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<p>A configurable U-Net architecture is trained to solve the multi-scale elliptical partial differential equations. The motivation is to improve the computational cost of the numerical solution of Navier-Stokes equations – the governing equations for fluid dynamics. Building on the underlying concept of V-Cycle multigrid methods, a neural network framework using U-Net architecture is optimized to solve the Poisson equation and Helmholtz equations – the characteristic form of the discretized Navier-Stokes equations. The results demonstrate the optimized U-Net captures the high dimensional mathematical features of the elliptical operator and with a better convergence than the multigrid method. The optimal performance between the errors and the FLOPS is the (3, 2, 5) case with 3 stacks of U-Nets, with 2 initial features, 5 depth layers and with ELU activation. Further, by training the network with the multi-scale synthetic data the finer features of the physical system are captured.</p>

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Accelerated Elliptical PDE Solver for Computational Fluid Dynamics Based on Configurable U-Net Architecture: Analogy to V-Cycle Multigrid

  • 期刊论文

  • 发表日期：

    2025年2月21日

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  • Kiran Bhaganagar,
  • David Chambers

摘要

A configurable U-Net architecture is trained to solve the multi-scale elliptical partial differential equations. The motivation is to improve the computational cost of the numerical solution of Navier-Stokes equations – the governing equations for fluid dynamics. Building on the underlying concept of V-Cycle multigrid methods, a neural network framework using U-Net architecture is optimized to solve the Poisson equation and Helmholtz equations – the characteristic form of the discretized Navier-Stokes equations. The results demonstrate the optimized U-Net captures the high dimensional mathematical features of the elliptical operator and with a better convergence than the multigrid method. The optimal performance between the errors and the FLOPS is the (3, 2, 5) case with 3 stacks of U-Nets, with 2 initial features, 5 depth layers and with ELU activation. Further, by training the network with the multi-scale synthetic data the finer features of the physical system are captured.