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Exterior sound field estimation based on physics-constrained kernel

Juliano G. C. Ribeiro, Ryo Matsuda, Jorge Trevino

TL;DR

This paper tackles exterior sound field interpolation by proposing a Gaussian process regression framework built on a multipole point source kernel with a trainable inner product, ensuring adherence to exterior Helmholtz solutions. The kernel is defined in a reproducing kernel Hilbert space with a parametric attenuation that automatically downweights higher-order modes, enabling data-driven adaptation to arbitrary microphone layouts. Compared to conventional spherical wave function expansion and a physics-informed point neuron network, the proposed method achieves lower interpolation error by about 2 dB on average across 100 Hz to 2.5 kHz and exhibits more consistent reconstructions within the target region. The approach offers a flexible, physically constrained interpolation tool with practical implications for exterior sound field synthesis, active noise control, and sound field reproduction, especially when microphone distributions are non-uniform or sparse.

Abstract

Exterior sound field interpolation is a challenging problem that often requires specific array configurations and prior knowledge on the source conditions. We propose an interpolation method based on Gaussian processes using a point source reproducing kernel with a trainable inner product formulation made to fit exterior sound fields. While this estimation does not have a closed formula, it allows for the definition of a flexible estimator that is not restricted by microphone distribution and attenuates higher harmonic orders automatically with parameters directly optimized from the recordings, meaning an arbitrary distribution of microphones can be used. The proposed kernel estimator is compared in simulated experiments to the conventional method using spherical wave functions and an established physics-informed machine learning model, achieving lower interpolation error by approximately 2 dB on average within the analyzed frequencies of 100 Hz and 2.5 kHz and reconstructing the ground truth sound field more consistently within the target region.

Exterior sound field estimation based on physics-constrained kernel

TL;DR

This paper tackles exterior sound field interpolation by proposing a Gaussian process regression framework built on a multipole point source kernel with a trainable inner product, ensuring adherence to exterior Helmholtz solutions. The kernel is defined in a reproducing kernel Hilbert space with a parametric attenuation that automatically downweights higher-order modes, enabling data-driven adaptation to arbitrary microphone layouts. Compared to conventional spherical wave function expansion and a physics-informed point neuron network, the proposed method achieves lower interpolation error by about 2 dB on average across 100 Hz to 2.5 kHz and exhibits more consistent reconstructions within the target region. The approach offers a flexible, physically constrained interpolation tool with practical implications for exterior sound field synthesis, active noise control, and sound field reproduction, especially when microphone distributions are non-uniform or sparse.

Abstract

Exterior sound field interpolation is a challenging problem that often requires specific array configurations and prior knowledge on the source conditions. We propose an interpolation method based on Gaussian processes using a point source reproducing kernel with a trainable inner product formulation made to fit exterior sound fields. While this estimation does not have a closed formula, it allows for the definition of a flexible estimator that is not restricted by microphone distribution and attenuates higher harmonic orders automatically with parameters directly optimized from the recordings, meaning an arbitrary distribution of microphones can be used. The proposed kernel estimator is compared in simulated experiments to the conventional method using spherical wave functions and an established physics-informed machine learning model, achieving lower interpolation error by approximately 2 dB on average within the analyzed frequencies of 100 Hz and 2.5 kHz and reconstructing the ground truth sound field more consistently within the target region.
Paper Structure (13 sections, 21 equations, 4 figures)

This paper contains 13 sections, 21 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Conventional methods
  4. Spherical wave function expansion
  5. Point neuron network
  6. Proposed method
  7. RKHS definition
  8. Model derivation and optimization
  9. Numerical experiments
  10. Evaluation metrics
  11. Training of the models
  12. Experimental results
  13. Conclusion

Figures (4)

  • Figure 1: Example of the problem statement showing the bounds of the target region and the source region enclosed by it.
  • Figure 2: Proposed method mode attenuation scheme as function of weight parameters $\alpha$ and $\beta$.
  • Figure 3: NMSE as a function of frequency for each method.
  • Figure 4: Real part of the ground truth as well as estimated sound fields. The white circles represent the bounds of the target region, while the dotted circle represents the bounds of the source region.