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arXiv 2409.04067math.NAcs.LGcs.NAphysics.flu-dyn
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基于预条件FEM的神经网络求解不可压缩流体流动及相关反问题

Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems

DLR（德国航空航天中心）

Franziska Griese, Fabian Hoppe, Alexander Rüttgers, Philipp Knechtges

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AI总结：

本文提出在物理信息神经网络训练中引入预条件子修正方程残差，用于求解稳态Stokes和Navier-Stokes方程，显著降低训练开销并提升精度与泛化能力，并应用于相关反问题。

摘要：

由偏微分方程描述的技术系统的数值模拟与优化代价高昂，尤其是在多查询场景中，底层方程必须针对不同参数进行求解。在这一背景下，一种相对较新的方法是将神经网络（用于参数依赖性）的良好逼近性质与经典有限元方法（用于离散化）相结合。然而，与其将PDE从参数空间到FEM离散化解空间的解映射视为纯粹的数据驱动回归问题，所谓的物理信息回归问题已被证明是有用的。在这些问题中，方程残差在神经网络训练过程中被最小化，即神经网络“学习”问题背后的物理。在本文中，我们将该方法分别扩展到鞍点和非线性流体动力学问题，即稳态Stokes方程和稳态Navier-Stokes方程。特别地，我们提出了对现有方法的一种改进：在训练过程中，不是最小化普通的方程残差，而是最小化经过预条件子修正的方程残差。与线性情形类似，这同样改善了当前非线性情形中的条件数。我们的数值示例表明，该方法显著降低了训练开销，并大幅提高了精度和泛化能力。最后，我们展示了所得参数化模型在一个相关反问题中的应用。

英文摘要：

The numerical simulation and optimization of technical systems described by partial differential equations is expensive, especially in multi-query scenarios in which the underlying equations have to be solved for different parameters. A comparatively new approach in this context is to combine the good approximation properties of neural networks (for parameter dependence) with the classical finite element method (for discretization). However, instead of considering the solution mapping of the PDE from the parameter space into the FEM-discretized solution space as a purely data-driven regression problem, so-called physically informed regression problems have proven to be useful. In these, the equation residual is minimized during the training of the neural network, i.e., the neural network "learns" the physics underlying the problem. In this paper, we extend this approach to saddle-point and non-linear fluid dynamics problems, respectively, namely stationary Stokes and stationary Navier-Stokes equations. In particular, we propose a modification of the existing approach: Instead of minimizing the plain vanilla equation residual during training, we minimize the equation residual modified by a preconditioner. By analogy with the linear case, this also improves the condition in the present non-linear case. Our numerical examples demonstrate that this approach significantly reduces the training effort and greatly increases accuracy and generalizability. Finally, we show the application of the resulting parameterized model to a related inverse problem.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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