Improve the scalability of B-Spline bases

Improve the scalability of tensor-product B-Spline bases for physics-informed learning as the dimension of the input coordinates or parameters and the number of represented physical fields increase.

Background

The B-Spline representation yields fast loss minimization and convergence rates consistent with approximation theory, but its tensor-product structure causes the number of degrees of freedom to grow rapidly with the dimension of the input or output. The paper identifies this curse of dimensionality as the principal limitation of B-Splines for larger or more complex physics-informed learning problems.

The authors suggest local-refinement approaches, such as LR-B-Splines and TH-B-Splines, and low-rank B-Spline representations as possible directions for reducing computational cost and dimensionality, but do not resolve the scalability problem.