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.

我们提出了几何信息神经算子（GINO），一种高效的方法，用于学习具有不同几何形态的大规模偏微分方程的解算器。该算子基于图和傅里叶结构，使用输入形状的有符号距离函数和点云表示来学习解算器。用于验证 GINO 方法在大规模模拟上的性能，我们生成了 3D 车辆几何形态的工业标准空气动力学数据集，雷诺数高达五百万。在这个大规模 3D 流体模拟中，传统数值方法计算表面压力的成本很高。我们成功地训练 GINO 仅使用五百个数据点来预测车身表面的压力。在成本 - 准确性实验中，与优化后的基于 GPU 的计算流体动力学（CFD）模拟器相比，GINO 的计算阻力系数速度提高了 26000 倍。在测试新的几何形态和边界条件（入口速度）组合时，GINO 相对于深度神经网络方法的误差率降低了四分之一。

基于几何信息的大规模三维偏微分方程神经算子

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

We show how to apply the recursive quantum approximate optimization algorithm
(RQAOA) to MAX-$k$-CUT, the problem of finding an approximate $k$-vertex
coloring of a graph. We compare this proposal to the best known classical and
hybrid classical-quantum algorithms. First, we show that the standard
(non-recursive) QAOA fails to solve this optimization problem for most regular
bipartite graphs at any constant level $p$: the approximation ratio achieved by
QAOA is hardly better than assigning colors to vertices at random. Second, we
construct an efficient classical simulation algorithm which simulates level-$1$
QAOA and level-$1$ RQAOA for arbitrary graphs. In particular, these hybrid
algorithms give rise to efficient classical algorithms, and no benefit arising
from the use of quantum mechanics is to be expected. Nevertheless, they provide
a suitable testbed for assessing the potential benefit of hybrid algorithm: We
use the simulation algorithm to perform large-scale simulation of level-$1$
QAOA and RQAOA with up to $300$ qutrits applied to ensembles of randomly
generated $3$-colorable constant-degree graphs. We find that level-$1$ RQAOA is
surprisingly competitive: for the ensembles considered, its approximation
ratios are often higher than those achieved by the best known generic classical
algorithm based on rounding an SDP relaxation. This suggests the intriguing
possibility that higher-level RQAOA may be a potentially useful algorithm for
NISQ devices.

本文探讨了 RQAOA 算法在 MAX-k-CUT 问题上的应用，进行了经典算法和量子经典混合算法的比较，并发现 level-1 的 RQAOA 算法可以优于现有 SDP 基础的经典算法，该算法可能是 NISQ 设备的有用算法。