Conceitos essenciais
The authors propose a new framework, Geo-FNO, that can efficiently solve partial differential equations (PDEs) on arbitrary geometries by learning to deform the input domain into a uniform latent space where the fast Fourier transform can be applied.
Resumo
The authors present a new framework called Geo-FNO to solve partial differential equations (PDEs) on arbitrary geometries. The key idea is to learn a deformation that maps the irregular input domain to a uniform latent space, where the efficient Fourier neural operator (FNO) can be applied.
The content covers the following:
Problem settings: The authors consider parametric PDEs defined on various domains, with the domain shape parameterized in different ways (e.g., meshes, functions, design parameters).
Neural operators: The authors provide background on neural operators, which generalize standard deep neural networks to learn operator mappings between infinite-dimensional function spaces.
Geo-FNO architecture:
The authors propose two scenarios: (1) using a given coordinate map to deform the domain, and (2) learning the deformation as part of the end-to-end training.
The deformation maps the irregular physical domain to a uniform latent space, where the standard FNO can be applied efficiently using the fast Fourier transform.
The deformation can be fixed or learned using a neural network.
Numerical experiments:
The authors evaluate Geo-FNO on various PDE problems, including elasticity, plasticity, advection on a sphere, airfoil flows, and pipe flows.
Geo-FNO outperforms existing methods like interpolation-based approaches and mesh-free methods, achieving up to 105 times acceleration compared to traditional numerical solvers while maintaining high accuracy.
The authors demonstrate Geo-FNO's flexibility in handling irregular geometries and non-uniform meshes.
Overall, the Geo-FNO framework combines the computational efficiency of the fast Fourier transform with the flexibility of learned deformations, enabling efficient and accurate solutions of PDEs on arbitrary geometries.
Estatísticas
Geo-FNO is up to 105 times faster than traditional numerical solvers on the airfoil problem.
Geo-FNO is twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.
Citações
"Geo-FNO combines both the computational efficiency of the FFT and the flexibility of learned deformations."
"Geo-FNO does not have the limitation of the traditional adaptive moving mesh method, where the system in the Fourier space is no longer equivalent to the original system."