Data-driven extrapolation via feature augmentation based on variably scaled thin plate splines

TL;DR

This paper introduces a novel data-driven extrapolation method using variably scaled thin plate splines and kernel models, demonstrating effective predictions without complex training, especially in Laplace transform inversion tasks.

Contribution

The paper proposes a new feature augmentation strategy with variably scaled kernels based on polyharmonic splines, providing error bounds and practical effectiveness over existing methods.

Findings

Effective in Laplace transform inversion tasks

Does not require complex architecture training

Provides theoretical error bounds in Beppo-Levi spaces

Abstract

The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the considered dynamics, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo-Levi spaces. The performances of the method are then tested on functions which are common in the framework of the Laplace transform inversion. Comparisons with Support Vector Regression (SVR) are also carried out and show that the proposed method is effective particularly since it does not require to train complex architecture constructions.