[Paper Review] Geometry-Informed Neural Operator for Large-Scale 3D PDEs

Zongyi Li, Nikola Kovachki+9 more|arXiv (Cornell University)|Sep 1, 2023

Model Reduction and Neural NetworksPhysics and Astronomy31 citations

TL;DR

GINO combines graph-based and Fourier neural operators with signed distance functions to learn solution operators for large-scale PDEs on varying geometries, delivering massive speedups with competitive accuracy.

ABSTRACT

We propose the geometry-informed neural operator (GINO), a highly efficient approach to learning the solution operator of large-scale partial differential equations with varying geometries. GINO uses a signed distance function and point-cloud representations of the input shape and neural operators based on graph and Fourier architectures to learn the solution operator. The graph neural operator handles irregular grids and transforms them into and from regular latent grids on which Fourier neural operator can be efficiently applied. GINO is discretization-convergent, meaning the trained model can be applied to arbitrary discretization of the continuous domain and it converges to the continuum operator as the discretization is refined. To empirically validate the performance of our method on large-scale simulation, we generate the industry-standard aerodynamics dataset of 3D vehicle geometries with Reynolds numbers as high as five million. For this large-scale 3D fluid simulation, numerical methods are expensive to compute surface pressure. We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points. The cost-accuracy experiments show a $26,000 imes$ speed-up compared to optimized GPU-based computational fluid dynamics (CFD) simulators on computing the drag coefficient. When tested on new combinations of geometries and boundary conditions (inlet velocities), GINO obtains a one-fourth reduction in error rate compared to deep neural network approaches.

Motivation & Objective

• Motivate learning solution operators for PDEs with varying geometries and meshes.
• Develop a geometry-informed architecture that handles irregular inputs and maintains discretization convergence.
• Achieve large-scale efficiency by coupling graph-based local kernels with global Fourier-based operators.
• Demonstrate speedups over GPU-based CFD solvers while preserving predictive accuracy on real 3D geometries.

Proposed method

• Encode irregular geometry (point cloud + signed distance function) with a graph neural operator (GNO) encoder to a regular latent grid.
• Apply a sequence of Fourier neural operator (FNO) layers on the latent grid for global kernel integration.
• Decode back to the target geometry with a GNO-based decoder to query outputs on irregular meshes.
• Train end-to-end to map geometry, forcing, and boundary data to the solution (e.g., surface pressure).
• Employ a GPU hash-table based neighbor search to accelerate graph construction and kernel integration.

Experimental results

Research questions

• RQ1 Can GINO learn a discretization-convergent operator that maps geometry to PDE solutions across varying grids and meshes?
• RQ2 How does the combination of local GNO kernels and global FNO kernels perform on large-scale 3D CFD data?
• RQ3 What are the speedups and accuracy trade-offs of GINO compared to traditional CFD solvers and other neural operators?
• RQ4 Does GINO support zero-shot super-resolution and efficient evaluation on unseen geometries?

Key findings

• GINO achieves up to 26,000x speed-up in computing drag coefficients over GPU OpenFOAM for large-scale Ahmed-body geometries.
• On Shape-Net Car data, GINO encoder-decoder achieves 7.95% training error and 9.47% test error; decoder-only variant achieves 6.37% training and 7.12% test error.
• On Ahmed-body data, GINO encoder-decoder achieves 9.36% training and 9.01% test error; decoder-only variant achieves 9.34% training and 8.31% test error.
• GINO attains 0.1 seconds per evaluation, enabling massive speed advantages over traditional CFD solvers that take hours per case.
• Discretization-convergence and zero-shot super-resolution analyses show stable errors across latent grid resolutions and input/output sampling rates, with robust performance when training on coarser meshes.