Fourier Neural Operator with Learned Deformations for PDEs on General Geometries (2207.05209v2)

Abstract: 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.

• The paper introduces Geo-FNO, which learns input domain deformations to map irregular geometries to a latent space for efficient FFT processing.
• It achieves significant computational speed-ups up to 10^5 times faster and doubles the accuracy compared to standard FNO methods.
• Geo-FNO supports diverse inputs like point clouds and meshes, enabling real-time simulations and advanced inverse design in complex engineering applications.

Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

The paper "Fourier Neural Operator with Learned Deformations for PDEs on General Geometries" introduces an innovative framework named Geo-FNO, which aims to solve complex Partial Differential Equations (PDEs) on arbitrary geometries with improved efficiency and accuracy. This framework builds upon the Fourier Neural Operator (FNO), inheriting its computational efficiency while expanding its applicability beyond the constraints of rectangular domains and uniform grids.

Background and Methodology

FNO previously demonstrated remarkable performance in solving PDEs across various domains, thanks to the Fast Fourier Transform (FFT) which accelerates computation. However, its application was limited to structured, rectangular domains. Geo-FNO overcomes these limitations by incorporating a learned deformation of the input domain. It transforms irregular geometries into a latent space with a uniform grid where FFT can be applied effectively. The transformation is either fixed or adaptable, with the latter achieved through end-to-end training using a neural network that models the deformation.

Geo-FNO accommodates several input formats, such as point clouds, meshes, and design parameters, providing versatility in addressing diverse engineering applications. It applies to both forward modeling and inverse design problems, exemplified in cases like Elasticity, Plasticity, Euler's, and Navier-Stokes equations.

Numerical Results

Geo-FNO showcases significant computational benefits. Numerical experiments reveal that Geo-FNO can achieve cost-accuracy advantages, being $10^5$ times faster than traditional numerical solvers and delivering twice the accuracy relative to current ML-based PDE solvers with direct interpolation, such as the standard FNO. This is particularly relevant in computational fluid dynamics and aerodynamics, where Geo-FNO provides complete PDE simulations over varying geometries, impacting notably the efficiency of the design optimization processes.

Implications and Future Directions

Geo-FNO presents strong potential for advancing the capabilities of AI in the simulation and analysis of PDEs with complex geometrical constraints. The framework could lead to deep learning models more intertwined with physics-based constraints (e.g., physics-informed neural operators), exploiting neural networks’ potential for real-time and adaptive problem-solving in high-dimensional spaces.

For future research, extending Geo-FNO’s framework to more complex topologies could further broaden its application range. Techniques like domain decomposition and Fourier continuation might facilitate such expansions, offering a universal approach for general PDE topologies.

The transformative approach proposed by Geo-FNO not only aligns with traditional numerical strategies but also opens new avenues for leveraging the power of machine learning in solving PDEs. This marks an important step towards integrating AI effectively into scientific computing and engineering design domains, which require high precision, flexibility, and computational efficiency.