ODeform: Learning Continuous 4D Motion
for Shape Deformation with Neural ODEs

IROS 2026

Yordanka (Dani) Velikova1,2Mahdi Saleh1Liming Kuang1,2Benjamin Busam1,2

1 Technical University of Munich2 Munich Center for Machine Learning

PaperCodeData & CheckpointsBibTeX

TL;DR

ODeform predicts continuous object deformation from an initial point cloud and physical conditions by learning separate latent neural ODEs for global motion and local deformation.

Qualitative Results

Abstract

Modeling continuous object deformation is important for many computer vision and robotics tasks, such as manipulation and simulation. Existing approaches rely on learning-based methods or physics simulators to model shape deformations. However, these approaches either use discrete time steps or are too computationally intensive for real-time applications. We present ODeform, a novel extension of Neural Ordinary Differential Equations to continuous 4D dynamics of deformable objects in 3D space. Our method transforms 3D point clouds and physical conditions (like material properties) into a unified latent space. By solving the resulting ordinary differential equations over time, we model deformations as continuous flows within this learned embedding, eliminating the need for discrete time steps while maintaining computational efficiency. We evaluate our approach on unseen physical parameter configurations, showing improved motion prediction accuracy over baseline methods. Our experiments further demonstrate a successful transfer to real 3D captured objects with novel shapes, along with effective interpolation and extrapolation of the learned dynamics.

Initial shape and physical parameters define a latent trajectory. Sparse observed states supervise continuous shape evolution.
Learn a continuous trajectory from discrete observations.

Continuous Latent Dynamics

Rather than predicting each frame independently, ODeform learns how latent states change. Integrating these dynamics links observed states and enables predictions at intermediate and future times.

dzg / dt = fg(zg, t)dzl / dt = fl(zl, t)

Schematic notation follows the paper. See the code for the conditioning and numerical integration details.

Method Overview

Separate global motion from local deformation, then reconstruct their combined effect.

Figure 2 from the paper: physical parameters and the initial point cloud feed rigid-motion and local-deformation encoders, parallel neural ODEs, and decoders that reconstruct the pose and shape at time t.
Figure 2. Overview of the ODeform architecture. Separate encoders and neural ODEs model global motion and local deformation; decoders reconstruct the object at the requested time.

Generalization to Unseen Physical Parameters

Evaluated on contact interactions and mass-dependent elastic deformation.

Unseen contact conditions · lower is better
MethodRMSE (mm) ↓MAE (mm) ↓MSE (mm²) ↓
Direct nODE6.7504.95682.710
RNN22.67016.076683.000
PE-GNN14.2209.837244.400
ODeform Ours6.6575.14058.430

ODeform achieves the lowest reported RMSE and MSE in both settings. Direct nODE has the lowest contact MAE. Values are from Table I of the paper.

Temporal Interpolation

Train with sparse temporal supervision and evaluate the intervening frames. The paper studies sampling intervals of 3, 5 and 8 frames.

Temporal Extrapolation

Extend learned dynamics into the future: five additional frames for Contact-Force and ten for Mass-Elastic.

Inverse Parameter Estimation

Freeze the learned dynamics and fit mass or bending stiffness to an observed deformation sequence.

Four time steps comparing direct nODE, RNN, ODeform and ground truth on a cat-shaped object.
Preserving shape through contact. Deformation under an unseen force vector, compared with the baselines and simulation reference.
Ball deformation with three bending values at fixed mass and three masses at fixed bending.
Changing the physical conditions. Varying bending stiffness at fixed mass (top), and mass at fixed bending (bottom).

Two complementary datasets

Contact-Force: 2,100 interaction sequences across seven object categories.

Mass-Elastic: 500 simulated drops with varying mass and bending stiffness.

Generalization to Unseen Geometry

Qualitative experiments explore learned deformation on new geometry and 3D representations.

Undeformed and deformed shoe meshes, including unseen shapes reconstructed from HouseCAT6D.

Unseen instances

Transfer deformation learned on simulated shapes to reconstructed shoe meshes from HouseCAT6D, within the same object category.

A reconstructed volleyball in three deformation states represented by 3D Gaussian Splatting.

3D Gaussian Splatting

Apply the learned dynamics to a reconstructed volleyball, bringing deformable motion into a captured scene.

BibTeX

Download .bib
@inproceedings{velikova2026odeform,
  title   = {ODeform: Learning Continuous 4D Motion for Shape
             Deformation with Neural ODEs},
  author  = {Velikova, Yordanka and Saleh, Mahdi and
             Kuang, Liming and Busam, Benjamin},
  booktitle = {Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)},
  year    = {2026}
}