标题： 几何感知神经算子用于大规模三维偏微分方程 显示英文标题

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

摘要： 我们提出了几何感知神经算子（GINO），这是一种高效的方法，用于学习具有不同几何形状的大规模偏微分方程的解算子。 GINO 使用输入形状的符号距离函数和点云表示以及基于图和傅里叶架构的神经算子来学习解算子。 图神经算子处理不规则网格，并将它们转换为常规潜在网格，在这些网格上可以高效地应用傅里叶神经算子。 GINO 是离散化收敛的，这意味着训练后的模型可以应用于连续域的任意离散化，并且随着离散化的细化，它会收敛到连续算子。 为了在大规模模拟中实证验证我们方法的性能，我们生成了行业标准的空气动力学数据集，包含3D车辆几何形状，雷诺数高达五百万。 对于这种大规模的3D流体模拟，数值方法计算表面压力的成本很高。 我们成功地使用仅五百个数据点训练了GINO来预测汽车表面的压力。 成本-精度实验显示，在计算阻力系数时，与优化的基于GPU的计算流体力学（CFD）模拟器相比，速度提高了$26,000 \times$。 当在新的几何形状和边界条件（入口速度）组合上进行测试时，GINO 相比深度神经网络方法将误差率降低了四分之一。 显示英文摘要

摘要： 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.