Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights
Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights
复制标题
离散弱公式的神经控制：伽辽金、最小二乘法
DOI：
10.1016/j.cma.2022.115716
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发表时间：
2022
影响因子：
7.2
通讯作者：
Brevis I
中科院分区：
文献类型：
--
作者：
Brevis I
利用神经网络优化数值方法有着巨大的潜力。在本文中，我们介绍和分析了一个框架的神经优化离散弱配方，适用于有限元方法。该框架的主要思想是包括一个神经网络函数作为弱形式的控制变量。找到神经控制，（准）最小化一个合适的成本（或损失）功能，然后产生一个数值逼近所需的属性。特别地，该框架允许以自然的方式并入精确解的已知数据，或者并入稳定机制（例如，我们分析的主要结果涉及到相关约束优化问题的适定性和收敛性。特别是，我们证明了在一定的条件下，离散弱形式是稳定的，并且存在准最小化神经控制，其收敛准最优。我们专门的分析结果Galerkin，最小二乘法和最小残差公式，其中的神经网络依赖出现在合适的权重的形式。基本的数值实验支持我们的研究结果，并展示了框架的潜力。
There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyze a framework for theneural optimization of discrete weak formulations, suitable for finite element methods. The main idea of the framework is to include a neural-network function acting as acontrolvariable in the weak form. Finding the neural control that (quasi-) minimizes a suitable cost (or loss) functional, then yields a numerical approximation with desirable attributes. In particular, the framework allows in a natural way the incorporation of known data of the exact solution, or the incorporation of stabilization mechanisms (e.g., to remove spurious oscillations).The main result of our analysis pertains to the well-posedness and convergence of the associated constrained-optimization problem. In particular, we prove under certain conditions, that the discrete weak forms are stable, and that quasi-minimizing neural controls exist, which converge quasi-optimally. We specialize the analysis results to Galerkin, least squares and minimal-residual formulations, where the neural-network dependence appears in the form of suitable weights. Elementary numerical experiments support our findings and demonstrate the potential of the framework.