Poisson-CNN:求解笛卡尔网格上Poisson方程的卷积神经网络

IF 2.4 Q3 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE
Ali Girayhan Ozbay, A. Hamzehloo, S. Laizet, Panagiotis Tzirakis, Georgios Rizos, B. Schuller
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引用次数: 24

摘要

摘要泊松方程在工程中很常见,例如,在计算流体动力学(CFD)中,需要计算压力场的修正,以确保速度场的不可压缩性。在目前的工作中,我们提出了一种新的全卷积神经网络(CNN)架构,在给定右侧项、任意边界条件和网格参数的情况下,以不同分辨率推断二维笛卡尔网格上泊松方程的解。它为处理偏微分方程的CNN方法提供了前所未有的通用性。通过将原始泊松问题分解为齐次泊松问题加上四个非齐次拉普拉斯子问题,使用一种新的方法来处理边界条件。该模型使用一个新的损失函数进行训练,该函数近似于预测和目标之间的连续${L}^p$范数。即使在密度比以前更大的网格上进行预测,我们的模型也证明了复制正确解决方案的令人鼓舞的能力。所提出的模型优于众所周知的神经网络模型,可以包含在CFD求解器中,以帮助求解泊松方程。分析测试案例表明,我们的CNN架构能够预测平均百分比误差低于10%的泊松问题的正确解,与传统迭代方法的第一步相比有所改进。我们模型的预测被用作Multigrid等迭代算法的初始猜测,与零初始猜测相比,单次迭代后的均方根误差可以减少90%以上。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh
Abstract The Poisson equation is commonly encountered in engineering, for instance, in computational fluid dynamics (CFD) where it is needed to compute corrections to the pressure field to ensure the incompressibility of the velocity field. In the present work, we propose a novel fully convolutional neural network (CNN) architecture to infer the solution of the Poisson equation on a 2D Cartesian grid with different resolutions given the right-hand side term, arbitrary boundary conditions, and grid parameters. It provides unprecedented versatility for a CNN approach dealing with partial differential equations. The boundary conditions are handled using a novel approach by decomposing the original Poisson problem into a homogeneous Poisson problem plus four inhomogeneous Laplace subproblems. The model is trained using a novel loss function approximating the continuous $ {L}^p $ norm between the prediction and the target. Even when predicting on grids denser than previously encountered, our model demonstrates encouraging capacity to reproduce the correct solution profile. The proposed model, which outperforms well-known neural network models, can be included in a CFD solver to help with solving the Poisson equation. Analytical test cases indicate that our CNN architecture is capable of predicting the correct solution of a Poisson problem with mean percentage errors below 10%, an improvement by comparison to the first step of conventional iterative methods. Predictions from our model, used as the initial guess to iterative algorithms like Multigrid, can reduce the root mean square error after a single iteration by more than 90% compared to a zero initial guess.
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来源期刊
DataCentric Engineering
DataCentric Engineering Engineering-General Engineering
CiteScore
5.60
自引率
0.00%
发文量
26
审稿时长
12 weeks
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