E2N: Error Estimation Networks for Goal-Oriented Mesh Adaptation
E2N: Error Estimation Networks for Goal-Oriented Mesh Adaptation
复制标题
E2N：面向目标的网格自适应的误差估计网络
DOI：
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
M. Piggott
中科院分区：
文献类型：
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作者：
Joseph G. Wallwork;Jing Lu;Mingrui Zhang;M. Piggott
给定偏微分方程（PDE），面向目标的误差估计使我们能够理解兴趣诊断量（QoI）或目标中的误差如何在数值近似中发生和累积，例如使用有限元方法。通过将误差估计分解为各个元素的贡献，可以制定适应方法，以最小化结果qi误差为目标修改网格。然而，标准误差估计公式涉及真伴随解，这在实际中是未知的。因此，通常的做法是用“丰富的”近似来近似它（例如在高阶空间或在精细网格上）。这样做通常会导致计算成本的显著增加，这可能是影响（面向目标的）自适应模拟竞争力的瓶颈。本文的中心思想是开发一种“数据驱动”的面向目标的网格自适应方法，通过选择性地用适当配置和训练的神经网络取代昂贵的误差估计步骤。在这样做时，误差估计量甚至可以不构造充实空间而得到。这里采用逐单元构造，将与网格几何和潜在问题物理相关的各种参数的局部值作为输入，并将相应的误差估计量的贡献作为输出。我们证明，对于与潮汐涡轮机周围流动相关的自适应网格测试用例，这种方法能够以更低的计算成本获得相同的精度，潮汐涡轮机通过下游尾迹相互作用，并且将农场的总功率输出作为qi。此外，我们证明了逐元素方法意味着合理的低培训成本。
Given a partial di ff erential equation (PDE), goal-oriented error estimation allows us to under-stand how errors in a diagnostic quantity of interest (QoI), or goal , occur and accumulate in a numerical approximation, for example using the finite element method. By decomposing the error estimates into contributions from individual elements, it is possible to formulate adaptation methods, which modify the mesh with the objective of minimising the resulting QoI error. How-ever, the standard error estimate formulation involves the true adjoint solution, which is unknown in practice. As such, it is common practice to approximate it with an ‘enriched’ approximation (e.g. in a higher order space or on a refined mesh). Doing so generally results in a significant increase in computational cost, which can be a bottleneck compromising the competitiveness of (goal-oriented) adaptive simulations. The central idea of this paper is to develop a “data-driven” goal-oriented mesh adaptation approach through the selective replacement of the expensive error estimation step with an appropriately configured and trained neural network. In doing so, the error estimator may be obtained without even constructing the enriched spaces. An element-by-element construction is employed here, whereby local values of various parameters related to the mesh geometry and underlying problem physics are taken as inputs, and the corresponding contribution to the error estimator is taken as output. We demonstrate that this approach is able to obtain the same accuracy with a reduced computational cost, for adaptive mesh test cases related to flow around tidal turbines, which interact via their downstream wakes, and where the overall power output of the farm is taken as the QoI. Moreover, we demonstrate that the element-by-element approach implies reasonably low training costs.
影响因子：
3.2
作者：
Colin J. Cotter;David A. Ham;Christopher C. Pain
通讯作者：
Colin J. Cotter;David A. Ham;Christopher C. Pain