DoMINO: A Decomposable Multi-scale Iterative Neural Operator
for Modeling Large Scale Engineering Simulations
DoMINO：一种可分解的多尺度迭代神经算子，用于建模大规模工程仿真

Rishikesh Ranade Mohammad Amin Nabian Kaustubh Tangsali Alexey Kamenev Oliver Hennigh Ram Cherukuri Sanjay Choudhry

Abstract 摘要

Numerical simulations play a critical role in design and development of engineering products and processes. Traditional computational methods, such as CFD, can provide accurate predictions but are computationally expensive, particularly for complex geometries. Several machine learning (ML) models have been proposed in the literature to significantly reduce computation time while maintaining acceptable accuracy. However, ML models often face limitations in terms of accuracy and scalability and depend on significant mesh downsampling, which can negatively affect prediction accuracy and generalization. In this work, we propose a novel ML model architecture, DoMINO (Decomposable Multi-scale Iterative Neural Operator) developed in NVIDIA Modulus to address the various challenges of machine learning based surrogate modeling of engineering simulations. DoMINO is a point cloud-based ML model that uses local geometric information to predict flow fields on discrete points. The DoMINO model is validated for the automotive aerodynamics use case using the DrivAerML dataset. Through our experiments we demonstrate the scalability, performance, accuracy and generalization of our model to both in-distribution and out-of-distribution testing samples. Moreover, the results are analyzed using a range of engineering specific metrics important for validating numerical simulations.
数值模拟在工程产品和流程的设计与开发中起着至关重要的作用。传统的计算方法，如计算流体动力学（CFD），可以提供准确的预测，但对于复杂几何形状来说计算成本很高。文献中已经提出了几种机器学习（ML）模型，以显著减少计算时间，同时保持可接受的准确性。然而，机器学习模型在准确性和可扩展性方面往往存在局限性，并且依赖于显著的网格下采样，这可能会对预测准确性和泛化能力产生负面影响。在这项工作中，我们提出了一种新的机器学习模型架构，DoMINO（可分解的多尺度迭代神经算子），该模型在 NVIDIA Modulus 中开发，旨在解决基于机器学习的工程仿真代理建模的各种挑战。DoMINO 是一种基于点云的机器学习模型，它使用局部几何信息来预测离散点上的流场。使用 DrivAerML 数据集对 DoMINO 模型在汽车空气动力学用例中的有效性进行了验证。通过我们的实验，我们证明了我们的模型在分布内和分布外测试样本上的可扩展性、性能、准确性和泛化能力。此外，使用了一系列重要的工程特定指标来分析结果，以验证数值模拟。

Local Learning, Engineering Simulation
局部学习，工程模拟

1 Introduction 1 引言

Simulation and modeling of engineering applications involve representing complex physical and chemical processes with partial differential equations (PDEs) followed by solving these equations on a computational domain with given boundary conditions. Engineering simulation tools are predominantly used in the development of innovative and optimized designs. Typically, simulations are based on numerical methods such as finite volume or element methods (LeVeque, 2002, 2007), which are used to discretize the computational domain into smaller elements known as a mesh, approximating the various partial differentials and solving the system of equations derived on the mesh to obtain the discrete PDE solutions. In many cases, the choice of discretization is a trade-off between accuracy and computational cost. The computational time and memory grows proportionally with the number of elements as it directly impacts the process of meshing the computational domain as well as solving the discrete equations on this mesh. As a result, there has been a constant drive to develop newer numerical methods that can provide greater accuracy as well as greater computational efficiency, i.e., faster and ideally at a lower computational cost.
工程应用中的仿真和建模涉及用偏微分方程（PDEs）表示复杂的物理和化学过程，然后在给定边界条件下的计算域上求解这些方程。工程仿真工具主要用于创新和优化设计的发展。通常，仿真基于有限体积或有限元素方法（LeVeque, 2002, 2007）等数值方法，这些方法用于将计算域离散化为更小的单元，称为网格，近似各种偏微分，并在网格上求解得到的方程组，以获得离散的 PDE 解。在许多情况下，离散化的选择是在精度和计算成本之间进行权衡。计算时间和内存与单元数量成正比增长，因为它直接影响计算域的网格划分以及在网格上求解离散方程的过程。因此，人们一直致力于开发更精确且计算效率更高的数值方法，即更快，并理想情况下以更低的计算成本。

A significant amount of work has been carried out in improving the computational efficiency of existing methods, development of new methods (Sharma et al., 2020; Angel et al., 2024; Ashton et al., 2025; Bernaschi et al., 2023), and also by utilizing the latest High Performance Computing (HPC) hardware such as GPUs, arm64-based processors etc (Stone et al., 2021; Appa et al., 2021; Piscaglia & Ghioldi, 2023; Xue et al., 2024). Despite these advancements, there has been a continual need to explore Machine Learning (ML) approaches to serve as surrogate models for numerical simulations and provide further acceleration. ML methods developed for numerical simulations have primarily fallen into 2 categories, neural operators and mesh (or graph-)-based approaches.
在提高现有方法的计算效率、开发新方法（Sharma 等人，2020 年；Angel 等人，2024 年；Ashton 等人，2025 年；Bernaschi 等人，2023 年）以及利用最新的高性能计算（HPC）硬件（如 GPU、基于 arm64 的处理器等）（Stone 等人，2021 年；Appa 等人，2021 年；Piscaglia & Ghioldi，2023 年；Xue 等人，2024 年）方面已经进行了大量工作。尽管取得了这些进步，但仍然需要探索机器学习（ML）方法作为数值模拟的替代模型，以提供进一步的加速。为数值模拟开发的机器学习方法主要分为两类：神经算子和基于网格（或图-）的方法。

Neural operators are a class of deep-learning methods that learn relationships functional spaces. These are mesh-free, infinite dimensional operators that are learned using neural networks (Lu et al., 2019; Li et al., 2020a, b; Patel et al., 2021). Recently, significant advancements have been made in applying these methods to large-scale simulations such as computational fluid dynamcics, weather modeling etc (Azizzadenesheli et al., 2024). Methods are being developed to learn and represent geometries to develop more accurate and generalizable methods. Li et al. (2024) adapted to neural operators to modeling directly on meshes. They learn a graph neural operator to project surface meshes of geometries on to structured latent spaces, apply fourier layers and decode the solutions back on the structured meshes. Zhong & Meidani (2025) developed a physics-informed geometry aware neural network where the geometry representations was learned from point cloud representation of the surfaces and combined that with physics losses to improve the robustness of the learning algorithm. He et al. (2024); Park & Kang (2024) used geometry parameters and signed distance field (SDF) information in their DeepONet architecture to model a structural problem.
神经算子是一类深度学习方法，用于学习函数空间中的关系。这些是无网格、无限维的算子，通过神经网络进行学习（Lu 等人，2019；Li 等人，2020a，b；Patel 等人，2021）。最近，在这些方法应用于大规模模拟（如计算流体动力学、天气建模等）方面取得了重大进展（Azizzadenesheli 等人，2024）。人们正在开发方法来学习和表示几何形状，以开发更准确和更具泛化能力的方法。Li 等人（2024）将神经算子应用于直接在网格上进行建模。他们学习一个图神经算子，将几何形状的表面网格投影到结构化的潜在空间上，应用傅里叶层，并将解决方案解码回结构化的网格上。Zhong & Meidani（2025）开发了一个物理信息感知的几何感知神经网络，其中几何表示从表面的点云表示中学习，并将其与物理损失相结合，以提高学习算法的鲁棒性。He 等人 (2024); Park & Kang (2024) 在他们的 DeepONet 架构中使用了几何参数和符号距离场 (SDF) 信息来模拟结构问题。

Mesh-based methods have been another area of focus in modeling numerical simulations. In these methods, ML models are directly trained on simulation meshes and learn to predict solution fields using local mesh information. Early works in this field, such as GraphNet (Allen et al., 2023) and Gated Graph Neural Networks (GGNNs) (Li et al., 2015), have been applied to a wide range of tasks, including fluid dynamics and structural mechanics, where they model physical interactions through message passing between mesh nodes. In recent years, these methods have been enhanced in works such as MeshGraphNet (Pfaff et al., 2020; Sanchez-Gonzalez et al., 2020), where the geometries are represented with nodal information and physical relationships are learned using mesh faces to train more accurate and generalizable ML models. While these GNN-based models have made significant advancement in modeling numerical simulations, they face challenges in terms of scalability and capturing long-range interactions. Several methods and frameworks have been developed to address the scalability challenge by decomposing the graphs into smaller subdomains and parallelizing the training algorithms on multi-gpu and multi-node architectures (Nabian, 2024; Kakka et al., 2024; Nastorg, 2024).
基于网格的方法一直是建模数值模拟的另一个重点领域。在这些方法中，机器学习模型直接在模拟网格上进行训练，并学习使用局部网格信息来预测解场。该领域的早期工作，如 GraphNet (Allen 等人，2023) 和门控图神经网络 (GGNNs) (Li 等人，2015)，已被应用于广泛的任务，包括流体动力学和结构力学，其中它们通过网格节点之间的消息传递来模拟物理交互。近年来，这些方法在 MeshGraphNet (Pfaff 等人，2020; Sanchez-Gonzalez 等人，2020) 等工作中得到了增强，其中几何形状用节点信息表示，物理关系通过网格面进行学习，以训练更准确和泛化的机器学习模型。虽然这些基于 GNN 的模型在建模数值模拟方面取得了显著进展，但它们在可扩展性和捕捉长程交互方面仍面临挑战。已经开发出几种方法和框架来解决可扩展性挑战，通过将图分解成更小的子域，并在多 GPU 和多节点架构上并行化训练算法（Nabian, 2024; Kakka 等人, 2024; Nastorg, 2024)。

The classes of ML methods described above provide different paradigms for modeling numerical simulations. However, a few shortcomings of these methods are outlined below, and subsequently are motivations for this work. Most ML methods struggle to scale to large-scale simulations with element counts ranging in hundreds of millions or billions due to significant memory and compute requirements. The methods that scale to large meshes demonstrate lower accuracy and generalizability due to their inability to efficiently represent the geometries and capture long-range interactions. Typically, ML methods try to learn global geometry representations which are then used to predict solution fields on the surface and in the volume. However, global geometry representations are high-dimensional and dense and are not efficient in accurately predicting solution fields especially in large computational domains and when geometries have finer features. Finally, many of the ML methods are dependent on the spatial structure of the input and do not generalize to other arbitrary distributions. For example, ML algorithms trained on simulation meshes do not generalize and have lower accuracy when evaluated on uniform point clouds or meshes.
上述描述的机器学习方法为数值模拟提供了不同的建模范式。然而，这些方法的几个缺点如下，并随后是这项工作的动机。大多数机器学习方法由于巨大的内存和计算需求，难以扩展到具有数亿个元素的大规模模拟。扩展到大型网格的方法由于无法有效地表示几何形状和捕捉长程相互作用，表现出较低的精度和泛化能力。通常，机器学习方法尝试学习全局几何表示，然后用于预测表面和体积上的解场。然而，全局几何表示是高维和密集的，并且在大型计算域中以及当几何形状具有更精细特征时，它们在准确预测解场方面效率不高。最后，许多机器学习方法依赖于输入的空间结构，并且无法泛化到其他任意分布。例如，在模拟网格上训练的机器学习算法在评估均匀点云或网格时泛化能力差，精度也较低。

In this work, we propose a novel model architecture, DoMINO, a decomposable, multi-scale, iterative neural operator, to provide scalable, accurate and generalizable surrogates to model large-scale numerical simulations. We demonstrate the DoMINO model using the external aerodynamics use case. External aerodynamics simulations on realistic car geometries involve significantly large surface and volume meshes to capture the near-wall effects and resolve the flow fields in the downstream wake and near other parts of the car such as underbody, side mirrors etc. Accurate and fast predictions of velocity fields, pressure and wall-shear-stresses obtained from aerodynamic simulations are important for providing designers with guidance in developing optimized and innovative vehicle designs. Due to the large-scale nature of this use case, it is ideal for validating the scalability, accuracy and generalizability of ML models. Additionally, as vehicle geometries are complex it requires ML models to efficiently represent the large and small features present in these geometries to enable accurate predictions. In the past, several studies have explored the development of ML surrogates for this application. Chen & Akasaka (2021) use a 3-D UNet architecture (Ronneberger et al., 2015) in combination with a custom loss function including the continuity equation loss to model flow fields around vehicles. They use SDFs as inputs to their model to represent and parameterize the car geometries. Jacob et al. (2021) follows a similar approach where they use a modified 3-D UNet architecture to learn a mapping between a SDF input and the output flow fields. However, they also use the bottleneck layer of the UNet to simultaneously predict the drag coefficient using a fully-connected neural network. Ananthan et al. (2024) provide a comprehensive study of modeling vehicle aerodynamics. They provide 4 different case studies, ranging from surface predictions using MeshGraphNets (Pfaff et al., 2020) on the Drivaer car dataset, volume predictions with UNet on a motorbike dataset generated OpenFoam tutorial (Jasak et al., 2007) to a stable diffusion model to generate different variations starting from a base car geometry. Elrefaie et al. (2024) proposed a RegDGCNN model to directly learn surface aerodynamic quantities from meshes by integrating encoding capability of PointNet with graph CNNs. These advances demonstrate the potential of machine learning in accelerating design cycles by providing rapid predictions of aerodynamics quantities. However, challenges still exist in terms of generalizing across diverse geometries, scaling to large simulation meshes and the ability to accurately predict both surface and volume aerodynamic quantities.
在这项工作中，我们提出了一种新的模型架构，DoMINO，一种可分解的多尺度迭代神经算子，以提供可扩展、准确和通用的替代模型来模拟大规模数值模拟。我们使用外部空气动力学用例来展示 DoMINO 模型。在真实汽车几何形状上的外部空气动力学模拟涉及非常大的表面和体积网格，以捕获近壁效应并解析下游尾流和汽车其他部分（如底盘、侧镜等）附近的流场。从空气动力学模拟中获得的速度场、压力和壁面剪切应力的准确和快速预测对于为设计师提供指导，以开发优化和创新车辆设计非常重要。由于这个用例的大规模性质，它非常适合验证机器学习模型的可扩展性、准确性和泛化能力。此外，由于车辆几何形状复杂，它需要机器学习模型有效地表示这些几何形状中存在的大和小特征，以实现准确预测。过去，一些研究探索了为该应用开发机器学习替代模型。Chen & Akasaka (2021) 使用 3-D UNet 架构（Ronneberger 等人，2015 年）与包含连续性方程损失的定制损失函数相结合，来模拟车辆周围的流场。他们使用 SDF 作为模型的输入，以表示和参数化汽车几何形状。Jacob 等人（2021 年）采用了类似的方法，他们使用修改后的 3-D UNet 架构来学习 SDF 输入和输出流场之间的映射。然而，他们还使用 UNet 的瓶颈层，同时使用全连接神经网络预测阻力系数。Ananthan 等人（2024 年）对车辆空气动力学建模进行了全面研究。他们提供了 4 个不同的案例研究，范围从在 Drivaer 汽车数据集上使用 MeshGraphNets（Pfaff 等人，2020 年）进行表面预测，到在 OpenFoam 教程生成的摩托车数据集上使用 UNet 进行体积预测（Jasak 等人，2007 年），再到使用稳定扩散模型从基础汽车几何形状开始生成不同变化。Elrefaie 等人 (2024) 提出了一种 RegDGCNN 模型，通过将 PointNet 的编码能力与图卷积神经网络相结合，直接从网格中学习表面空气动力学量。这些进步展示了机器学习在通过提供快速空气动力学量预测来加速设计周期方面的潜力。然而，在跨多种几何形状泛化、扩展到大型模拟网格以及准确预测表面和体积空气动力学量方面仍然存在挑战。

In this paper, we demonstrate the DoMINO model for the external aerodynamics use case to address these challenges. We train the model on a select set of samples from the DrivAerML dataset (Ashton et al., 2024) and validate the model on the remaining samples. The DrivAerML dataset is chosen for this experiement because it consists of high-fidelity simulation results on large meshes ranging in hundreds of millions. Moreover, the geometric variations in this dataset are significant, resulting in drag force variations from 300 N to 600 N across the various designs. Through the experiments we demonstrate the ability of the DoMINO model to capture both surface and volume flow fields as well as other relevant engineering metrics such as design trends and drag force comparisons. Additionally, we showcase that DoMINO model is mesh independent by validating the surface predictions on a uniformly sampled point cloud instead of the simulation mesh. The results highlight the potential of DoMINO as a practical and efficient surrogate models for aerodynamic evaluation, supporting real-time applications in automotive design and optimization.
本文展示了 DoMINO 模型在外部空气动力学应用案例中解决这些挑战。我们在 DrivAerML 数据集（Ashton 等人，2024 年）的选定样本上训练该模型，并在剩余样本上验证该模型。选择 DrivAerML 数据集进行此实验，是因为它包含在数亿级别的网格上的高保真模拟结果。此外，该数据集中的几何变化很大，导致不同设计的阻力从 300 N 变化到 600 N。通过实验，我们证明了 DoMINO 模型能够捕捉表面和体积流场以及其他相关工程指标，如设计趋势和阻力比较。此外，我们展示了 DoMINO 模型与网格无关，通过在均匀采样的点云上验证表面预测，而不是模拟网格。结果突出了 DoMINO 作为空气动力学评估的实用且高效的替代模型的潜力，支持汽车设计和优化的实时应用。

The paper is structured as follows. In Section 2, we provide an overview of the DoMINO model architecture and delve deeper into the details of the architecture. Section 3 presents additional details related to the experiment, problem setup, dataset etc. Following this, we present the results and analysis in Section 4.
本文结构如下。在第二节中，我们概述了 DoMINO 模型架构并深入探讨了架构的细节。第三节介绍了与实验、问题设置、数据集等相关的更多细节。随后，我们在第四节中展示了结果和分析。

2 DoMINO Model Overview 2DoMINO 模型概述

In this section, we introduce the DoMINO approach to learn local geometry encodings from point cloud representations to predict PDE solutions on discrete points sampled in computational domain using dynamically constructed computational stencils in local regions around it. We present how the DoMINO model leverages local features within sub-regions of the computational domain to predict solutions on both the surface of geometry as well as in the computational domain volume. Predicting both sets of quantities is extremely important in industrial applications for making crucial choices and decisions regarding product and process design.
在本节中，我们介绍了 DoMINO 方法，该方法从点云表示中学习局部几何编码，以使用在局部区域周围动态构建的计算模板来预测离散点上的 PDE 解。我们展示了 DoMINO 模型如何利用计算域子区域内的局部特征来预测几何表面以及计算域体积上的解。预测这两组量在工业应用中至关重要，因为它关系到产品和过程设计的关键决策。

An overview diagram for the DoMINO model architectures is shown in Fig. 1. Although we present an automotive aerodynamics example for explanation purposes, the approach is applicable to other applications in engineering simulation as well. DoMINO takes the 3-D surface mesh of the geometry as input. A 3-D bounding box is constructed around the geometry to represent a computational domain that captures the most important solutions fields required for design guidance. The arbitrary point cloud representation is transformed into a N-D fixed, structured representation of resolution $m\times m\times m\times f$ defined on the computational domain. A multi-resolution, iterative approach described in Section 2.1, is used to propagate geometry representation into the computational domain and to learn short- and long-range dependencies. The N-D structured representation is a unique global encoding for a given geometry and encapsulates the relevant features of the geometry. However the global encoding is high-dimensional and dense, while the solution at any point in the computational domain is heavily influenced by the information in its locality. As a result, a local encoding needs to be extracted from the global geometry encoding to enable accurate predictions of solutions.
DoMINO 模型架构的概述图如图 1 所示。尽管我们为了解释目的展示了一个汽车空气动力学示例，但该方法也适用于工程模拟中的其他应用。DoMINO 将几何形状的三维表面网格作为输入。在几何形状周围构建一个三维边界框，以表示一个计算域，该计算域捕获设计指导所需的最重要解域。任意的点云表示被转换为在计算域上定义的分辨率为 $m\times m\times m\times f$ 的 N 维固定结构表示。使用第 2.1 节中描述的多分辨率迭代方法，将几何形状表示传播到计算域，并学习短程和远程依赖关系。N 维结构表示是给定几何形状的唯一全局编码，封装了该几何形状的相关特征。然而，全局编码是高维且密集的，而计算域中任何点的解严重受到其局部信息的影响。因此，需要从全局几何编码中提取局部编码，以实现对解的准确预测。

Next, we sample a batch of discrete points in the computational domain where we want to evaluate the solution. When the model is training, these can be sampled from points where the solution is known, for example the nodes of the simulation mesh, while during inference they can be sampled randomly as the simulation mesh may not be available. For each of the sampled points in a batch, a subregion of size $l\times l\times l$ is defined around it and a local geometry encoding is calculated. The local encoding is essentially a subset of the global encoding depending on its position in the computational domain and is calculated using point convolutions. Additional details about the local encoding are provided in Section 2.2.
接下来，我们在计算域中采样一批离散点，在这些点上我们需要评估解。当模型训练时，这些点可以从未知解的点中采样，例如模拟网格的节点，而在推理期间，它们可以随机采样，因为模拟网格可能不可用。对于一批中采样的每个点，在其周围定义了一个大小为 $l\times l\times l$ 的子区域，并计算了局部几何编码。局部编码本质上是全局编码的一个子集，取决于其在计算域中的位置，并使用点卷积计算。关于局部编码的更多细节在 2.2 节中提供。

Furthermore, for each of the sampled points in a batch $p$ neighboring points are sampled in the computational domain to form a local computational stencil of $p+1$ points. A batch of computational stencils are represented by their local coordinates in the domain and normals whenever available, especially for points sampled on the surface of the geometry. An aggregation network is trained to to predict the solutions on each of the discrete points in the batch using the local geometry encoding and the input features of the computational stencil. More details about the aggregation network are provided in Section 2.3.
此外，对于批次 $p$ 中的每个采样点，在计算域中采样其 $p+1$ 个邻近点以形成一个局部计算模板。一批计算模板由其在域中的局部坐标和法线表示，尤其是在几何表面上采样的点。一个聚合网络被训练用来使用局部几何编码和计算模板的输入特征来预测批次中每个离散点的解。聚合网络的更多细节在 2.3 节中提供。

The solution variables on surface typically differ from those calculated in the volume. For example, in the external aerodynamics use case, the solutions variables on surface include pressure and the wall shear stress vector and the volume variables include pressure, velocity and turbulence parameters. As a result, the aggregation neural network needs to be defined separately for predicting volume and surface variables but the global geometry encoding network can be shared between them.
表面上的解变量通常与体积中计算的变量不同。例如，在外部空气动力学应用案例中，表面上的解变量包括压力和壁面剪切应力矢量，而体积变量包括压力、速度和湍流参数。因此，聚合神经网络需要分别定义以预测体积和表面变量，但全局几何编码网络可以在它们之间共享。

Refer to caption — Figure 1: DoMINO architecture
图 1：DoMINO 架构

The code for the DoMINO is available in the Modulus repository (Contributors, 2023) on Github (https://github.com/NVIDIA/modulus/tree/main/examples/cfd/external_aerodynamics/domino).
DoMINO 的代码可以在 Modulus 仓库（Contributors，2023）中的 Github 上找到（https://github.com/NVIDIA/modulus/tree/main/examples/cfd/external_aerodynamics/domino）。

2.1 Global geometry representation
2.1 全局几何表示

In this section, we describe the details of the geometry representation network used to calculate the global geometry representation from a point cloud. Fig. 2 describes the various components of the geometry representation neural network. First, a tight-fitted bounding box is constructed around the surface in addition to a bounding box representing the computational domain as shown in Fig. 2A. The features of the geometry point cloud, such as spatial coordinates, are projected on to a N-D structured grid of resolution $m\times m\times m\times f$ overlaid on the surface bounding box using learnable point convolution kernels. The point convolution kernel used for projecting point clouds onto N-D structured grids is described in Eq. 1, where $\vec{x_{i}}$ , $\vec{x_{j}}$ represent different sets of point clouds, $d_{ij}$ the distance metric between them and $f$ represents a fully connected deep neural network. In this work, the point convolution kernel is implemented using custom, GPU accelerated ball query layers using NVIDIA Warp.
在本节中，我们描述了用于从点云计算全局几何表示的几何表示网络的细节。图 2 描述了几何表示神经网络的各个组件。首先，除了图 2A 所示代表计算域的包围盒外，在表面周围构建了一个紧密贴合的包围盒。几何点云的特征（如空间坐标）使用可学习的点卷积核投影到覆盖在表面包围盒上的分辨率为 $m\times m\times m\times f$ 的 N 维结构化网格上。用于将点云投影到 N 维结构化网格上的点卷积核在公式 1 中描述，其中 $\vec{x_{i}}$ 、 $\vec{x_{j}}$ 代表不同的点云集， $d_{ij}$ 表示它们之间的距离度量， $f$ 表示一个全连接的深度神经网络。在本工作中，点卷积核使用基于 NVIDIA Warp 的自定义、GPU 加速的球查询层实现。

y_{i}=\sum_{j=0}^{j=n_{y}}f(\vec{x_{i}},\vec{x_{j}},d_{ij})

(1)

The point convolution kernel depends on 2 additional factors, a radius of influence and number of points specified in the kernel ( $n_{y}$ ). The radius of influence determines the physical size of the kernel and the extent of information learned from the neighbors. A larger radius signifies a bigger kernel and means that the geometry information propagates further into the domain. A smaller radius learns smaller kernels and captures finer geoemtry details. In the DoMINO architecture, we adopt a multi-scale approach by using a range of kernel sizes controlled by the radius parameter to capture both finer geometry features and long-range interactions of geometry.
点卷积核取决于 2 个附加因素，即影响半径和在核中指定的点数（ $n_{y}$ ）。影响半径决定了核的物理大小以及从邻居处学习的信息范围。较大的半径表示更大的核，这意味着几何信息会进一步传播到域中。较小的半径学习较小的核，并捕获更精细的几何细节。在 DoMINO 架构中，我们通过使用由半径参数控制的多种核大小来采用多尺度方法，以捕获更精细的几何特征和几何的长程相互作用。

A structured grid of the same resolution is also constructed in the computational domain bounding box. Geometry features are propagated into the computational domain bounding box using 2 methods, 1) a separate set of multi-scale point convolution kernels are learned to project geometry information on to the computational domain grid and 2) the features on the surface bounding box grid ( $G_{s}$ ) are propagated to the computational domain bounding box grid (( $G_{c}$ ) using CNN blocks containing convolution, pooling, and unpooling layers. The CNN blocks are evaluated iteratively for a specified number of iterations. Although not explored in this work, a stopping criterion (such as $L_{2}(G_{s},G_{c})<1e^{-3}$ ) can be specified to break the iterations using a fixed-point iteration strategy. The $m\times m\times m\times f$ features calculated on the grid in the computational domain represent the geometry point cloud. In addition this, a SDF and its gradient components are also calculated on the computational domain grid and appended to the learned features to provide additional information about the topology of the geometry.
在计算域边界框中也构建了一个相同分辨率的结构化网格。使用 2 种方法将几何特征传播到计算域边界框中，1)学习一组多尺度点卷积核，将几何信息投影到计算域网格上，2)使用包含卷积、池化和解池化层的 CNN 块，将边界框网格（ $G_{s}$ ）上的特征传播到计算域边界框网格（（ $G_{c}$ ））。CNN 块在指定次数的迭代中被评估。尽管在本工作中未进行探索，但可以指定一个停止标准（例如 $L_{2}(G_{s},G_{c})<1e^{-3}$ ），使用定点迭代策略来中断迭代。在计算域网格上计算的 $m\times m\times m\times f$ 特征表示几何点云。此外，还在计算域网格上计算了 SDF 及其梯度分量，并将它们附加到学习到的特征中，以提供有关几何拓扑的附加信息。

2.2 Local geometry representation
2.2 局部几何表示

The local geometry representation depends on the physical location within the computational domain where the solutions fields are evaluated. Hence, before computing the local geometry representation, a batch of points is sampled in the computational domain. For each sampled point in the batch, $p$ neighboring points are sampled randomly around them to form a computational stencil of points similar to finite volume and element methods. Fig. 1 represents a single sampled point and $p$ neighboring points randomly sampled around it. The local geometry representation is learned by drawing a subregion around the computational stencil of $p+1$ points. A point convolution kernel, as described in Eq. 1, is used to extract the local features of resolution, $l\times l\times l\times n_{f}$ , in this subregion from the global geometry representation on the computational domain. The extracted local features are further transformed using fully connected neural networks. This local geometry representation is used to evaluate the solution fields on the sampled points using the aggregation network described Section 2.3.
局部几何表示取决于在计算域中求解场被评估的物理位置。因此，在计算局部几何表示之前，会在计算域中采样一批点。对于批次中的每个采样点，会随机采样 $p$ 个邻近点，以形成类似于有限体积和有限元方法的计算模板点。图 1 表示一个单独的采样点及其周围随机采样的 $p$ 个邻近点。局部几何表示是通过在 $p+1$ 个计算模板点周围绘制一个子区域来学习的。使用公式 1 中描述的点卷积核，从计算域的全局几何表示中提取该子区域中分辨率为 $l\times l\times l\times n_{f}$ 的局部特征。提取的局部特征进一步通过全连接神经网络进行转换。这个局部几何表示用于使用 2.3 节描述的聚合网络在采样点上评估求解场。

2.3 Aggregation network 2.3 聚合网络

The local geometry representation represents the learned features of the geometry and solution in the vicinity of the computational stencil of the sampled points and its neighbors. Each of the points in this computational stencil are represented by their physical coordinates in the computational domain, SDF at these coordinates, normal vector from the center of mass of the domain, and surface normals (only if the point is on the surface). These input features are passed through a fully connected neural network, known as the basis function neural network, where a latent vector is computed representing these features for each of the points in the computational stencil. Each latent vector is concatenated with the local geometry encoding and passed through another set of fully-connected layers to predict a solution vector on each of the points in the computational stencil. The solution vector is averaged using an inverse distance weighing scheme to predict the final solution vector at the sampled point. A separate instance of the aggregation network is used for each solution variable but the local geometry representation across these remains the same.
局部几何表示代表在采样点及其邻域的计算模板附近学习到的几何和求解特征。计算模板中的每个点都由其在计算域中的物理坐标、这些坐标处的 SDF、从域质心出发的法向量以及表面法向量（仅当点位于表面时）来表示。这些输入特征通过一个全连接神经网络（称为基函数神经网络）进行处理，其中计算出一个潜在向量，代表计算模板中每个点的特征。每个潜在向量都与局部几何编码连接起来，并通过另一组全连接层来预测计算模板中每个点的解向量。解向量使用逆距离加权方案进行平均，以预测采样点的最终解向量。每个解变量都使用一个单独的聚合网络实例，但它们之间的局部几何表示保持相同。

3 Experiments details 3 实验细节

In this section, we demonstrate the application of DoMINO to predict key aerodynamic quantities on the surface of cars and in the volume around the cars. Most ML models are in literature exhibit a trade-off between scalability, accuracy and generalizability. ML models are typically trained on significantly down sampled meshes or point clouds. However, the proposed DoMINO model architecture provides a possible solution to simulation of large-scale aerodynamic simulations without sacrificing accuracy.
在本节中，我们展示了 DoMINO 在预测汽车表面和汽车周围空间的关键空气动力学量方面的应用。文献中的大多数机器学习模型都表现出可扩展性、准确性和泛化能力之间的权衡。机器学习模型通常在显著下采样的网格或点云上进行训练。然而，所提出的 DoMINO 模型架构为不牺牲准确性的大规模空气动力学模拟提供了一种可能的解决方案。

3.1 Problem setup 3.1 问题描述

In this study, the DoMINO model is trained to predict volume and surface fields in the computational domain. The volume fields considered are, velocity vector, $\vec{u}$ , pressure, $p$ and turbulent viscosity, $\nu_{t}$ and the surface fields are pressure, $p$ , and the wall-shear-stress vector, $\vec{\tau}$ . The input to the model is an STL of the car geometry which is a triangulated representation of the surface. The coordinates of the nodes of the STL are used as input to learn the encoded geometry representation. In the aggregation network, the coordinates of the sampled points, SDF and the normal vector from the center of mass of the car geometry are considered as input features. When points are sampled on the surface of the car geometry, the surface normals at those points are also considered as part of the input features. Hidden representations learned from these input features, using neural networks, combined with the local encoded geometry representation are used to predict the respective fields on the sampled points. If the points are sampled on the surface, then the surface fields are predicted and compared with ground-truth values available on those sampled points to calculate loss. On the other hand, if the points are sampled in the volume around the car geometry then the volume fields are computed and compared with the corresponding ground truth values to calculate loss. The computational domain is trimmed for training the DoMINO model such that 2 car lengths in the downstream flow direction, 1 car length in the upstream direction and 1 car width and height in the other axial directions are considered. Finally, it is important to note that the simulation data is used as is to train the model and is not down-sampled or pre-processed in any manner.
在本研究中，DoMINO 模型被训练用于预测计算域中的体积场和表面场。考虑的体积场包括速度矢量、 $\vec{u}$ 、压力、 $p$ 和湍流粘度、 $\nu_{t}$ ，表面场为压力、 $p$ 和壁面剪切应力矢量、 $\vec{\tau}$ 。模型的输入是汽车几何形状的 STL 文件，它是表面的三角表示。STL 节点的坐标被用作学习编码的几何表示的输入。在聚合网络中，采样点的坐标、SDF 以及汽车几何形状质心处的法向量被作为输入特征。当在汽车几何形状的表面上采样点时，这些点的表面法线也被视为输入特征的一部分。使用神经网络从这些输入特征中学习到的隐藏表示，与局部编码的几何表示相结合，用于预测采样点上的相应场。如果点是在表面上采样的，那么预测表面场，并将其与采样点上的真实值进行比较以计算损失。另一方面，如果点在汽车几何形状周围的体积中采样，则计算体积场并将它们与相应的真实值进行比较以计算损失。计算域被修剪用于训练 DoMINO 模型，以便在下游流动方向上考虑 2 个车长，在上游方向上考虑 1 个车长，在其他轴向方向上考虑 1 个车宽和高度。最后，重要的是要注意，模拟数据被原样用于训练模型，并且没有被下采样或以任何方式进行预处理。

3.2 Dataset 3.2 数据集

In this study, we use the DrivAerML dataset, a publicly available, high-fidelity dataset comprising of volume and surface data for 500 geometrically morphed variants of the DrivAer Notchback car (Ashton et al., 2024). Figure 4 shows different examples of the car geometry variations performed in this dataset. The simulations in this dataset were carried out using the hybrid RANS/LES scale-resolving method. The time-averaged data is stored in the form of VTP and VTU files for each geometry variant. The VTP files contain the surface fields, namely pressure and wall-shear-stresses whereas the VTU files contain the volume fields, namely velocity, pressure and turbulent viscosity. The volume meshes across the 500 simulations cases are of the order of 150 million elements and the surface meshes are of the order 10 million elements. We split the simulations into a train and test set based on the drag force trends. 10% of the samples are reserved in the test set with about 20% of the test set consisting of out-of-distribution samples based on the ranges of the drag force. The out-of-distribution samples contain the cases with some of the lowest and the highest drag forces and are not seen by the model during training.
在本研究中，我们使用 DrivAerML 数据集，这是一个公开可用的高保真度数据集，包含 DrivAer Notchback 汽车 500 种几何形态变体的体积和表面数据（Ashton 等人，2024 年）。图 4 展示了该数据集中执行的不同汽车几何形态变化示例。该数据集中的模拟是使用混合 RANS/LES 尺度解析方法进行的。时间平均数据以 VTP 和 VTU 文件的形式存储在每个几何变体中。VTP 文件包含表面场，即压力和壁面剪切应力，而 VTU 文件包含体积场，即速度、压力和湍流粘度。500 个模拟案例中的体积网格约为 1.5 亿个单元，表面网格约为 1000 万个单元。我们根据阻力力趋势将模拟分为训练集和测试集。10%的样本被保留在测试集中，其中大约 20%的测试集由基于阻力力范围的分布外样本组成。分布外样本包含一些阻力和最高阻力的情况，在训练期间模型没有看到。

3.3 Model training 3.3 模型训练

The DoMINO model is trained using the Adam optimizer. A reduce-on-plateau learning rate scheduler is used such that the initial learning rate, $1e^{-3}$ , progressively drops down to a final learning rate, $1e^{-6}$ , with the training epochs. The model is trained for a total of 500 epochs. Training is performed in float-16 precision format using Automatic Mixed Precision (AMP). The loss function used for training includes a mean-squared-error loss calculated on the sampled mesh points evaluated by the model on the surface of the car geometry and in the volume around it. The loss is calculated separately on each solution variable and then added together. In addition a mean-squared-loss is calculated for the area-weighted surface predictions for each surface variable and added to the overall loss formulation. During each epoch of training, the loss is calculated only on a subset of points randomly sampled from the simulation mesh and a new set is sampled each time. An area-weighted sampling algorithm is used for the sampling points on the surface, whereas a random uniform sampling algorithm is used for the volume points. Similarly, nodes of the STL are also sampled using a random uniform sampling algorithm for learning the geometry encoding. The number of points that can be sampled every epoch is dependent on the GPU memory availability. Additional details related to the model hyperparameters and training routine can be found in the Modulus Github repo (Contributors, 2023).
DoMINO 模型使用 Adam 优化器进行训练。采用学习率衰减调度器，初始学习率 $1e^{-3}$ ，随着训练轮次的增加，逐步衰减到最终学习率 $1e^{-6}$ 。模型总共训练 500 个轮次。训练过程使用 float-16 精度格式，并采用自动混合精度（AMP）技术。训练中使用的损失函数包括在模型对汽车几何表面及其周围体积的采样网格点上计算的均方误差损失。每个解变量上的损失单独计算后相加。此外，对每个表面变量进行面积加权表面预测，并计算均方损失，加到总损失公式中。在每次训练轮次中，仅在从模拟网格中随机采样的点子集上计算损失，并且每次采样都使用新的点集。对于表面上的采样点，使用面积加权采样算法，而对于体积点，使用随机均匀采样算法。同样地，STL 节点的采样也使用随机均匀采样算法来学习几何编码。每个 epoch 可以采样的点数取决于 GPU 内存的可用性。有关模型超参数和训练流程的更多详细信息，请参阅 Modulus Github 仓库（Contributors，2023）。

4 Results and analysis 4 结果与分析

The trained DoMINO model is validated on a test set across several metrics for both the surface and volume quantities. The test set consists of both in-distribution and out-of-distribution samples, and the categorization is carried out based on the minimum and maximum value of the drag force in the training set. For example, if a test sample has a drag force that falls within the min-max range of the training set, then it is considered an in-distribution sample and if outside, then it is an out-of-distribution sample. In addition to the average error metrics and design trends, in this section, we will also present the contour and line plots for both surface and volume quantities for 2 out-of-distribution test samples, ids $419$ and $439$ from the dataset. These samples correspond to designs resulting the smallest and largest drag force in the complete DrivAerML dataset and are not a part of the training set. It may be observed from Fig. 5 that there is substantial geometric variation between these samples resulting in around 300 N difference in the calculated drag force.
训练好的 DoMINO 模型在多个指标上对表面和体积量进行了测试集验证。测试集包含分布内和分布外样本，分类基于训练集中阻力力的最小值和最大值。例如，如果一个测试样本的阻力力落在训练集的最小值-最大值范围内，则被视为分布内样本；如果超出范围，则被视为分布外样本。除了平均误差指标和设计趋势，在本节中，我们将为 2 个分布外测试样本（数据集中的 id $419$ 和 $439$ ）展示表面和体积量的轮廓和线图。这些样本对应于完整 DrivAerML 数据集中产生最小和最大阻力力的设计，并且不是训练集的一部分。从图 5 可以看出，这些样本之间存在显著的几何变化，导致计算出的阻力力差异约为 300 N。

4.1 Surface quantities 4.1 表面量

First we present the results on the prediction of surface quantities. In Table 1 we show the relative $l_{2}$ and area weighted relative $l_{2}$ errors for pressure and wall-shear stress fields calculated on the surface of the car and averaged over all the test cases. The relative $l_{2}$ errors are calculated using Eq. 2.
首先，我们展示了表面量预测的结果。在表 1 中，我们展示了计算在汽车表面并针对所有测试案例进行平均的压力和壁面剪切应力场的相对 $l_{2}$ 和面积加权相对 $l_{2}$ 误差。相对 $l_{2}$ 误差是使用公式 2 计算的。

\epsilon=\frac{\sqrt{\sum y_{T}^{2}-y_{P}^{2}}}{\sqrt{\sum y_{T}^{2}}}

(2)

where, $y_{T}$ and $y_{P}$ correspond to the true and predicted fields on cell centers of surface elements. For area weighted errors the field values are multiplied by the corresponding facet areas of the elements. The area-weighted relative errors are lower for all solution variables indicating that most of the errors occur on surface facets with small areas. The relative errors for Y and Z wall-shear-stresses are comparatively worse than pressure and X wall-shear-stress because of their small magnitudes.
其中， $y_{T}$ 和 $y_{P}$ 对应于表面单元中心处的真实场和预测场。对于面积加权误差，场的值乘以单元的相应面面积。所有解变量的面积加权相对误差都较低，表明大部分误差发生在面积较小的表面单元上。Y 和 Z 壁面剪切应力的相对误差比压力和 X 壁面剪切应力差，因为它们的幅值较小。

Table 1: Average surface error metrics on test set
表 1：测试集上的平均表面误差指标

Field 场	Rel. L-2 相对 L-2	Area-weighted rel. L-2 面积加权相对 L-2
Pressure 压力	0.1505	0.1181
X-Wall-Shear X 向壁面剪切	0.2124	0.1279
Y-Wall-Shear Y 向壁面剪切	0.302	0.2769
Z-Wall-Shear Z 向壁面剪切	0.3359	0.229

In Fig. 6 we present the surface contour comparisons between simulated results and DoMINO predictions and the error between them. It maybe observed that both pressure and wall-shear-stress magnitude are captured reasonably well on different surfaces of both the test designs including the windshield, side mirrors, underbody etc.
如图 6 所示，我们展示了模拟结果与 DoMINO 预测之间的表面等值线比较以及它们之间的误差。可以观察到，在测试设计的不同表面上，包括挡风玻璃、侧镜、底盘等，压力和壁面剪切应力的大小都被合理地捕捉到了。

Fig. 7 shows the drag force design trends and the regression plot between the simulated and DoMINO predicted drag force for the various test designs. The dashed blue line in the regression plot represents the ideal case where, predictions match the true values perfectly. We conduct this experiment with 2 scenarios, 1) solution fields, pressure and x-wall-shear-stress are evaluated on the surface mesh and integrated to calculate the drag force and 2) solution fields, pressure and x-wall-shear-stress are evaluated on a uniform point cloud with 10 million points generated on the surface from the STL representation and integrated to calculate the drag fore. Each point in the uniform point cloud has the same surface area and the normals are derived from the STL facets. In both the scenarios, it may be observed that the coefficient of determination $R_{2}$ is 0.96 indicating a reasonable match between simulated drag force and DoMINO predictions. The design trends are calculated by arranging the drag forces of the test designs in an ascending order and comparing them against the drag force predictions for the same designs. The design trends are captured well, however smaller directional changes between successive designs shows some oscillatory behavior in the DoMINO predictions and resolving this will be a focus of future efforts. Finally, this experiment shows the invariance of the DoMINO model architecture to the spatial distribution of points. In both scenarios, the predicted regression coefficient and the drag force trends match very closely. The independence and insensitivity between the model and the spatial distribution of evaluation points is a an extremely important feature of this architecture especially for modeling engineering simulations where meshing can be expensive.
图 7 显示了不同测试设计的阻力力设计趋势以及模拟的阻力力与 DoMINO 预测阻力力之间的回归图。回归图中的蓝色虚线表示理想情况，即预测值与真实值完全匹配。我们进行了两个场景的实验，1) 求解场、压力和 x 壁剪切应力在表面网格上被评估并积分以计算阻力力，2) 求解场、压力和 x 壁剪切应力在由 STL 表示在表面上生成的 1000 万个点的均匀点云上被评估并积分以计算阻力力。均匀点云中的每个点具有相同的表面积，法线是从 STL 面推导出来的。在两个场景中，都可以观察到确定系数 R²为 0.96，表明模拟的阻力力与 DoMINO 预测之间有合理的匹配。设计趋势是通过将测试设计的阻力力按升序排列并与相同设计的阻力力预测进行比较来计算的。设计趋势捕捉得很好，然而连续设计之间较小的方向变化显示出 DoMINO 预测的振荡行为，解决这一问题将是未来工作的重点。最后，这项实验表明 DoMINO 模型架构对点的空间分布具有不变性。在两种情况下，预测的回归系数和阻力力趋势都非常吻合。模型与评估点的空间分布之间的独立性和不敏感性是这种架构的一个极其重要的特征，特别是在建模工程仿真时，网格划分可能很昂贵。

4.2 Volume quantities 4.2 体积量

Table 2: Average volume error metrics on test set
表 2：测试集上的平均体积误差指标

Field 场	Rel. L-2 相对 L-2
Pressure 压力	0.2193
X-velocity	0.2397
Y-velocity Y-速度	0.5025
Z-velocity Z 向速度	0.4567
Turb-viscosity 湍流粘度	0.2175

Next, we present the volumetric results obtained from the DoMINO model evaluated in a bounding box around the car. Table 2 shows the relative $l_{2}$ errors, as described in 2, observed between DoMINO predictions and the simulated results averaged over the test set for different volume variables including, velocity, pressure and turbulent viscosity. A higher relative error is observed for Y and Z velocity components due to their sparse distribution and lower magnitudes in large parts of the computational domain. Fig 8 shows the comparisons between simulated results and DoMINO predictions for the pressure, velocity magnitude and turbulent-viscosity fields plotted on the X-Z and X-Y planes. The X-Y plane is cut through the center of the car (z = 0) while the X-Z plane represents the flow dynamics in the region between the floor and the car underbody. Additionally in Fig. 9, we plot the predicted and simulated velocities at 5 locations, in the wake at downstream distances of x=4 and x=5, on the rear and front wheel and along the centerline through the car. It can be observed from the contours as well as the line plots that the predicted fields capture the flow dynamics reasonably accurately, especially in regions of the wake, underbody, and near the hood of the car. Moreover, sparse fields such as turbulent viscosity are also modeled accurately in the wake downstream of the car. The contours for all fields show an odd behavior near the edges of the bounding box. Although this behavior does not affect the important regions near the car where the flow field predictions are reasonably accurate, the reason for it is not clearly understood and will be investigated further.
接下来，我们展示了在围绕汽车的边界框中从 DoMINO 模型评估得到的体素结果。表 2 显示了在 2 中描述的相对误差，这是在 DoMINO 预测和模拟结果在测试集上的平均不同体积变量（包括速度、压力和湍流粘度）之间观察到的。由于 Y 和 Z 速度分量在计算域的大部分区域分布稀疏且幅值较低，因此观察到更高的相对误差。图 8 显示了在 X-Z 和 X-Y 平面上模拟结果和 DoMINO 预测的压力、速度大小和湍流粘度场的比较。X-Y 平面通过汽车中心（z=0）切割，而 X-Z 平面表示地板和汽车底盘之间区域的流动动力学。此外，在图 9 中，我们在下游距离 x=4 和 x=5 的尾流中的 5 个位置、后轮和前轮以及沿汽车中心线的位置绘制了预测和模拟速度。从等高线和线形图可以观察到，预测场合理地捕捉了流动动力学，特别是在汽车的尾流区、底盘和车顶附近。此外，诸如湍流粘性等稀疏场也在汽车尾流下游被准确地建模。所有场的等高线在边界框的边缘附近表现出一种奇特的特性。虽然这种行为不会影响汽车附近的重要区域，那里的流场预测是相当准确的，但它产生的原因尚不清楚，并将进一步研究。

5 Conclusion 5 结论

In this paper, we introduced a new ML-architecture DoMINO, designed to address key challenges in the use of ML for large-scale numerical simulations. The model has several unique features that enable accurate modeling of large-scale numerical simulations.
在这篇论文中，我们介绍了一种新的机器学习架构 DoMINO，旨在解决使用机器学习进行大规模数值模拟中的关键挑战。该模型具有几个独特的特性，能够对大规模数值模拟进行精确建模。

1.

The model operates directly on point cloud representations of geometries. It uses dynamic ball query kernels implemented using NVIDIA Warp to represent geometry point clouds into global representations of geometry. NVIDIA Warp provides significant GPU acceleration as compared to using other PyTorch based alternatives. Moreover, this makes the model very flexible in utility as it does not require generation of a mesh during inference as often times mesh generation is an expensive process for large-scale industrial simulations.

1. 该模型直接作用于几何体的点云表示。它使用基于 NVIDIA Warp 实现的动态球查询核，将几何点云表示为几何的全局表示。与使用其他基于 PyTorch 的替代方案相比，NVIDIA Warp 提供了显著的 GPU 加速。此外，这使得模型在实用性上非常灵活，因为它在推理过程中不需要生成网格，因为网格生成对于大规模工业模拟来说通常是一个昂贵的流程。
2.

A range of kernel sizes are used to propagate the geometry information into the computational domain using an iterative process thereby enabling efficient modeling of both short- and long-range interactions.

2. 使用一系列的核大小通过迭代过程将几何信息传播到计算域中，从而能够有效地对短程和远程相互作用进行建模。
3.

The global geometry representations are dense and high-dimensional and cannot be used as is to accurately predict solutions in all regions of the computational domain. As a result, local geometry encodings are extracted using dynamic ball query points around the points where the solution is calculated. Additionally as these local geometry representations are learned during training they extract the essential information from the geometry required to accurately predict solution fields in different regions of the computational domain. For example, in predicting the flow fields in the wake, only the local geometry features required for learning these are extracted from the global geometry encoding during training. The local geometry encoding also improves the generality of the model to different geometries.

3. 全局几何表示是密集和高维的，不能直接用于准确预测计算域中所有区域的解。因此，使用动态球查询点提取局部几何编码，这些点位于计算解的位置周围。此外，由于这些局部几何表示在训练过程中学习，它们从几何中提取了必要的信息，以准确预测计算域不同区域的解场。例如，在预测尾流中的流场时，在训练过程中仅从全局几何编码中提取学习这些所需的局部几何特征。局部几何编码还提高了模型对不同几何的泛化能力。
4.

The solution at any point in the computational domain is heavily influenced by the local stencil of points around it. A local stencil of points, similar to a finite volume or element method, is constructed around each sampled point, processed using a basis function neural network and aggregated to calculate the solutions on the sampled points. This enables capturing local information important for accurate learning of the solution fields.

4. 计算域中任意点的解受到其周围局部点模板的严重影响。类似于有限体积或单元方法，在每个采样点周围构建一个局部点模板，使用基函数神经网络进行处理，并聚合以计算采样点的解。这能够捕捉对准确学习解场重要的局部信息。
5.

Finally, since the model can predict solutions on arbitrarily sampled points, it is not dependent on how these points are sampled. For example, the sampled points may be the nodes of a simulation mesh or randomly sampled in the computational domain. Additionally, the model evaluation is not limited in memory or compute by the number of sampled points as the evaluation can happen in batches. As a result, the DoMINO model can easily scale to large computational domains and meshes.

5. 最后，由于该模型可以在任意采样的点上预测解，因此它不依赖于这些点的采样方式。例如，采样点可以是模拟网格的节点，或是在计算域中随机采样的。此外，模型评估不受采样点数量的内存或计算限制，因为评估可以分批进行。因此，DoMINO 模型可以轻松扩展到大型计算域和网格。

The DoMINO model is demonstrated on the external automotive aerodynamics use case, which is challenging because of the large-scale nature of the simulations (meshes of the order of hundreds of millions to billions), difficulty in representing finer features of car geometries and accurately modeling both surface and volume field quantities. The experiments show that the model can accurately capture both the flow fields on the volume and surface, as well as other engineering metrics such as drag force regression and design trends. We also show the generalization of the model to prediction on out-of-distribution test samples and different mesh or point-cloud configurations. Finally, the experiments also showcase the model’s ability to scale to large meshes and point clouds with real time inference making it suitable for other engineering applications.
DoMINO 模型在外部汽车空气动力学应用案例中得到了验证，该案例具有挑战性，因为模拟具有大规模特性（网格数量达到数亿），难以表示汽车几何形状的更精细特征，以及准确建模表面和体积场量。实验表明，该模型可以准确捕捉体积和表面的流场以及其他工程指标，如阻力力回归和设计趋势。我们还展示了模型对分布外测试样本和不同网格或点云配置的泛化能力。最后，实验还展示了模型扩展到大型网格和点云并实现实时推理的能力，使其适用于其他工程应用。

Future work will focus on extending the model architecture to improving accuracy and performance, especially in resolving the oscillatory predictions using smoothing constraints in the design trends and line comparisons. We will also explore the impact of adding fixed point iterations in our iterative learning strategy. The convergence of the fixed point iteration can depend on the ratio of the physical size of the surface bounding box and computational domain bounding box. If this ratio is too small then a multi-layered approach can be used by adding bounding boxes of different sizes between the surface and computational domain and solving fixed point iterations between each transformation similar to a multi-grid approach. Currently, the model relies on fixed resolution latent spaces but in the future we would like to add multi-resolution latent spaces, in an attempt to better capture the finer features in the geometry as well as further improve the long-range interactions. Additionally, we will explore the effect of hybrid training pipelines involving constraining the model training with PDE based losses computed using automatic differentiation. Finally, we would like to extend the model to transient problems and other large-scale applications in engineering simulation.
未来的工作将集中于扩展模型架构以提高准确性和性能，特别是在使用平滑约束解决设计趋势和线比较中的振荡预测方面。我们还将探索在迭代学习策略中添加定点迭代的影响。定点迭代的收敛可能取决于表面包围盒的物理尺寸与计算域包围盒尺寸的比率。如果这个比率太小，则可以通过在表面和计算域之间添加不同尺寸的包围盒，并在每次变换之间解决定点迭代，类似于多网格方法，来使用多层方法。目前，该模型依赖于固定分辨率的潜在空间，但在未来，我们希望添加多分辨率潜在空间，以更好地捕捉几何中的更精细特征，并进一步提高长程交互。此外，我们将探索混合训练管道的效果，该管道涉及使用自动微分计算的基于 PDE 的损失来约束模型训练。最后，我们希望将模型扩展到瞬态问题和工程模拟中的其他大规模应用。

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