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A Recipe for Learning Variably Scaled Kernels via Discontinuous Neural Networks

Gianluca Audone, Francesco Della Santa, Emma Perracchione, Sandra Pieraccini

TL;DR

This work tackles the challenge of selecting a scaling function for Variably Scaled Kernels (VSKs) in meshfree interpolation. It provides theoretical insight through error bounds that tie accuracy to how closely the scaling function $\bar{f}$ matches the target $f$, augmented by the Lebesgue function, and introduces a data-driven approach using Discontinuous Neural Networks ($\delta$NNs) to learn nearly optimal $\bar{f}$. The authors present two practical pathways: $\delta$NN-VSKs, which jointly learn $\bar{f}$ and interpolation coefficients to achieve interpolation-based optimization, and VSK-$f$, which uses an approximation of the target as a scaling function. Numerical experiments on synthetic functions and a real acetone phase-transition dataset show that the proposed methods outperform traditional fixed-scale kernels, with the learned scaling functions effectively capturing key target features and discontinuities. This data-driven, adaptive framework enhances the flexibility and accuracy of VSKs for discontinuous or highly heterogeneous targets and has potential applications in PDE collocation and sparse sampling strategies.

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 ($δ$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.

A Recipe for Learning Variably Scaled Kernels via Discontinuous Neural Networks

TL;DR

This work tackles the challenge of selecting a scaling function for Variably Scaled Kernels (VSKs) in meshfree interpolation. It provides theoretical insight through error bounds that tie accuracy to how closely the scaling function matches the target , augmented by the Lebesgue function, and introduces a data-driven approach using Discontinuous Neural Networks (NNs) to learn nearly optimal . The authors present two practical pathways: NN-VSKs, which jointly learn and interpolation coefficients to achieve interpolation-based optimization, and VSK-, which uses an approximation of the target as a scaling function. Numerical experiments on synthetic functions and a real acetone phase-transition dataset show that the proposed methods outperform traditional fixed-scale kernels, with the learned scaling functions effectively capturing key target features and discontinuities. This data-driven, adaptive framework enhances the flexibility and accuracy of VSKs for discontinuous or highly heterogeneous targets and has potential applications in PDE collocation and sparse sampling strategies.

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 (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.
Paper Structure (11 sections, 2 theorems, 31 equations, 11 figures, 3 tables)

This paper contains 11 sections, 2 theorems, 31 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Brief review of kernel-based interpolation
  3. Investigating and learning the scaling function for VSKs
  4. VSKs: investigating the scaling function via Lagrange bases
  5. $\delta$NN-VSKs: learning nearly-optimal scaling functions via $\delta$NNs
  6. VSKs-$f$: using an approximation of $f$ as scaling function
  7. Numerical experiments
  8. Tests with synthetic datasets
  9. Real-world test case
  10. Conclusions
  11. $\delta$NN Architecture in the Numerical Experiments

Key Result

Proposition 3.1

Let ${\bar{f}}$ be a scaling function and let $\kappa_{\bar{f}}: \Omega \times \Omega \longrightarrow {\mathbb R}$ be the VSK corresponding to the strictly positive definite and symmetric kernel $\kappa: \tilde{\Omega} \times \tilde{\Omega} \longrightarrow {\mathbb R}$. Let $P_{f}^{\bar{f}}: \Omega where is the Lebesgue function for the VSK interpolant, and $\boldsymbol{f}-\boldsymbol{\bar{f}} =

Figures (11)

  • Figure 1: Flowchart 1: Diagram for the VSK-based algorithm.
  • Figure 1: Examples of discontinuous layers ($n_{\rm in}=n_{\rm out}=1$) with different activation functions.
  • Figure 2: Interpolation results for $f_1$. Case $n=33^2=1089$ interpolation points, Gaussian Kernel. Shared color bars for subfigures ($a$)-($d$); custom color bars for the scaling functions (subfigures ($e$) and ($f$)).
  • Figure 3: Interpolation results for $f_2$. Case $n=33^2=1089$ interpolation points, Matérn-$C^2$ Kernel. Shared color bars for subfigures ($a$)-($d$); custom color bars for the scaling functions (subfigures ($e$) and ($f$)).
  • Figure 4: Interpolation results for $f_3$. Case $n=33^2=1089$ interpolation points, Matérn-$C^2$ Kernel. Shared color bars for subfigures ($a$)-($d$); custom color bars for the scaling functions (subfigures ($e$) and ($f$)).
  • ...and 6 more figures

Theorems & Definitions (5)

  • Remark 3.1
  • Proposition 3.1
  • Corollary 3.1
  • Remark 3.2
  • Remark 3.3