Advection-dominated problems are commonly noticed in nature, engineering systems, and a wide range of industrial processes. For these problems, linear approximation methods (proper orthogonal decomposition and reduced basis method) are not suitable, as the Kolmogorov $n$-width decay is slow, leading to inefficient and inaccurate reduced order models. There are few non-linear approaches to accelerate the Kolmogorov $n$-width decay. In this work, we use a neural-network shift augmented transformation technique, that employs automatic-shit detection and detects the optimal non-linear transformation of the full-order model solution manifold $\mathcal{M}$. We exploit a deep-learning framework to derive parameter-dependent bijective mapping between the manifold $\mathcal{M}$ and the transformed manifold $\tilde{\mathcal{M}}$. It consists of two neural networks, 1) ShiftNet, to employ automatic-shift detection by learning the shift-operator, which finds the optimal shifts for numerous snapshots of the full-order solution manifold, to accelerate the Kolmogorov $n$-width decay, and 2) InterpNet, which learns the reference configuration and can reconstruct the field values of the same, for each shifted grid distribution. We construct non-intrusive reduced order models on the resulting transformed linear subspaces and employ automatic-shift detection for predictions. We test our methodology on advection-dominated problems, such as 1D travelling waves, 2D isentropic convective vortex and 2D two-phase flow test cases. This work leads to the development of the complete NNsPOD-ROM algorithm for model reduction of advection-dominated problems, comprising both offline-online stages.

翻译：平流主导问题在自然界、工程系统及广泛的工业过程中普遍存在。对于此类问题，线性近似方法（如本征正交分解和降基方法）并不适用，因为其Kolmogorov $n$-宽度衰减缓慢，导致降阶模型效率低下且精度不足。目前加速Kolmogorov $n$-宽度衰减的非线性方法较为有限。本研究采用一种神经网络平移增强变换技术，通过自动平移检测实现对全阶模型解流形$\mathcal{M}$的最优非线性变换。我们利用深度学习框架推导流形$\mathcal{M}$与变换后流形$\tilde{\mathcal{M}}$之间的参数依赖双射映射。该框架包含两个神经网络：1) ShiftNet，通过学习平移算子实现自动平移检测，为全阶解流形的众多快照寻找最优平移以加速Kolmogorov $n$-宽度衰减；2) InterpNet，通过学习参考构型并针对每个平移后的网格分布重构其场值。我们在所得变换线性子空间上构建非侵入式降阶模型，并采用自动平移检测进行预测。我们在平流主导问题上验证了该方法，包括一维行波、二维等熵对流涡旋和二维两相流测试案例。本研究最终形成了完整的NNsPOD-ROM算法，包含离线与在线阶段，专门用于平流主导问题的模型降阶。