
1a.vios (-1,
103
a Corotated acresa Lezn
104
corotated 2.0
(P
(B)
105
Volumetzic streas Lezn
107
velumetris la + 13 1) (i. 1, 12 (B)
108
Pirst Kiola-Kirchhaff cozotaced velunotric (B,2,3)
111
112
1 Kirchhoff aczesa tao P.
P.cranggose disshbof_xxtewe Lorch-netimal (P, (8,3,3)
115
116
return rirchhoff. Saathhoff_stteeeee rerss
H.2 CLAYCAT
In the ClayCat scenario, the constitutive law inferred by our method is presented.
Amport toech
import torch.cn as an
PlascicityModd:lmm
7
def init_ (sols, ylold_stress: float 2.15, sheas_nedelus: float 28.01:
8
Dafine trainable continaoua physlcal pazanacers difforantiable optinisation. Initializa with best valcas from iterative feodback.
10
11
Aros:
yisid_atress (flont) yisld stress thrashoid for plastic fiow. shaor_nedolus [float]: sheat modulus for plastic correction,
supar in:==_
welf.yielddttess mn.esaameie (tarch. Lemsus (yield_streas) self, shar_ an.Farameter (torch, tenser (ohearr
19
Eornardiseit, Tenaue] Lensh.
Computa carrected deformation gradient from defermation gradient senser using von Mises plasticity
logarithnic deviatoric priacipal strains.
Ares:
26
(sorch.Fensee) Censer) deformation gradiant tenser (8, 3, 3).
Returns:
P_soxrected Tensol: corracted deformation cradient tensor (B, 3, 31.
deformation ion oradient siema, Vh. torch.1iaciq.svee (V)
(B,2)3), aimma: (B,3), Vhi (B,3,3)
signe chagggginnnnnnneeeeee (signs, ls 61 4 clamp Le prwvent loy101, 18,3)
34
Compute principal logerithnin strains
spailon Locch.loy (xigna) (B,3)
4 Volumetric (seen) strain spsilon_mean spailan.manttiamll kespdin=(euue)
a Devistorie straing spallon_dev epsilon spailor_pean 4 (8,3)
42
Norn deviatorie strein
dewiloo__de__nuse spai donn_dde. (dinel, keepdineTrue) r Sen12 1B,
plasticity paramsters
nunerical
yield_stewa Lozch. clanp (et.:/el..ylllllaeeeeee 1s=6] shear modu us earch. cllap, min (sell dself.ate syshear medeeee.ndolllss nodelus, 1e-63
muitiplier
del germa epai dev. norn yield stress / shear modulux) # (h,12
28

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Published as a conference paper at ICLR 2026
selta_gama__or ceech.clanppniie aateeppaaad/aaaaaaaaaaaaaaaaa ganna, 0.0) (2,1)
55
1 Correct deviatoric streine by return mapping it yiclding oestionncorreeted
[delta_gammapes essilon_dex_nom apailon_dev
Where not yiclding, keep original strain
39
yieldingnnask (delta_aamaa vievv/1-,, 3) [2,1) spstion_tinal
jyaid sptloor_cotrsetdd, epsilon) (2,3)
1 Raconstruct corrected siacular values and defermation gradiant
63
ciema_corrocted torch.cxp (spriiio__tiaaa)) 1 (B,3)
sleg_ Lord. www.gaaadddigggn......eeoooaoe
(8,3,3)
E_corraeted torch.nstmei (0, astaal (dwop_ Vhji (8,3,3)
E_corrected
70
71 clags Elasticityltass Inn. :odule);
__int___oel,, younga nedulca__gg float 11.7, poisaona_ io logic: float -0.7)
75
Define irainable cont inauua physical pacenelacs for di lable opt iniinisation. Initialize with values inforred fzon analyala.
78
Arga:
yoddes _od (float) log of Young"s a nodulus. poissone, loggi (Float) pre-aigno'd paraneter For Paresonfa ratio.
super ( B. inis_
self youngggnggguu. modul ux. log or ar.Paraneeer (torch, tenssr lyoungs uodallu,, swlf. wtw....ossuusaaa.
logit. na.Pasateees Lenauu (poissuns pocioo______oooti
def feewardize f, Fs torch.Tensprl >> porch,/enxor ane
Compute Kirchhoff strees Kirchbaff clatticity.
deformation gradient tensor using
39
(sorchh Pensor) deformation gradient (8, 3, 3)
93
Returns:
94
2i kizchhoff atress Itorch, Teraus) Kizchhuft strwas Lensor (8, 3)
device = F.device
F.dtype
Compute Youngfa modul from log
101
younge_moote modolus torch.expies expieelttiieeee. younggs..t.o............ www..us..uuuee. log) scalar
102
Compute Poisson's s retio erom simmed scaled to (0,0.09)
104
pot pciesons_reee retio torch, si woelddeel.....o.m t poi setio_logit * 0,49 scalar (0,0.49)
105
106
msangs_ooculus (2 youngs. moculus poiesons, salssonssrationn scolar
poine ratiol
poise ratiol]
108
100
Identity expanded batch size
torch.eyo12, 12, dtype-dtype, deviccl Junqqueze (0) (B, -1, -1) 1,,3,3)
111
112
Right Cauchy-Green tensor - P1T
113
(. 3,33
tosch.natmul (8,3,3)
115
116
Gzocn-Lagrange strain B 0.5 (C
117
119
Tzacs of conputed by summing disronal aiemente viex 12 (3,111)
120
trB E.ciagonal faiml=l,
121
Sscond Fiola-Zicchoote strose cencor S nu E trB + (8,3,3)
123
124
125
Pirst Flola-Kirchhatt torah.natmul1f, 1,,3,
tensoz F.8.
127
128
+ Kirchhott screes tensor tau P F'T kirchbatt_streaa terch.natmal (P, (8,
131
return xischhott_stxees
H.3 HONEYBOTTLE
In the HoneyBottle scenario, the constitutive law inferred by method is presented.
1 Import Eorch
import torch.on
29

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Published as a conference paper at ICLR 2026
cless Plasticityyfdci (nn.Kodiee))
init_!
self,
youngs_rodulue moduiusuodos log: :loat 6.C, peisaona_tation http:/aaaii.unonaaaaaaeeoo float -1.0,
10
yield_strass: float 2.2,
1:
14
Flasticity model with loçaritanic scroin return napping.
Ares:
modiles 1eg (flozt): log Young's modules. goiasons_rate wenoonenneaaaan (floal)s unconatiad trained acalac for Poisson'a satio. ytsld_streas (flost) stress threshoid.
saper (3 i (nis_t)
22
solf mtt..yyanppnnouuu_..oo ar. (torch. tensor peung__nedblu__lgg) scalar sorrch.tessus self.yielLatrees an.Prramczoe (torch wtorch.tensoo (traas
self. wow.ssus utt.
scalas
25
Fornardtanit, Lorch.Tenaoz] Pozeh,
Dampate oor reuted defornat fon gradient Sron defornal lon gradieat Lemoor.
Argsz
(Locch, Demaar) deformation gradieni Lenser (B, 3, 3).
Returne:
P_ouroected fLuzch. [Lnach)Temaaz]] cocrected defornation gradient 3,
moungs_ndddlug sel youngs youngg modulus log. log.exp() soslar
poisaons__atii Luzch. signoid/ pow.
0.49 scalar in (0,
stred_ self-yield_stress scaler
moculus (2.0 (1.0 ssons, ratio]]
signa, Vh Lorch.,inalg.vvdee Gell_matriowweabsee (B, signa: 18,3), Vh: 3,
43
Clarp singular values
collapse
signe el slamped torch.elang, mir (signs, 1e-41 (0,3)
4 Logarsthnis strain
50
4 Volumetric strain (trace) spai loncerace epai inn.aunjdimee kempdine"rue) (h,1)
Devistoric strain losilon bar bar egailon apsi / 3.0 (9,31
epai bar_ bar_ torch, norm torch tornl lon bar, dinel, Keegdin=Truel 1e-12 08.1)
4 Plastic nultiplier
tomt apciion_bar_nocee yield_stress (2.0 (B,1)
4 Plastic factor (clanped)
plastiz_factor torch.ciamp.nin taanp_niiitolittsemmm bor_eon, 0.01 (B,3)
Correct 'ogarithnie strair
66
opsiion_carroeted apsilca plastic_tactor cpsilon_bor (B,3)
singular
coma__orroctoe exp
(B,3)
70
Recerpose corrected deformation gradient
[_carroeted torch.matmul (U, torch.matmol (terch. disq_onbesiicion cor.eectte. VE) +3,31
73
F_correctod
77
class Elasticitylode inn, Module);
init_i
selt.
81
monouus_ddoluu__ooo log: float 11.7, petconn__attoo_monntttot float
85
Corstated Elasticity model wish eroinable shysical parenstare.
Ares:
youngs youngs_nodules moduiessdddesseecooo iio (floztl: log Young's modules.
poiasons_ratieeeooo wnw.oouunnonttraancooooo (float)s unconatrained acalas for Poisson's ratio.
cupar() -)
30

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Published as a conference paper at ICLR 2026
SaaE-yyongssnodaee unoduuts 1og
ar. (Lorch, Lensor fyeungp_noduuus _odullus_ scalar
self. sttoono__sss_____oo__o_ ion.soooooooonseeeaed na. Pa.Paranttor (torch.tensoo tensor(
poissene_ salssone_ettoo_uncconee wtt..ettioouccoosreeesssii scalar
forwardteali, torch.Jensoz)
TenaeE:
97
Computa Kirchhoff acresa tersoz fzon decornation gradient tensor.
Azgaz
100
(torch.Pensee) deformazion gradient tenser (B, 3, 3)
101
102
103
Xizchhoff_stree (torch. [torch.Toraoz] Rizchhofe stress tensor (B) 3, 3)-
101
105
youngs sel:. yooogss_ndddl wodulas wa log.exp() acalaz
106
peisaons_ratio torch, aignoid 0.49)
0.49 sealar
107
108
younas_nodails / (2.0 (1.0 polssons_ration
100
youngesnnnuulu poissons_rattio poissons_ratiof)
polseonsraatio)
110
111
signe, Vh. 3incll (oll_netrionanaalbel 4 18,3,31, (8,3), (6,3,3)
clanp singuiar values for namerical stability
114
plarped Lorch..languuin Ceign, Geigna, la 51 (B,
115
117
Loceh.matnul (U, (h 8 (8,3,3)
118
119
P.tranagose(), (5,3,31
121
Corotated aczesa:
122
porotated at 2.0 ma torch (P B, Ve) (0,3,3)
123
Compute determinent product singuler pordh.prod//iggna.. alggadgd dinel] 4 18,1
126
3 J.vient-1, 1-, 1, 1) 18,1,1)
127
129
Lozch.wye13, diyyee7 diypp,, devicwe ddevice)) unaqueese saw.eeeeeeeeeeeossseeeeee. (1,3,3)
130
131
volume_xtioee la (J 1)-vinet 1, (9,3,3)
133
Piola-Kirchhoff atreas cocul atdd vol vollmmattess 4 18, 3,
134
135
kirchhoff torehnnatmal (P,
137
138
return circhhost, stress
H.4 JELLYDUCK
In the JellyDuck scenario, the constitutive law inferred by our method is presented.
Lozch
import torch.cn
class PlasticityWadel Kodalees
7
dez Initt_ fswlc, yisld_atresa: float 0.1, hardening: Cloat 0.C)
8
Define urainable continaus physical paranelers different iable inizaaaii Initialize yield atzesa and laotropic hardaning paranstars.
12
Argas
yield_atzess (float) yiald threshoid plaatic cosrection. hardening (cleat) isotropic hardoning paranator.
16
super () init_
self self.yield.no atrees bard.bagg mn.Perameee (Larch. Lensus (hardening))
(torch. (yiald
paraneter scalas paranstes
19
20
def forxardizn" f, Fr Tensor) torch,"enxon Penxor:
are
Computa corracted
gradient using von Hisea plastieaty zecurn naspine.
24
Args:
deformstion gradient (B, 3, 3)
Belurns:
are Pocorrected [torch.Tensor]; Tensori corrected information gradient tensor (B, 3, 31.
P.shapw[0;
4 500 deformet ion gradients P Bllggtiggal Vi
nigno, torch.lineig (V) 0/ (3,3,3), signa: (B,3)
31

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Published as a conference paper at ICLR 2026
Clanp singular
10g(0)
sisma_stamped terch. terch.ckamp.nit (signa, 1e-51 (B, 2)
37
Legarichnie straic opsiion (stg:llsosssssttttaio
(B, 31
41
1 Deviatoric strain: subtract mean (volunetric) scrain
spstion_msan epasien.nean ksepdin
upation_ãev epeilon spailor_nean 1B,
45
1 Norn of deviatorie strein
hespdinnTrual
Eirestive yicld threshold with hardening, clamped to positive thlld_treehhoe
49
clan.c.lannnnneeell (ta.e/lllyyyltatteeee. sell,hacdening, 1=-8) soalar
Plastic cocrection fector (return nepping!
52
ganna = www.caanggiieepaaooo...... dev Chreathh 0.0) (epsilon_dee_nosnn le
-12) (3,1)
Correct apsiion__dvooorrettte epsilor_ddv (1 canna) 1 (B, 3)
56
Reconatruct corrected logarithnic opsiion_corracted apsiicn_dev_correee
epsiion_nean [B,
59
6
1 Exponentiate to get corrected singulaz values
sign__ooreeten torch.exp (epeleeelloo oogeilooo.coo concorrected) (8,
1 Raccopose corrected deformation gradient Encorrected torch.matmol (0, torch.netmal (torch.die diw...dddig.... diw..d inbeddiiggnnn.....mmm... inttedieigne.coreetteiie corrected), VH31 (%,
64
66
return corrected
67
cleas Flast tylde Kodule)
70
71
def init............................ Innlf__ youngs, modu log: Flost 11,49, poi soinssons ratio signoid: flost 1,00)
befine teainable continavua physlcal pazaneters differant iablu optinisation.
Initialize tre with previous best val values.
Arge:
youngs modulos log (flost): log of Young's modulus.
78
poixions_rat watpe/saaaaaaaaaaaeaaaae (floas)s Pul ratle before xignold Lrana Cormt
super
well.yoouns.mmuull log an. (turch, Lensur (youngs moggllx._ogg)) soalar pelt.po.esons_ cessoosssattiiooo sestorrs_attoo_sigooiti. signoid En. (torch. teneor tener (poiseone, aacoooeatto_eggeeeeeeeeeee soaler
det forxardiee.t, Ft torch.Tensori torch, Jensors
Compute Kirchhett stress torsox using Corotated clasticity model.
Arges
39
(sorch.Ceneorl) detornation gradient tenser (B, 3, 3).
Roturns: kirehhott_stree Itorea. Torso)) Rizchhott strese tencor (B, 3, 3).
93
7.512310)
Recover physicsl parameters
97
goungs_nodoluz yodngg leg. sezlar pasitive rettionssrrtioo sel:. pott-poossos_rttee crroons_rrttoooooo pat.osrrttto_oomii...oooo sigmoid() 0,49 scalar in 10, 0.49)
100
yourgs_ modulus (2 11 poissons_ration f (seslari
101
youngs moculus poissons_ratio / (11 polseone_rstiann
ratio))
(scalar)
102
103
SVD
104
U, siema, Vh. = torch, (vd (F) 16-51
(B,3,3), (3,31, (B,3,3) singulas values
105
carch..iamp_niniiiioe
107
Rotation matrix B U.+
108
R tosch.matmul (U, Vh) + 18, 3,
109
Desecminant product singelaz values
111
3 poochhproddfiggm, dim-l), viowi-,,, 4, 1) t (3, 1,
112
113
Identity natrix
torch.eye 13, dtype Rewice-F.eevien unsqueeze ixpand -1, -11 t (B, 3, 3)
115
116
Corotated clast Kichhoki
P_ooret
117
au_axpandad mixviex 1) [B, 11
118
F_corot 2 miLoxpanded (3 R) 1 (8, 3, 3)
119
120
Fliny torch. invi: (B, + (
4 Volume
32

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Published as a conference paper at ICLR 2026
P_inv_t (L,nvtteanspose,,, 2)
123
velure_tastor (-., 1, 11 + +d - 21.viavt-1, 4, 1) (2,
124
F_yol " volume_tactor + < F_ing_7 + (B, 3,
125
Biolo-Rizahhorz
127
F P_oozot ELVc1 (B, 3,
128
129
Kirchhofr
0 FT
130
Le (8, 2,
131
kirchboct_atteaa torch.natnal (P, Ft) (8,
132
kiachhuIC_attess
H.5 RUBBERPAWN
In the RubberPawn scenario, the constitutive law inferred by our method is presented.
import torch import torch.on as nn
cless PlasticityMode Inn. innuledul
_niit_isols. Iselt, yicld_etress: tloat 0.22, mu_icg: flost 6,011
Define nable cont contimmus physical paraneters di ffrrent leble opt ininat fon. Initializs yieid_stress and plastic thear mosulus (mc) in log space.
10
Args:
yisic_ctress (tlectls yiels stress contrelling plastic shrechold. mo_log (tlcat): log shear modelus for plastic correction.
...
supert) 1nnitt
self-yield stress nn. PmrPaameeerrttorr wtwrchittrsssseeeeeeeee tyield stress) scaler seli.mu_iag Po.oorancorr (torch, tensor (mu_icg1) scalar
18
forwardtanlf, torch.Tensoz! tozch.
22
Compate corrected decormation grasient crom setornation gradient tensor via logarithnic plasticity.
Args:
25
(corch. Censer) doformation gradiant tenser (8, 3, 3).
Returns:
2_sorrected (tosch.Tensoel: coeracted deformation cradient tonsor [B, 3, 31.
29
s.shape[0;
30 solt.mu col..mm_leg.cxp( scalar
33
decampositien
signe, Vi. 1orch,
3,,,, aignas (B,3), Vhr (B,3,3)
37
1 Clamp singular volues
Lorch.clangi Larch.clanguninnsiign 1w-61
Logarsthmia principel strotchos
41
spallon corch.log (xigna) (B,3)
Compute velumetric near opsilon
mpailon_man apailon.manndinee ksepdim=Zruu) (B,1)
45
Decistorie log
bastion_bar apsilon spailor_nean (B,3)
1 Born of deviatocie strain opstion_bar_noo
aar.e...iiia....oooooiottooo dimei, keepdineZrue) (3,11
a Me iptipplis
53
gamma apsilon_bar_norn self.yiold_atzes / (2 ma) a (B;1)
Clamp to non negat ive
baita_gam__caappe torch, Corch.ciamppiiniillta...oom 0.0) (B,1)
Avaid division by
danon epsilon_bar waw_ooo__rrommmoaaee wtw.. (1a-8) (3,1)
61
1 Compute cocrect lon seals factor
germa_alealoo alpee
1 No corsection fon 12 yield cordition not suxpeased scale xheehe
65
> sesle, toreh.ones 1teeeoooee_..keeeeellllee liketecelet) (3.13
Apply corsection
apsi lon bar bar corrected lor har scale (9,31
Recorpose
log
epsi lon corrected epxilen corrected epaillon nean (B,3)
33

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Published as a conference paper at ICLR 2026
Invezse leg to get corracted singuler values corrected. torch.cxp corsecttt (8,3)
74
Roconstructod corrected deforaation gradient
E_correetdd witcch.dacccmmeecc (eigmnscceeenneereeeess Vh. (B,2,3)
E_corrected
82
class Elasticitylta:: Inn, :odule)
__ii__ Iselt, nounsa_ndovca__eee float :2.9, peissona_.aciio.sggook float 2.01:
Define izainable continauua physlcal paseneters for different lable optinisation. Initializs
from best prioz estimates.
Args:
90
moddes_nodlle__oon (flost): log of Young's a nodulus. polieons_ratio_siin
(flows)s Poisson's paranecer before signoid lasell Ligg
super () () ini::
94
self well--yoogguuunii modul ar.Parameer (Lurch, Lensur tyoungs_nodul moduulo logl) soalar self walf.posssons.ee iot_ggggoo en.Faranstez (torch. wor.iasos_ratttsggnneetto.
97
def Torsard Et torch.Tenaoe] Zenace:
Compate Kirchhoff abeeaa Fron corswoted deCurnation gradiens Lenauz using wlastinig
100
101
102
(sorch.Fenser) deformation gradiant tenser (B, 3, 3)
103
104
Belurns:
105
(9) kizchhoff. strress Tersor) Rirchhoff
107
B [.s
108
Physical paraneters
youngs_nodulux wtwp://g___olluwooooooooooooo
acalaz
111
112
1 Signood napping to 10, 0,499) for tains ratle
poissons_rette
signoic)) 0,499 scalar
115
youngs modul / (2.0 (1.0 pai rati auslar
116
youngs_ moculus poissons_ / (11.0 poissoos_raatis poi ratio)]
118
I toroh.eye 13,
seriiceFF.deiion unsqueeze 4 11, 3,
119
E.eranapose (B, 3,
121
122
Right Cauchy Green tensor torch.natmullFs
Cruchy-Green
123
Green Green-llarangg lagrange strain tensor
126
E 0.5 10 It 1B, 3,
127
129
trs E.ciagonal Idiml-1,
(tnee))))mmmmmmm.n....mm vieexiB 1, 11 (B, 1,
130
131
Szcond Ficla-Rtrchhort mu
133
134
First Hlols-Kirchhett stress tensoz torch.natmul1f,,
135
3,
137
Kirchhoft stress bensor: tau F FI Eirshbort_streen torcn.matmal (P, Fti (8,
138
139
seturn kirchhozr_ptree
H.6 SANDFISH
In the SandFish scenario, the constitutive law inferred by our method is presented.
Import Eorch
import Lorch.on
class MistiicityMaae Kodule);
6
-intt_ [swlf, ritee Float 0.07)
9
Define trsinable plastic yield streas paraneter with enforced numerical stability.
10
Args:
yield strees (flastle yield strees contrelling devistoria plastin flov tagn
34

---

Published as a conference paper at ICLR 2026
super 11 10---11___111__
15
celf steisssteee wt. tensoor (plcld_strese)
fornardicali, torch.Jensoe) Tonaee:
19
Computa paaticity--orrectee deformation gradient by ahrinking deviatoric tonarithnic
strain.
Ares:
22
Storch.Ccnsere doformation gradient tenser (8, 3, 3).
Returns:
P_sorroctod, Storch.Tensocl, corracted deformation: gradient tonsor [B, 3, 31.
26
*.*
decanpositien
U, sagne, Vh torch,1inalg.vve()) 3)
(B, 3, 31, (B, 3), (B, 3,
29
singular sianped cerch.elarp_nin (signa, 1u-6]
stability
(B, 3)
4 Compute itgee principal strain opsiion corch. (aigra_
33
36
1 Volumitzi pact (nean)
apstion_mann spailon.meanddinr. keepdimeCrue)
4 (B, 13
Devistoric strain
40
epsiion_dav = apsilon epallor__ean
(B, 3)
41
Norn deviatocie strain
epation_day_nom torch. wwrall.nornnnnneeoeooooo (ornnnpsiooooo dim=l, Krepdinetrue) (2,
45
1 Enfocoe ninimum yleld stresa avoid muter inal Los Lobiabili yield
torch.olanp min feyelll streas, 0.051
scalar
1 Clamp norn for division.
lon dev safe torch.clarg min (eps! lon, dev 1e-12)
Compute plastic cocrect nagnitude daltagganna
de it game = epailan lon dey sorn - yield stress palta_gemm__aaapee
4 18, 11 4 18,
53
0.0)
(8, 33 Scaling factor for deviatoric strain correction scale = 1.0
mer.gemmmasaaaeee / wpw.llon_deennorn..ee
scale torch.elang clanp_minrecale 0.0)
Apply plastie perrection to deviatoris strain
60
epsilon cev_corrected epsilor_dnee dev ssale
f (B, 31
61
Recombine voluetricc deviatorio parts
epsi correctcd speilln_mean epsi Los__eoeeev dev_corrected
18,
Csiculato corrected signa correctes torch.exp. lepeilon corrected)
68
Reconstruct corrected decormation gradient corrected torch.diag ented corrected) 8 Vh
return F_corrected
72
ElasticityMode :
76
dof init__isels, (selt, youngs_nodules_log float 9,35, siissons_catio_ssgmoote flost 2,30)
are
befine trainable Young's
Poisson's with phys Ty resliatic boards.
80
Arge:
younes_nodiies_tt _id (flozt): Logasithn of Youne's modulus.
poiseans_ratie ratte_rrtt_____sis____ (eloat) saw pasemeter be passed through signoid Polesor's a ratio.
83
super i)
selt.youngs, moculus, log ar. itornnc (torch. tensor lyoungs modolus logl)
Fn.Farancter teasch. tenser (ptiocon._otttsssseeeooo
87
fersardiaatt, Tonao. Cosch.Jenses.
Computo Kirchhett stress tersoz Ezon deformation gradient with corctated clasticity.
91
Ares:
(socch. setormation gradient tenser (B, 3, 3).
95
Returns:
kizehhoff_strese [torch, [Loach.Tenaoz]: Kizelhoft atress
B s.chapo[0)
99
100
Recover naterial Dolf.youngs_ncdu11a
scalar
35

---

Published as a conference paper at ICLR 2026
BaLzav aeI_-polasons_ratae ratio (imio
(0,2)
103
na ERLE3M + 0.45
scalo to nax 0.45
Poisson ratio ("staolc and compressibist
104
105
(2.0 11.0 ad):
acalar ocolar
106
lan E nu / (11.0 ac) (1.0 2.0 no)
107
108
109
sagae, Vh Lozch, (ardh.1inag..ee(( 3)
(B, 3, 31, (B, 3), (B, 3,
110
111
singulas stama_stamped sarch.elarp_miic (signa, 16-6)
peevent munesical issuua
112
(B, 3)
113
114
1 Compute rolation part R U.R. R
115
3,
116
117
3 Expand au toz broadcaating
118
12 ma.din(1 > 0:
119
muuaapended mu.vient 1, elaes
(8,
120
121
miLoxpanded nu
scalaz
122
123
Corotated cerstatod_attosp 2.0 aia_axpanded (2
124
[B, 3,
125
126
1 Compute deterninant and clamp foz stability torgh." t det I_elanped clach.clanp.nintt.
127
128
(B.)
129
130
4 Identity tensor 1 (,, 3, 1ysdh diypeed ddype,, devicwe devvcee) conauueeee
131
3,
132
133
3 Rxpand and reshage parametere for broadoast ing
134
if lan.dint) 5 0z
lam expanded lan.vieut-ly
EB, 1,
elaes
137
lantespanded lan
3 scalaz
138
Expppnded d_elanped 1, Iininu expanded (J_clamped 1.0) viiettl,, 1,
1, (B, 1,
140
141
142
4 Volumetele streens lambda 3 (
143
wwwetrri__tteeeeeee lan_expanded O_oxpandess J_mänus Lxxpanced (3, 3,
144
145
Pirst Piola Kizchhoff streas corotated stress volimerrie_attee
146
3,
148
Transpose defornation ion gradient
149
Pt F.transpose 11, 21
(B2 3, 33
150
Kirchhott serees boncor: kirchhott_co strres 8
F'T
152
(B, 3, 33
153
154
kirohoot__trress
H.7 BUN
In the Bun scenario, the constitutive law inferred by our method is presented.
import toech
impost corch..n as na
class Plaacieit Kodale;;
6
def init__ssess, ytald_stress: float 0.30)
7
Trainable concinuous yleld streaa pasanatez for Misea plassielty correction.
10
Args:
11
yiold_atreas (Cloat) yield atress thrashold for plaatic correction.
super (n
14
seif.yioldLsteess an.Pasanezer (torch.tensor (yleld_streas)
15
fornardionlt, Lurch.Tenanee Penser:
18
Compate corrected deformat ion gradient From Input, deformat fon gradient benaor.
19
Arge:
(Gorch.eemaer) Denwar); i deformesion gradient tensor (B, 3, 3).
23
Returne:
P_ourrected Mtorch.Tenane) corrected defomatiion ion gradient tensar 3, 31.
36
4 Compute sVn of Y: U. signe, Vh. vagne, Vh. torch, lin.lg. (P) (8,3,3), signas (8,3), Vhr (8,3,5)
Clemp wingol ar velues to avoid (og(0)
36

---

Published as a conference paper at ICLR 2026
stana_etamped tarch. clarp (#igna, 16-61
(6,3)
32
+ Compute Legezithn ot singelaz valuos (principal logarithase straine)
opsiion torch. tagtainnnsaaapeee
Compute velunetric neat strain
36
upstion_man spailon.mean (dantdimi,, keepdimeCrue) F (2,1)
Devialoric strain (deciation awan)
apstlon_dav apsilon epallor_mean 1 (B,3)
40
Norn deviatorie strain, dey norhh.oom(esziion... dinel, kespdin- cta. (B,1)
avuid namerccal
44
1 Compute plastic meltippier (excesa over yleld acresa)
dalta_gama aosilon_dor_oom self.yield_stress
a Asply plastic correction only if exceeding yleld stresa
48
dalta pamaaammm_aaaaapa corch.clamp_ttnreatte.
0.0) 4 (8,2)
shzink factos daviacorio straina cheiak_faccer 1.0 gatta_amaa_caampe epstlon_dev_oon (B,11
52
Correct deviatorie strain projecting vield auzfase ssstion_doycoorrettte epollor_dov shrink_factor (2,31
1 Beassenble corrected total logarichnis straina lon con_cceedt apailon_mean epai don_dey_ dev_ecoree (h
Exponentiate to gst corrected singriaz values com_ecrre torch.exp weplepalii lepeiloooonnnoo con_eonnettee (8,31
Raconatruct corrected deformation gradient: P_corrected compooooimooorrrrsss Vh
63
Pecorsected 6 Lozch.ddaag diw... (signn 8 Vh. (,,3,3)
return P_oorrected
yoite Kodulee;
Init Snnit_/sllll [sulC, young nodulcua__oago
9,82, coisaonn float :.07):
70
71
Trainable oom insoux peramebers for Xeo Bookean leity.
Arges
youngs modullu Loy (Clost): log of Young's x nodulus.
poiseene ratio_ ontio_oignoid Itloat): Foisson's ratic parsneter betore nignoid scaling. s.e
super Anis_
78
selt selt.youne youngs moculus, log = Pr..aareeeettorreee... wtttoohteessoorrr...ee. tensor lyoungs, medulued__oogoo self
signoid ParParaeeter (torch, .tenaor (poiseone, ratio_ xiggniid)
det forwardiee.t, Ft torch.Tensori ForchhJensoo
Computo Kirchhett torsos tron decormation gradient tensor Noo-Hookean elasticity.
Arge:
(torch, Zeneari deformstion gradient tensor (B, 3, 3).
Returns:
zizchhoff stress Terson) Ri Ricchhofé stress teneor (3, 3, 39.
...
F.s1ze10)
93
Compute Young's medolue 3 and Poicson's catio nu selt.younge_nne wtw._ncddus__oooo f.poissons signoid.eigmidd))
soster in (0,0.49)
97
/ / @
mull i scalar
scclar
100
Isentity tensor 1 (,,3,33
101
I tyzch. dtypor?. devicerF. dovicce) (asguesze cxp -1, -11
102
103
Compute determinent of (B,]
104
3 inelg..iee (F) waa....oannnnaaeaae wit.:le-12.. 1, 18,2,11
105
leeJ iogch. (,,
107
Compute inverse tronspose ot (2,3,3)
108
PLiny iarch. è (3,3,3)
109
F_iny_r E_inv.transppsel1,
(B,3/3)
111
Compute fiest Piola-Rirchteff stress tensor P (B F_inn_T IenrlogE-a_iod B mu (F F_inv_Ii lan 169J F_inw_I (2,3,3)
112
113
Compute Rizchhoze stress
115
Ft i.rronepoce 21 3,,3,
116
KirshlaCF_acteas tarch.natnal (P, PL) (8,3,3)
117
soturn xischhott_ptxees
37

---

Published as a conference paper at ICLR 2026
H.8 BURGER
In the Burger scenario, the constitutive law inferred by our method is presented.
import torch import torch.cn as nm
cleas PlasticityMode R
_iiii____onn isoltii
...
Identity Pi ass no correct for to deformat gradient,
10
super init__.
det formoriien.t, Fi torch.Tensorl torohiTenser:
Args:
(corch. Teneaer) setormation gradient teneer (B, 3, 3)
18
Returns:
(torch.Tensor]: corrected deforaaiinn gracient tensor 3, 31.
No plastic ccerection
22
return F (B, 3, 3)
cless ElasticityMode Module)
26
inst Iselt.
nounla_nedolus float 8.37,
polsscns_ratior floaz 0.491:
30
Corotated clasticity with trainable
Args:
34
nounes_nodilog_ogg (float): log of Young's a nodulus.
poissans_ratie (float): Poisson's ratia (clamped (0,00.))).
super () inis_()
38
solf mow.:uunooooccuu_oorrr Paremetrt (torch. tensor yeungs_noduusse_ogg (poisscra_ratiat)
swif wel..po.ssons_satiio Paranetee
def fomardiesnit F: Tensoe +> Tenser::
42
Computa Kirchherr tersoz fzon decormation gradient
Ares:
46
(corch.Censee) Conser) dofornation gradiant tenser (8, 3, 3).
xizehhoff_stres (torch. Torsoz) Rizchhote stress tonsor (B, 3, 3).
*.*
P.shape[0;
Physical paramelera E rounes_nedolus_log toochh (sel w.poissonss_reeee
54
scalaz
0.49) scelar
nu -EV (2.0 11.0 ac)) scalas
R. nu / (0..
(1.0 2.0 nut) scelar
Fx Signa, Va D, signe, Wh = torch, Hvv avd (P) U: 3,5),, signa: (8, Vh: 18,3,3)
(Biqna)
62
signa corch.clang.nintee (inngnnnannnnn 1w 4 18,3) snsure posilivity
Rotation
R tosch.natnul fil, (h) (8,3,3)
Corotated part:
6.
3, t = 2.0 mn borch (F B, Pt) # 18,3,3)
Volcmeteie part: tad.v Lamoda torch.prad (iidg simel).xiewe 1, 19,1,1)
Losch. 13, diygee? daype, devicw=F DevioweF.deviooe sunagueeee (0), exgand 1B, 1, 11
v3,31
(N,3,3)
Kirchhoft streas
kirchhoff Sttthoff_xttteee rettress tAiLO
3,33
return kirchhoff. stress
38


Reviews:
[{"review_id": "ospPyuwakA", "reviewer": "Reviewer_HDyU", "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.", "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.\n- **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.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 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.\n- **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.\n- **S.3:** Reproducibility. I appreciate that the authors released their source code, additional videos, and all LLM prompts.", "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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "umuwzVFtMi", "reviewer": "Reviewer_aJkm", "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.", "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.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The presentation is straightforward and easy to understand. \n\n- The discussed topic—inferring interpretable intrinsic physics—is fascinating.\n\n- The experiments conducted are thorough and comprehensive.", "weaknesses": "Please take the time to read the following two papers:  \n[1] \"LLM and Simulation as Bilevel Optimizers: A New Paradigm to Advance Physical Scientific Discovery.\"  \n[2] \"Neuma: Neural Material Adaptor for Visual Grounding of Intrinsic Dynamics.\"  \n\nIn 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].", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OvzPqc9MVm", "reviewer": "Reviewer_9Waz", "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.", "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.\n\n- 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.", "limitations": "", "rating": 2, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 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.", "weaknesses": "- Since the upper-level optimization is the main contribution, a more thorough examination is needed:\n    - How many upper-level evolutions are need to converge to the correct one?\n    - 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.\n    - What if the LLM outputs python code with bugs that can crush the program?\n    - 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.\n\n\n- Although the high-level ideas are clear, some details are lacking for reproducibility:\n    - What (modalities) are included in the feedback from the lower-level optimization?\n    - How is the first batch of constitutive model candidates are initialized? Are they generated by LLM based solely on multiview videos?\n    - How are the initial physical parameters set when the predicted constitutive models change?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kHeymIcifM", "reviewer": "Reviewer_cSP2", "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.", "questions": "Do the learned laws transfer across different objects with similar physical properties in real scenes without re-search?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 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.\n2. Implementation details are transparent and the source code is provided, largely promoting reproducibility.\n3. The paper is clearly written and easy to understand.\n4. The comprehensive experimental results are solid, proving the effectiveness very well.", "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.\n2. 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)