GMLS-Nets: A framework for learning from unstructured data
This provides a method for data-driven model development in scientific machine learning applications, addressing a domain-specific need for handling unstructured data in sciences and engineering.
The authors tackled the problem of learning from unstructured data, such as irregularly spaced point clouds, by generalizing Convolutional Neural Networks using Generalized Moving Least Squares (GMLS) to create GMLS-Nets, which enable functional regression and identification of differential operators on unstructured physical datasets.
Data fields sampled on irregularly spaced points arise in many applications in the sciences and engineering. For regular grids, Convolutional Neural Networks (CNNs) have been successfully used to gaining benefits from weight sharing and invariances. We generalize CNNs by introducing methods for data on unstructured point clouds based on Generalized Moving Least Squares (GMLS). GMLS is a non-parametric technique for estimating linear bounded functionals from scattered data, and has recently been used in the literature for solving partial differential equations. By parameterizing the GMLS estimator, we obtain learning methods for operators with unstructured stencils. In GMLS-Nets the necessary calculations are local, readily parallelizable, and the estimator is supported by a rigorous approximation theory. We show how the framework may be used for unstructured physical data sets to perform functional regression to identify associated differential operators and to regress quantities of interest. The results suggest the architectures to be an attractive foundation for data-driven model development in scientific machine learning applications.