FGeo-HyperGNet

This project is the official implementation of the IJCAI 2025 paper "FGeo-HyperGNet: Geometric Problem Solving Integrating FormalGeo Symbolic System and Hypergraph Neural Network". We built a neural-symbolic system, called FGeo-HyperGNet, to automatically perform human-like geometric problem solving.

The symbolic component is a formal system built on FormalGeo, which can automatically perform geometric relational reasoning and algebraic calculations and organize the solution into a hypergraph with conditions as hypernodes and theorems as hyperedges.

The neural component, called HyperGNet, is a hypergraph neural network based on the attention mechanism, including an encoder to effectively encode the structural and semantic information of the hypergraph, and a solver to provide guidance of solving geometric problem.

The neural component predicts theorems according to the hypergraph, and the symbolic component applies theorems and updates the hypergraph, thus forming a predict-apply cycle to ultimately achieve readable and traceable automatic solving of geometric problems. Experiments demonstrate the correctness and effectiveness of this neural-symbolic architecture. Experiments demonstrate the correctness and effectiveness of this neural-symbolic architecture. We achieved a TAP of 93.50% and a PSSR of 88.36% on the FormalGeo7K-v2 datasets.

More information about FormalGeo will be found in homepage. FormalGeo is in its early stages and brimming with potential. We welcome anyone to join us in this exciting endeavor.

Installation

Clone project:

$ git clone --depth 1 https://github.com/BitSecret/HyperGNet.git

Create Python environments using Conda:

$ conda create -n HyperGNet python=3.10
$ conda activate HyperGNet

Install Python dependencies:

$ cd HyperGNet
$ pip install -e .
$ conda install pytorch torchvision torchaudio pytorch-cuda=12.1 -c pytorch -c nvidia

Enter the code path (all subsequent code will be executed in this path):

$ cd src/gps

We provide a short code to test if the environment setup is successful:

$ python utils.py --func test_env

Modify the addresses of datasets datasets_path (for windows or macOS) and datasets_path_linux (for linux) in the data/config.json file, and then run the script to download the FormalGeo7K-v2 datasets:

$ python utils.py --func download_dataset

Download our trained model and generated data through Google Drive or Baidu NetDisk and put them into data/ path.

Now, your working directory should looks like this:

|--data
|  |--checkpoints
|  |--outputs
|  |--training_data
|  |--config.json
|
|--src
|  |--gps
|     |--__init__.py
|     |--data.py
|     |--model.py
|     |--pac.py
|     |--train.py
|     |--utils.py
|
|--architecture.png
|
|--LICENSE
|
|--pyproject.toml
|
|--README.md

Preparing Training Data

We use FormalGeo as the formal environment for solving geometric problems and generate training data using the FormalGeo7K-v2 dataset. The dataset is split into training:validation:test=3:1:1. Each annotated theorem sequence can be organized into a directed acyclic graph (DAG). We randomly traverse this DAG and apply each step's theorem while obtaining the state information of the problem. Each problem will repeat 5 times (the number of repetitions is defined in the data/config.json file).

This process ultimately generated 71,234 training data entries (from 4,200 problems), 23,878 validation data entries ( from 1,400 problems), and 23,522 test data entries (from 1,400 problems). Each data entry can be viewed as an input-label pair ((nodes, edges, structural_encoding, goals), theorems).

We have provided the generated data, located at the path data/training_data. You can also use a multi-process approach to generate training data:

$ python data.py --func make_data
$ python data.py --func make_onehot

In the end, you will obtain the files nodes_pretrain_data.pkl, edges_pretrain_data.pkl, and train_data.pkl in the data/training_data path.

Training

We first pretrain the hypernode embedding networks using a self-supervised method.

$ python train.py --func pretrain_nodes
$ python train.py --func pretrain_edges

Then, train the theorem prediction network:

$ python train.py --func train

The training log will be saved in data/outputs.

Testing

Test theorem prediction accuracy (TPA):

$ python train.py --func test

Test problem solving success rate (PSSR):

$ python pac.py

The test results of the model will be saved in HyperGNet/data/outputs.

Results

Show experimental results:

$ python utils.py --func show_contrast_results
$ python utils.py --func show_ablation_results

Comparative Experiment

Table 1: PSSR of different methods on the FormalGeo7K dataset.

Method	Total	L1	L2	L3	L4	L5	L6
Forward Search	39.71	58.47	41.01	34.16	16.40	5.45	4.79
Backward Search	35.44	66.43	34.98	11.78	6.56	6.09	1.03
T5-small with FGeo	36.14	49.90	34.84	34.59	23.57	8.06	3.33
BART-base with FGeo	54.00	73.90	56.12	50.38	26.75	16.13	8.33
DeepSeek-v3	60.79	75.99	56.38	63.91	43.31	32.26	28.33
Inter-GPS	60.50	76.2	63.30	60.90	39.49	17.74	15.00
NGS	62.60	62.22	64.97	72.79	57.47	56.41	36.59
DualGeoSolver	62.11	62.96	67.80	65.44	60.92	53.85	34.15
FGeo-TP	80.86	96.43	85.44	76.12	62.26	48.88	29.55
FGeo-DRL	80.85	97.61	91.88	70.82	57.55	36.17	27.59
FGeo-HyperGNet	88.36	96.24	91.76	87.59	82.17	56.45	56.67

Table 2: PSSR of existing state-of-the-art methods and FGeo-HyperGNet on different datasets.

Method	Geometry3K	GeoQA	FormalGeo7K
GeoDRL	89.40	-	-
E-GPS	90.40	-	-
SCA-GPS	-	64.10	-
DualGeoSolver	-	65.20	-
FGeo-TP	-	-	80.86
DFE-GPS	-	-	82.38
FGeo-HyperGNet	91.99	85.64	88.36

Ablation study

Table 3: Ablation study results of HyperGNet on the FormalGeo7k dataset.

Method	Beam Size	TPA	PSSR
	1	71.58	44.86
FGeo-HyperGNet	3	88.91	62.93
	5	93.50	67.79
	1	70.73	41.57
-w/o Pretrain	3	87.36	59.36
	5	92.21	64.43
	1	70.33	39.64
-w/o SE	3	88.14	60.21
	5	92.48	64.14
	1	68.11	36.93
-w/o Hypertree	3	87.38	57.57
	5	92.00	63.07

Acknowledge

This project is maintained by FormalGeo Development Team.
Please contact with Xiaokai Zhang if you encounter any issues.

Citation

To cite HyperGNet in publications use:

Zhang X, Li Y, Zhu N, Qin C, Zeng Z, and Leng T. FGeo-HyperGNet: Geometric Problem Solving Integrating FormalGeo Symbolic System and Hypergraph Neural Network[C]. Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence. 2025.

A BibTeX entry for LaTeX users is:

@inproceedings{zhang2025fgeohypergnet,
title={FGeo-HyperGNet: Geometric Problem Solving Integrating FormalGeo Symbolic System and Hypergraph Neural Network},
author={Zhang, Xiaokai and Li, Yang and Zhu, Na and Qin, Cheng and Zeng, Zhengbing and Leng, Tuo},
booktitle = {Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence},
year={2025},
}