This paper proposes a novel approach for learning a data-driven quadratic manifold from high-dimensional data, then employing this quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping consists of two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projection-based model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace.
翻译:本文提出一种新的方法,用于从高维数据中学习数据驱动的二次方块,然后利用这一二次方块来获得高效的物理减序模型。该方法的关键组成部分是高维状态和低维嵌入层之间的多位绘图。该绘图由两部分组成:线性次空间的表示(在这项工作中使用正正正正正正正正正正正正正正分解剖)和四面方块。该方法可被视为一种数据驱动的封闭模型形式,因为四方块组件向直线子空间的正近似方向引入方向,但不对低维代表层引入任何额外的自由度。将四方形多重近似与基于预测的模型减少的推论操作者组合在一起,可形成一种可伸缩的非侵入性方法,用于学习动态系统的减序模型。应用以传输为主的局部差异方程式的新方法,说明在线性子空间中可超过近近度的效率方面取得的成绩。