We design a physics-aware auto-encoder to specifically reduce the dimensionality of solutions arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by a large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Moreover, the realized latent variables are often hard to interpret. Therefore, many of these methods are often dismissed in the reduced order modeling of dynamical systems governed by the partial differential equations (PDEs). Accordingly, we propose an auto-encoder type nonlinear dimensionality reduction algorithm. The unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that registers the output sequence of the PDEs on a non-uniform parameter/time-varying grid, such that the Kolmogorov n-width of the mapped data on the learned grid is minimized. We demonstrate the efficacy and interpretability of our approach to separate convection/advection from diffusion/scaling on various manufactured and physical systems.
翻译:我们设计了一个物理觉察自动编码器,以具体减少对流主导非线性物理系统产生的解决方案的维度。虽然现有的非线性多元学习方法似乎是减少以大型科尔莫戈洛夫为特征的数据的维度的令人信服的工具,但它们通常缺乏从潜在空间到高维物理空间的直截了当的绘图;此外,已实现的潜伏变量往往难以解释。因此,许多这些方法往往在部分差异方程(PDEs)所规范的动态系统缩小型号模型中被忽略。因此,我们建议采用自动编码器型的非线性维度减少算法。未受监督的学习问题将电磁波-时空电网列成,将PDE的输出序列记录在非统一的参数/时间变换格上,因此,在所学的电网上已绘制数据的科尔莫戈洛夫 nwidth 被最小化。我们展示了将各种物理系统与扩散/剖析系统分开的调/调整/调整方法的功效和可解释性。