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.

翻译：我们提出几何信息驱动的神经算子（GINO），一种高效学习变几何条件下大规模偏微分方程解算子的方法。该方法通过符号距离函数和点云表示输入几何形状，并基于图神经网络与傅里叶架构的神经算子学习解算子。图神经算子处理非规则网格，将其转换至傅里叶神经算子可高效计算的规则潜在网格，亦可反向转换。GINO具有离散化收敛特性，即训练后的模型可应用于连续域任意离散方式，且随离散化精细度提升而收敛至连续算子。为实证验证该方法在大规模仿真中的性能，我们生成了工业标准的雷诺数高达五百万的三维汽车几何气动数据集。在此大规模三维流体仿真中，数值方法计算表面压力的代价高昂。我们成功使用仅五百个数据点训练GINO预测车体表面压力。成本精度实验表明，在计算阻力系数时，该方法相较基于GPU的优化计算流体动力学仿真器实现$26,000 \times$加速。当测试新几何形状与边界条件（入口速度）组合时，GINO相比深度神经网络方法误差率降低四分之一。