Transolver (ICML 2024 Spotlight)

:triangular_flag_on_post:News (2026.02) We present a new member of the Transolver Family, named Transolver-3, which can handle 100-million-scale geometries with SOTA results in full-size DrivAerML.

:triangular_flag_on_post:News (2025.07) We have released the code of Transolver++. Please check this GitHub Repository.

:triangular_flag_on_post:News (2025.04) We have released Neural-Solver-Library as a simple and neat code base for PDE solving. It contains 17 well-reproduced neural solvers. Welcome to try this library and join the research in solving PDEs.

:triangular_flag_on_post:News (2025.02) We present an upgraded version of Transolver, named Transolver++, which can handle million-scale geometries in one GPU with more accurate results.

:triangular_flag_on_post:News (2024.10) Transolver has been integrated into NVIDIA physicsnemo.

Transolver: A Fast Transformer Solver for PDEs on General Geometries [Paper] [Slides] [Poster]

In real-world applications, PDEs are typically discretized into large-scale meshes with complex geometries. To capture intricate physical correlations hidden under multifarious meshes, we propose the Transolver with the following features:

Going beyond previous work, Transolver calculates attention among learned physical states instead of mesh points, which empowers the model with endogenetic geometry-general capability.
Transolver achieves 22% error reduction over previous SOTA in six standard benchmarks and excels in large-scale industrial simulations, including car and airfoil designs.
Transolver presents favorable efficiency, scalability and out-of-distrbution generalizability.

<img src=".\pic\Transolver.png" height = "250" alt="" align=center /> Figure 1. Overview of Transolver.

Transolver v.s. Previous Transformer Operators

All of the previous Transformer-based neural operators directly apply attention to mesh points. However, the massive mesh points in practical applications will cause challenges in both computation cost and capturing physical correlations.

Transolver is based on a more foundational idea, that is learning intrinsic physical states under complex geometrics. This design frees our model from superficial and unwieldy meshes and focuses more on physics modeling.

As shown below, Transolver can precisely capture miscellaneous physical states of PDEs, such as (a) various fluid-structure interactions in a Darcy flow, (b) different extrusion regions of elastic materials, (c) shock wave and wake flow around the airfoil, (d) front-back surfaces and up-bottom spaces of driving cars.

<img src=".\pic\physical_states.png" height = "300" alt="" align=center /> Figure 2. Visualization of learned physical states.

Get Started

Please refer to different folders for detailed experiment instructions.
List of experiments:

Core code: see ./Physics_Attention.py
Standard benchmarks: see ./PDE-Solving-StandardBenchmark
Car design task: see ./Car-Design-ShapeNetCar
Airfoil design task: see ./Airfoil-Design-AirfRANS

Results

Transolver achieves consistent state-of-the-art in six standard benchmarks and two practical design tasks. More than 20 baselines are compared.

<img src=".\PDE-Solving-StandardBenchmark\fig\standard_benchmark.png" height = "300" alt="" align=center /> Table 1. Results on six standard benchmarks. <img src=".\Airfoil-Design-AirfRANS\fig\results.png" height = "300" alt="" align=center /> Table 2. Results on two design tasks: Car and Airfoild design.

Showcases

<img src=".\pic\showcases.png" height = "300" alt="" align=center /> Figure 3. Comparison of Transolver and other models.

Application of Transolver for crash dynamics modeling: https://arxiv.org/pdf/2510.15201

Citation

If you find this repo useful, please cite our paper.

@inproceedings{wu2024Transolver,
  title={Transolver: A Fast Transformer Solver for PDEs on General Geometries},
  author={Haixu Wu and Huakun Luo and Haowen Wang and Jianmin Wang and Mingsheng Long},
  booktitle={International Conference on Machine Learning},
  year={2024}
}

Contact

If you have any questions or want to use the code, please contact wuhx23@mails.tsinghua.edu.cn.

Acknowledgement

We appreciate the following github repos a lot for their valuable code base or datasets:

https://github.com/neuraloperator/neuraloperator

https://github.com/neuraloperator/Geo-FNO

https://github.com/thuml/Latent-Spectral-Models

https://github.com/Extrality/AirfRANS

Transolver

Install / Use

README