In this study, we present a novel computational framework that integrates the finite volume method with graph neural networks to address the challenges in Physics-Informed Neural Networks(PINNs). Our approach leverages the flexibility of graph neural networks to adapt to various types of two-dimensional unstructured grids, enhancing the model's applicability across different physical equations and boundary conditions. The core innovation lies in the development of an unsupervised training algorithm that utilizes GPU parallel computing to implement a fully differentiable finite volume method discretization process. This method includes differentiable integral and gradient reconstruction algorithms, enabling the model to directly solve partial-differential equations(PDEs) during training without the need for pre-computed data. Our results demonstrate the model's superior mesh generalization and its capability to handle multiple boundary conditions simultaneously, significantly boosting its generalization capabilities. The proposed method not only shows potential for extensive applications in CFD but also establishes a new paradigm for integrating traditional numerical methods with deep learning technologies, offering a robust platform for solving complex physical problems.
Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Microsoft Research（微软研究院）；Bielefeld University（比勒费尔德大学）
纳维-斯托克斯方程是描述诸如液体和空气的流体运动的重要偏微分方程。由于纳维-斯托克斯方程的重要性，发展高效的数值格式对科学和工程都具有重要意义。近年来，随着人工智能技术的发展，人们设计了多种方法将深度神经网络集成到不可压纳维-斯托克斯方程的流体动力学模拟和推断中，这些方法可以以无网格和可微的方式加速模拟或推断过程。本文指出现有的深度Navier-Stokes方法在处理非光滑或分数阶方程时能力有限，而这两种情况在现实中是非常关键的。为此，我们提出了\emph{Deep Random Vortex Method}（DRVM），它结合了神经网络与随机涡动力学系统相当于纳维-斯托克斯方程。具体而言，随机涡流动力学激励基于蒙特卡罗的损失函数用于训练神经网络，这避免了通过自微分计算导数。因此，DRVM不仅可以高效地求解包含粗糙路径、不可微初始条件和分数阶算子的纳维-斯托克斯方程，而且继承了基于深度学习的求解器的无网格和可微的优点。我们进行实验的柯西问题，参数求解器学习，以及2-D和3-D不可压缩的纳维-斯托克斯方程的反问题。该方法对纳维-斯托克斯方程的数值模拟和推导都取得了较好的结果。特别是在初始条件奇异的情况下，DRVM的性能明显优于现有的PINN方法。
NeuralFluid: Neural Fluidic System Design and Control with Differentiable Simulation
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；Georgia Institute of Technology（佐治亚理工学院）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Tsinghua University（清华大学）；Shanghai Qi Zhi Institute（上海期智研究院）
Generalizable data-driven turbulence closure modeling on unstructured grids with differentiable physics
School of Mechanical Engineering, Purdue University, West Lafayette, 47907, IN, USA（机械工程学院，普渡大学）；Mechanical Engineering Department, Carnegie Mellon University, Pittsburgh, 15213, PA, USA（机械工程系，卡内基梅隆大学）；Department of Aerospace Engineering, University of Michigan, Ann Arbor, 48109, MI, USA（航空航天工程系，密歇根大学）；College of Information Sciences and Technology, Pennsylvania State University, University Park, 16802, PA, USA（信息科学与技术学院，宾夕法尼亚州立大学）；Computer Science Division, Argonne National Laboratory, Lemont, 60439, IL, USA（计算机科学 division，阿贡国家实验室）
Harnessing Equivariance: Modeling Turbulence with Graph Neural Networks
Institute of Aerodynamics and Gas Dynamics, University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Centre for Analysis, Scientific Computing and Applications, Eindhoven University of Technology（埃因霍温理工大学分析、科学计算与应用中心）
Mean flow data assimilation using physics-constrained Graph Neural Networks
Dipartimento di Meccanica, Matematica e Management Politecnico di Bari（巴里理工学院机械、数学与管理系）；LISN-CNRS Université Paris-Saclay（巴黎萨克雷大学LISN-CNRS）；Digital Sciences & Technologies Department SafranTech（SafranTech数字科学与技术部门）
本文提出将 RANS 伴随梯度作为物理约束嵌入 GNN 训练，并结合 FEM 处理非结构化 CFD 数据，以在稀疏、含噪或缺损条件下提升钝体平均流重建精度。
Cross-Model Verification of Wall-Bounded Flows using Finite-JAX
Texas A&M University - Kingsville（德克萨斯农工大学金斯维尔分校）；Casady School（卡萨迪学校）；Moreau Catholic High School（莫罗天主教高中）；University of Texas at El Paso（德克萨斯大学埃尔帕索分校）
Project and Generate: Divergence-Free Neural Operators for Incompressible Flows
Incubation Institute, Fudan University, Shanghai, China（复旦大学上海孵化院）；Shanghai Academy of Artificial Intelligence for Science, Shanghai, China（上海人工智能科学研究院）
