基于有限元的物理信息算子学习框架,用于任意域上的时空偏微分方程

IF 8.7 2区 工程技术 Q1 Mathematics Engineering with Computers Pub Date : 2024-08-02 DOI:10.1007/s00366-024-02033-8
Yusuke Yamazaki, Ali Harandi, Mayu Muramatsu, Alexandre Viardin, Markus Apel, Tim Brepols, Stefanie Reese, Shahed Rezaei
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引用次数: 0

摘要

我们提出了一种新颖的基于有限元的物理信息算子学习框架,可以预测由偏微分方程(PDE)控制的时空动态。我们采用 Galerkin 离散化弱公式将物理学纳入损失函数,称为有限算子学习(FOL),同时采用隐式欧拉时间积分方案进行时间离散化。FOL 将当前时间步的温度场作为输入,并预测下一时间步的温度场。经过训练后,该网络能成功预测任何初始温度场的温度随时间的变化情况,与有限元法(FEM)的解法相比,精度很高,即使是在异质热传导和任意几何形状的情况下也是如此。FOL 的优势可归纳如下:首先,训练以无监督的方式进行,避免了从昂贵的模拟或实验中准备大量数据的需要。相反,由高斯随机过程和傅里叶级数生成的随机温度模式与恒温场相结合,被用作训练数据,以涵盖可能的温度情况。此外,还利用形状函数和后向差分近似进行域离散化,从而得到纯代数方程。这就提高了训练效率,因为在优化权重和偏置时避免了耗时的自动微分,同时还能接受可能的离散化误差。最后,得益于有限元的插值能力,任何具有异质微观结构的任意几何形状都可以用 FOL 处理,这对于解决各种工程应用场景至关重要。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains

We propose a novel finite element-based physics-informed operator learning framework that allows for predicting spatiotemporal dynamics governed by partial differential equations (PDEs). The Galerkin discretized weak formulation is employed to incorporate physics into the loss function, termed finite operator learning (FOL), along with the implicit Euler time integration scheme for temporal discretization. A transient thermal conduction problem is considered to benchmark the performance, where FOL takes a temperature field at the current time step as input and predicts a temperature field at the next time step. Upon training, the network successfully predicts the temperature evolution over time for any initial temperature field at high accuracy compared to the solution by the finite element method (FEM) even with a heterogeneous thermal conductivity and arbitrary geometry. The advantages of FOL can be summarized as follows: First, the training is performed in an unsupervised manner, avoiding the need for large data prepared from costly simulations or experiments. Instead, random temperature patterns generated by the Gaussian random process and the Fourier series, combined with constant temperature fields, are used as training data to cover possible temperature cases. Additionally, shape functions and backward difference approximation are exploited for the domain discretization, resulting in a purely algebraic equation. This enhances training efficiency, as one avoids time-consuming automatic differentiation in optimizing weights and biases while accepting possible discretization errors. Finally, thanks to the interpolation power of FEM, any arbitrary geometry with heterogeneous microstructure can be handled with FOL, which is crucial to addressing various engineering application scenarios.

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来源期刊
Engineering with Computers
Engineering with Computers 工程技术-工程:机械
CiteScore
16.50
自引率
2.30%
发文量
203
审稿时长
9 months
期刊介绍: Engineering with Computers is an international journal dedicated to simulation-based engineering. It features original papers and comprehensive reviews on technologies supporting simulation-based engineering, along with demonstrations of operational simulation-based engineering systems. The journal covers various technical areas such as adaptive simulation techniques, engineering databases, CAD geometry integration, mesh generation, parallel simulation methods, simulation frameworks, user interface technologies, and visualization techniques. It also encompasses a wide range of application areas where engineering technologies are applied, spanning from automotive industry applications to medical device design.
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