Geometry-Informed Neural Operator for Large-Scale 3D PDEs

TL;DR

We introduce GINO, a geometry-informed neural operator that combines graph-based local kernel integration with Fourier-based global kernel learning to predict PDE solutions on large-scale, irregular 3D geometries. By encoding input surfaces with a GNO-based encoder and refining outputs on a latent regular grid via an FNO block, GINO achieves discretization-convergent predictions and substantial speedups over traditional CFD solvers. Empirical results on Ahmed-body and Shape-Net Car datasets show GINO delivering up to ~26,000× faster drag computations with competitive surface-pressure accuracy, and robust zero-shot super-resolution capabilities. The work highlights a scalable, geometry-aware framework for 3D PDE operator learning with strong potential for design optimization and rapid simulation in engineering contexts.

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 $26,000 \times$ 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.

Figures (6)

• Figure 1: The architecture of GINO. The input geometries are irregular and change for each sample. These are discretized into point clouds and passed on to a GNO layer, which maps from the given geometry to a latent regular grid. The output of this GNO layer is concatenated with the SDF features and passed into an FNO model. The output from the FNO model is projected back onto the domain of the input geometry for each query point using another GNO layer. This is used to predict the target function (e.g., pressure), which is used to compute the loss that is optimized end-to-end for training.
• Figure 2: Visualization of a ground-truth pressure and corresponding prediction by GINOfrom the Shape-Net Car (top) and Ahmed-body (bottom) datasets, as well as the absolute error.
• Figure 3: Cost-accuracy trade-off analysis for the drag coefficient.
• Figure 4: Discretization-convergent studies and zero-shot super-resolution.
• Figure 5: Comparison of GNN and GNO as the discretization becomes finer. GNN is discretization dependent, while GNO is discretization convergent.