神经网络移位正交分解：一种双曲型方程非线性约简的机器学习方法
The Neural Network shifted-Proper Orthogonal Decomposition: a Machine Learning Approach for Non-linear Reduction of Hyperbolic Equations
摘要：
具有主导平流的模型总是对基于投影的降阶建模提出了困难的挑战。最近提出的许多方法是基于预处理的全阶解决方案，以加速柯尔莫哥洛夫N-宽度衰减，从而获得更小的线性子空间，提高精度。然而，这些方法必须依赖于知识的特征速度在相空间的解决方案，限制其适用范围的问题与明确的功能形式的平流场。在这项工作中，我们通过实现深度学习架构来解决在统计学习框架中自动检测正确的预处理转换的问题。纯粹的数据驱动的方法，使我们能够推广现有的线性子空间操作的非线性双曲问题与未知的平流场的方法。所提出的算法已被验证对简单的测试用例基准其性能，后来成功地应用到多相仿真。
英文摘要：
Models with dominant advection always posed a difficult challenge for projection-based reduced order modelling. Many methodologies that have recently been proposed are based on the pre-processing of the full-order solutions to accelerate the Kolmogorov N-width decay thereby obtaining smaller linear subspaces with improved accuracy. These methods however must rely on the knowledge of the characteristic speeds in phase space of the solution, limiting their range of applicability to problems with explicit functional form for the advection field. In this work we approach the problem of automatically detecting the correct pre-processing transformation in a statistical learning framework by implementing a deep-learning architecture. The purely data-driven method allowed us to generalise the existing approaches of linear subspace manipulation to non-linear hyperbolic problems with unknown advection fields. The proposed algorithm has been validated against simple test cases to benchmark its performances and later successfully applied to a multiphase simulation.
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