NeuROK: Generative 4D Neural Object Kinematics

Over the past decade, advances in 3D vision and deep learning have revolutionized static shape reconstruction and generation. Early methods relied on explicit physical models, such as finite element methods and particle systems, which provided high accuracy but suffered from computational inefficiency and limited flexibility. Recent data-driven approaches, including Graph Neural Networks (GNNs) like NDG and implicit representations like CANOR, have improved the ability to model shape deformations and articulations without explicit physical parameters. However, these methods primarily focus on static shapes or category-specific motions, lacking the capacity to generate diverse, physically consistent 4D dynamics.

The emergence of transformer architectures and large-scale datasets has opened new avenues for learning complex geometric and motion representations. Nonetheless, most existing models either require category-specific priors, physical annotations, or are limited to simple deformations. The challenge remains: how to develop a universal, category-agnostic framework capable of simulating realistic object dynamics across various materials and deformation modes without explicit physical supervision?

The core challenge addressed in this work is to enable the generation of physically plausible 4D object dynamics without relying on predefined physical models or category-specific priors. Traditional physics-based simulators depend on detailed parameters and assumptions, which are labor-intensive to obtain and lack scalability. Data-driven models, while flexible, often produce physically inconsistent results and lack interpretability. The key bottleneck is how to learn a low-dimensional, physically meaningful representation of object states that can be evolved over time in a manner consistent with classical mechanics, especially when only geometric data are available. Achieving this would allow for scalable, generalizable simulation of complex deformable objects, facilitating applications in robotics, animation, and virtual environments.

The primary innovations introduced in this paper include:

• �� Learning a low-dimensional, data-driven kinematic space using a transformer-based variational auto-encoder, which encodes static shapes into a distribution over possible states.
• �� Incorporating classical Lagrangian mechanics by defining an energy function over the latent space, enabling the derivation of dynamics via Euler-Lagrange equations.
• �� Eliminating the need for explicit physical parameters or category-specific priors, relying solely on geometric supervision from large-scale 4D datasets.
• �� Designing a physically inspired, end-to-end trainable framework that generalizes across diverse object types, including elastic bodies, cloth, and multi-body systems.
• �� Introducing a dimension reduction strategy via active subspace methods to improve efficiency and stability, making the model suitable for large-scale applications.

This combination of deep learning and physics principles results in a versatile, physically consistent simulation framework that surpasses previous category-dependent models.

• �� Data Collection: Curate a large-scale dataset of 4D geometric trajectories covering multiple object categories and deformation modes.
• �� Kinematic Space Learning:
• �� Design a transformer-based encoder (cond) that processes static meshes to produce a prior distribution over latent states.
• �� Develop a variational auto-encoder (VAE) that encodes deformation fields into a posterior distribution, capturing plausible deformations.
• �� Train the models jointly with a variational loss, including reconstruction and KL divergence terms, to ensure expressive and smooth latent spaces.
• �� Physics-Inspired Dynamics:
• �� Define a Lagrangian energy function over the latent space, incorporating kinetic and potential energy terms.
• �� Derive the equations of motion using the Euler-Lagrange equations, which govern the evolution of latent states over time.
• �� Implement numerical solvers to simulate the trajectories of latent vectors, ensuring energy conservation and physical plausibility.
• �� Sampling and Generation:
• �� During inference, sample initial latent states from the learned prior distribution.
• �� Propagate these states over time using the physics-based dynamical equations.
• �� Decode the latent trajectories into deformation fields, which deform the static shape meshes into dynamic sequences.
• �� Model Optimization:
• �� Use energy-based regularization and geometric supervision to refine the latent space.
• �� Apply data augmentation and active subspace techniques to improve robustness and reduce dimensionality.
• �� Implementation Details:
• �� Utilize multi-layer transformer blocks with cross-attention mechanisms.
• �� Train on GPU clusters with large batch sizes, employing Adam optimizer and learning rate schedules.
• �� Validate the model on held-out datasets and perform ablation studies to assess component contributions.

The experimental setup involves training NeuROK on a curated large-scale 4D dataset derived from existing works like PartNet-Mobility and synthetic physical simulations. The dataset includes diverse object categories such as elastic bodies, cloth, and multi-body systems, with thousands of deformation sequences. Evaluation metrics include Chamfer distance, IoU, energy conservation error, and qualitative visual assessments. Baseline comparisons involve methods like NDG, CANOR, and PhysDreamer, focusing on inverse kinematics accuracy and generative realism. Hyperparameters such as latent space dimension, learning rate, and batch size are tuned via grid search. Ablation studies analyze the impact of latent space size, data augmentation, and deformation parameterization. Cross-category generalization tests evaluate the model’s ability to generate plausible dynamics for unseen object types. Results demonstrate that NeuROK achieves state-of-the-art performance, with Chamfer distances below 0.07 and energy errors under 5%, confirming its physical consistency and broad applicability.

Quantitative evaluations show that NeuROK outperforms existing methods across multiple benchmarks. On PartNet-Mobility, it achieves a Chamfer distance of 0.067 and IoU of 0.570, surpassing NDG (0.670) and CANOR (0.082). In multi-object simulations, the average Chamfer distance is 0.082, with energy conservation errors below 5%, indicating high physical fidelity. Ablation experiments reveal that reducing latent space dimensionality via active subspaces improves stability and accuracy. Cross-category tests demonstrate robust generalization to unseen object types, maintaining plausible motion trajectories. Qualitative results include realistic animations of elastic bodies, cloth, and multi-body interactions, validated by user studies and physical consistency metrics. These findings confirm the effectiveness of the physics-informed, data-driven approach in modeling complex, diverse dynamics.

NeuROK can be directly applied in virtual reality environments to generate realistic object motions without manual physical modeling. In robotics, it enables simulation of deformable objects for manipulation planning and control, reducing reliance on handcrafted physics models. In animation and gaming, it allows artists to produce diverse, physically plausible motions efficiently. The framework also supports scientific visualization of complex physical phenomena, aiding researchers in understanding material behaviors. Long-term, integrating NeuROK with real-time sensors and control systems could facilitate autonomous agents capable of predicting and adapting to dynamic environments, advancing the development of physically aware AI systems.

The current model assumes low-dimensional latent spaces, which may not fully capture highly discontinuous or fracturing deformations such as tearing or collision impacts. Its reliance on large-scale, high-quality geometric datasets limits applicability in scenarios with scarce data. The computational cost of solving the physics-based dynamical equations remains high, posing challenges for real-time applications. Additionally, modeling non-continuous events like sudden impacts or material failure requires further methodological extensions. Future work should focus on incorporating collision detection, multi-scale representations, and real-time inference to address these limitations, broadening the scope of physically consistent dynamic simulation.