This paper considers the creation of parametric surrogate models for applications in science and engineering where the goal is to predict high-dimensional spatiotemporal output quantities of interest, such as pressure, temperature and displacement fields. The proposed methodology develops a low-dimensional parametrization of these quantities of interest using space–time bases combined with machine learning methods. In particular, the space–time solutions are sought in a low-dimensional space–time linear trial subspace that can be obtained by computing tensor decompositions of usual state-snapshots data. The mapping between the input parameters and the basis expansion coefficients (or generalized coordinates) is approximated using four different machine learning techniques: multivariate polynomial regression, $k$-nearest-neighbors, random forests and neural networks. The relative costs and effectiveness of the four machine learning techniques are explored through three engineering problems: steady heat conduction, unsteady heat conduction and unsteady advective–diffusive–reactive system. Numerical results demonstrate that the proposed method performs well in terms of both accuracy and computational cost, and highlights the important point that the amount of model training data available in an engineering setting is often much less than it is in other machine learning applications, making it essential to incorporate knowledge from physical models.

本文考虑为科学和工程中的应用创建参数替代模型，其目标是预测感兴趣的高维时空输出量，例如压力、温度和位移场。所提出的方法使用时空基础与机器学习方法相结合，开发了这些感兴趣量的低维参数化。特别是，在低维时空线性试验子空间中寻找时空解，该子空间可以通过计算通常状态快照数据的张量分解来获得。输入参数与基展开系数之间的映射（或广义坐标）使用四种不同的机器学习技术来近似：多元多项式回归，$克$-最近邻、随机森林和神经网络。通过三个工程问题探索四种机器学习技术的相对成本和有效性：稳态热传导、非稳态热传导和非稳态对流-扩散-反应系统。数值结果表明，所提出的方法在准确性和计算成本方面都表现良好，并强调了工程设置中可用的模型训练数据量通常比其他机器学习应用程序少得多的重要一点，使其整合来自物理模型的知识必不可少。