发表机构
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh（爱丁堡大学数学学院与麦克斯韦数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种卷积神经网络架构及改进的同化训练算法，可从涡度数据鲁棒预测聚合物构象，在二维Kolmogorov流的多流态下均有效，且训练出的网络可泛化至更大计算域。
AI 中文摘要
聚合物溶液平行剪切流中弹性湍流的数值模拟表明，该现象由精确相干态的形成与不稳定性主导，而这些相干态又以聚合物应力的薄层为核心。然而，实验中至今尚未直接观测到这类“箭头状”结构，因为同时获取速度与聚合物构象的测量数据极具挑战性。受这些挑战的驱动，本文提出了一种利用涡度测量时间序列预测聚合物构象场的方法。该方法包含两个核心部分：第一是卷积神经网络架构，其输入为涡度场，输出为正定构象张量；第二是基于同化的训练算法（Zhu & Page, 2026）的改进版本，该算法无需预先生成的“离线”参考构象张量库，仅通过涡度测量数据进行训练。这一点在粘弹性问题中尤为重要，因为与实验对比的合适模型及参数可能需要作为解的一部分来确定。训练过程中，要求随时间推进的网络预测所得到的测量值与保存的时间序列匹配，同时要求求解器的输出与网络在后续时刻的预测保持自洽。我们将上述思路应用于二维Kolmogorov流，覆盖从简单行波到完全混沌态的多个流态区间。在所有案例中，本文方法均能对聚合物拉伸量进行鲁棒预测，而标准的未正则化变分同化方法则无法奏效。在混沌流态下，我们还证明了所训练的网络无需进一步优化，即可泛化至比其训练时所用“最小”单元大得多的计算域。
英文摘要
Numerical simulations of elastic turbulence in parallel shear flows of polymer solutions indicate that the phenomena is associated with the formation and instability of exact coherent states dominated by thin sheets of polymer stress. However, these ``arrowhead'' structures are yet to be seen directly in experiments, where simultaneous velocity and polymer conformation measurements are challenging to obtain. Motivated by these challenges, we introduce a method for the prediction of the polymer conformation field given a time series of vorticity measurements. Our approach consists of two components: the first is a convolutional neural network architecture which takes vorticity fields and outputs a positive definite conformation tensor. The second is the adaptation of an assimilation-based training algorithm (Zhu \& Page, 2026) which does not require a pre-generated `offline' library of reference conformation fields, but is trained only using the vorticity measurements. This is particularly important in viscoelastic problems, where the appropriate model and parameters to compare to the experiments may need to be determined as part of the solution. In training, measurements made on a time-marched network prediction are required to match the saved time series, while the output of the solver and network predictions at later times are required to be self-consistent. We apply these ideas to two-dimensional Kolmogorov flow in a range of regimes, from simple traveling waves to a fully chaotic state. In all cases, our method produces robust predictions of the polymer stretch, while standard, unregularised variational assimilation is ineffective. In the chaotic case we show that our networks generalise to much larger domains -- without further optimisation -- than the `minimal' units in which they were trained.
