Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

By Zong-Yi Li, Daniel Zhengyu Huang, Burigede Liu, Anima Anandkumar. Journal of machine learning research, 2022

645 citations recorded by the source corpus.

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

View Paper: https://www.semanticscholar.org/paper/b768011c1dd42b13ae8a1c34b99ac4228304dabd

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

1. Jianmin Wang
2. Mingsheng Long
3. Haixu Wu
4. Hang Su
5. Yinpeng Dong
6. Huakun Luo
7. Jian Liang
8. Haowen Wang

Citations in the corpus

• Transolver: A Fast Transformer Solver for PDEs on General Geometries (2024, methodology): “To clearly benchmark our model among multifarious neural operators, we first experiment on six well-established datasets, which can conveniently build a complete leaderboard from previous papers (Li et al., 2022; Wu et al., 2023; Hao et al., 2023).”
• Geometry-Informed Neural Operator for Large-Scale 3D PDEs (2023, methodology): “A recent model, termed GeoFNO, learns a deformation from a given geometry to a latent regular grid [14] so that the FFT can be applied in the latent space.”
• GNOT: A General Neural Operator Transformer for Operator Learning (2023, methodology): “Transformer models (Cao, 2021; Li et al., 2022b), based on attention mechanism, are proposed since they have a larger model capacity.”
• Physics-Informed Neural Operator for Learning Partial Differential Equations (2021, unknown): “If the input and output query points are sampled from non-uniform mesh, we can use the graph kernel operator as the first and last layer for continuous evaluation, while using the Fourier layers in the middle for efficient computation, similar to [9].”
• Factorized Fourier Neural Operators (2021, methodology): “On Plasticity (d), the operator learns the displacement of each mesh point given an initial boundary condition. transform-based approaches (Li et al., 2021a; 2022) (Li et al., 2021a) for flow simulation.”