NCDL: A Framework for Deep Learning on non-Cartesian Lattices

The use of non-Cartesian grids is a niche but important topic in sub-fields of the numerical sciences such as simulation and scientific visualization. However, non-Cartesian approaches are virtually unexplored in machine learning. This is likely due to the difficulties in the representation of data on non-Cartesian domains and the lack of support for standard machine learning operations on non-Cartesian data. This paper proposes a new data structure called the lattice tensor which generalizes traditional tensor spatio-temporal operations to lattice tensors, enabling the use of standard machine learning algorithms on non-Cartesian data. However, data need not reside on a non-Cartesian structure, we use non-Dyadic downsampling schemes to bring Cartesian data into a non-Cartesian space for further processing. We introduce a software library that implements the lattice tensor container (with some common machine learning operations), and demonstrate its effectiveness. Our method provides a general framework for machine learning on non-Cartesian domains, addressing the challenges mentioned above and filling a gap in the current literature.