A Recipe for Learning Variably Scaled Kernels via Discontinuous Neural Networks

TL;DR

This paper introduces a method to learn effective scaling functions for Variably Scaled Kernels using Discontinuous Neural Networks, backed by theoretical proofs and numerical experiments demonstrating improved approximation accuracy.

Contribution

It provides the first theoretical proof and a practical neural network-based method for constructing scaling functions for VSKs, enhancing their approximation capabilities.

Findings

Learning the scaling function improves approximation accuracy.

Discontinuous Neural Networks effectively recover target function features.

Numerical experiments validate the proposed method's effectiveness.

Abstract

The efficacy of interpolating via Variably Scaled Kernels (VSKs) is known to be dependent on the definition of a proper scaling function, but no numerical recipes to construct it are available. Previous works suggest that such a function should mimic the target one, but no theoretical evidence is provided. This paper fills both the gaps: it proves that a scaling function reflecting the target one may lead to enhanced approximation accuracy, and it provides a user-independent tool for learning the scaling function by means of Discontinuous Neural Networks ($\delta$NN), i.e., NNs able to deal with possible discontinuities. Numerical evidence supports our claims, as it shows that the key features of the target function can be clearly recovered in the learned scaling function.