A meshless method to compute the POD and its variants from scattered data*

The Proper Orthogonal Decomposition (POD) is a widely adopted method for identifying patterns in fluid mechanics data. When data is organized on a fixed structured grid, such as in cross-correlation-based particle image velocimetry or in numerical simulations, POD is essentially equivalent to performing a cell-weighted Singular Value Decomposition (SVD) on the snapshot matrix. However, when data sampling locations change over time, as with mobile monitoring stations in meteorology and oceanography or with particle tracking velocimetry, interpolation is required to project the data onto a fixed grid before factorization. This interpolation is often both expensive and inaccurate.

In this work, we propose a method to compute POD from scattered data that eliminates the need for interpolation. Our method uses physics-constrained Radial Basis Function (RBF) regression to compute inner products in space and time. This approach provides an analytical and mesh-independent decomposition in space and time, demonstrating higher accuracy than traditional methods. Our results show that it is possible to extract the most relevant "components" even from measurements where the natural output is a distribution of data scattered in space, maintaining high accuracy and mesh independence. Since it does not require mesh definition and produces analytic, mesh-independent results, we refer to our method as meshless POD.