While extracting information from data with machine learning plays an increasingly important role, physical laws and other first principles continue to provide critical insights about systems and processes of interest in science and engineering. This work introduces a method that infers models from data with physical insights encoded in the form of structure and that minimizes the model order so that the training data are fitted well while redundant degrees of freedom without conditions and sufficient data to fix them are automatically eliminated. The models are formulated via solution matrices of specific instances of generalized Sylvester equations that enforce interpolation of the training data and relate the model order to the rank of the solution matrices. The proposed method numerically solves the Sylvester equations for minimal-rank solutions and so obtains models of low order. Numerical experiments demonstrate that the combination of structure preservation and rank minimization leads to accurate models with orders of magnitude fewer degrees of freedom than models of comparable prediction quality that are learned with structure preservation alone.

翻译：尽管通过机器学习从数据中提取信息发挥着日益重要的作用，但物理定律及其他第一性原理仍持续为科学与工程领域所关注的系统及过程提供关键见解。本文提出一种方法，该方法通过融入以结构形式编码的物理先验知识来从数据中推断模型，并在最小化模型阶数的同时，使训练数据得到良好拟合，且自动消除缺乏约束条件及充足数据来确定的冗余自由度。模型通过广义Sylvester方程特定实例的解矩阵进行构建，该方程强制执行训练数据的插值条件，并将模型阶数与解矩阵的秩相关联。所提方法通过数值求解Sylvester方程以获得最小秩解，从而得到低阶模型。数值实验表明，结构保持与秩最小化的结合能够生成精确模型，其自由度数量比仅通过结构保持学习且具有可对比预测质量的模型低数个量级。