Towards Unified Constitutive Laws for Time-Evolving Material Field

          
_{¹ University of Science and Technology of China (USTC)     |     ² University of Illinois Urbana-Champaign (UIUC)}

Authors: Haoqin Hong†¹, Din Fan†¹, Fubing Dou¹, Zhili Zhou², Haoran Sun¹, Congcong Zhu‡¹, Jingrun Chen‡¹.
†Equal contribution. ‡Corresponding author.

Overview

This is the official code implementation of the paper "Physics-Informed Deformable Gaussian Splatting: Towards Unified Constitutive Laws for Time-Evolving Material Field". We will open source each module in stages and release the complete code and dataset after the paper is accepted.

News

• PIDG is accepted by AAAI-26! See you in Singapore!
• Welcome to see more related cutting-edge research in Awesome-Physics-Inspired-Vision-Understanding.

Demo videos of dynamic reconstruction results from our PIDG method

Representative Scenes: Dry Ice (Fluid Simulation), Balls Reaction (Elastic Mechanics) and Mechanics Cloth (Cloth Simulation)

To do list:

• Release Training Demo code.
• Release Inference (rendering) code.
• Release the network architecture (including 4D decomposed hash encoding with dynamic and static decoupling, physical information material fields, and Lagrangian particle flows).
• Release CUDA/C++ optical-based rasteriser.
• Release the representative subset of fluid simulation scenes. (compressed by approximately 30× into the WebP format.)
• Release the dynamic reconstruction demo videos in representative scenes.
• Release Training code.
• Release experimental analysis code (velocity field and material field visualisation, Gaussian particle centre distribution, Gaussian particle variation residual analysis).
• Release full PIDG custom physics-driven synthetic dataset. (due to double-blind review constraints and space limitations, we are currently unable to anonymously upload the dataset)

Datasets

Our experiments employ three monocular datasets:

Dataset	Type	Scenes	Source
D-NeRF	Synthetic	8, monocular	Official release
HyperNeRF	Real-world dynamic	7, Subset of monocular sequences	Official release
PIDG	Synthetic, physics-driven	5, Custom	Generated in Blender

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HyperNeRF (Real-world)

1. Geometry extraction: Point clouds are reconstructed with COLMAP following the protocol in E-D3DGS Per-Gaussian Embedding-Based Deformation for Deformable 3D Gaussian Splatting (Bae et al.).

2. Auxiliary supervision

   Quantity	Method	Checkpoint
   Optical flow & occlusion	UniMatch (Xu et al.)	GMFlow-scale2-regrefine6-sintelft
   Motion mask	SAM-v2 (Ravi et al.)	sam2.1_hiera_large.pt
   Depth map	Distill Any Depth (He et al.)	Distill-Any-Depth-Multi-Teacher-Large

   File-format conventions

   • Optical flow is stored in Middlebury.flo files.
   • Naming rule: Forward flow of frame t encodes motion t → t + 1;
   • Backward flow of frame t encodes motion t → t − 1.

3. For the HyperNeRF vrig scenes, we apply ./tools/hyper_filter.py to filter the dataset such that only the left-view (monocular) images are retained for training, validation, and testing, without altering the original data split logic.

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PIDG (Synthetic)

Because learning-based models trained on real imagery perform poorly on synthetic PIDG data, alternative preprocessing is applied:

Quantity	Method
Optical flow	Dual TV-L1 implementation in OpenCV
Motion mask	Extracted from RGBA alpha channel (background → mask)
Depth map	SameDistill Any Depth pipeline as for HyperNeRF

We use ./tools/dualtvl1.py to extract the corresponding forward and backward optical flow between consecutive frames using the Dual TV-L1 algorithm. This script generates both .flo files for downstream processing and .png visualizations in HSV format for qualitative inspection.

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Directory layout (after preprocessing)

├── data
│   | HyperNeRF
│     ├── broom
│       ├── colmap
│       ├── rgb
│           ├── 2x
│               ├── left1_000000.png
│               ├── left1_000001.png
│               ├── ...
│       ├──flow 
│           ├── 2x
│               ├── left1_000000_flow_fwd.flo
│               ├── left1_000002_flow_fwd.flo
│               ├── left1_000002_flow_bwd.flo
│               ├── ...
│       ├──resized_mask 
│           ├── 2x
│               ├── left1_000000.png
│               ├── left1_000001.png
│               ├── ...
│       ├──depth-distill
│           ├── 2x
│               ├── left1_000000.npy
│               ├── left1_000001.npy
│               ├── ...
│     ├── split-cookie
│     ├── ...
│
│   | PIDG
│     ├── dry_ice
│       ├── train
│           ├── 0001.png
│           ├── 0002.png
│           ├── ...
│       ├──flows_flo
│           ├── flow_bwd_0002.flo
│           ├── flow_bwd_0003.flo
│           ├── ...
│       ├──motion_mask 
│           ├── 0000.png
│           ├── 0001.png
│           ├── ...
│       ├──depth-distill
│           ├── 0001.npy
│           ├── 0002.npy
│           ├── ...
│     ├── balls-reaction
│     ├── ...

Network Architecture Navigation

Section A: Dynamic-Static Decoupled 4D Hash Encoding: ./hashencoder & ./train_pidg.py & ./scene_PIDG/gaussian_model.py and so on.

Section B: Physics-Informed Gaussian Representation: ./motion_utils/time_evolving_material_field.py & ./scene_PIDG/gaussian_model.py & ./scene_PIDG/deform_model.py & ./train_pidg.py and so on.

Section C: Lagrangian Particle Flow Matching：./submodules/flow-based-diff-gaussian-rasterization & ./utils/flow_utils.py & ./utils/flow_vis_utils.py and so on.