[Paper Review] Physics-informed convolutional neural network with bicubic spline interpolation for sound field estimation

Kazuhide Shigemi, Shoichi Koyama+2 more|arXiv (Cornell University)|Jul 22, 2022

Flow Measurement and Analysis4 citations

TL;DR

This paper proposes a physics-informed convolutional neural network (PICNN) with bicubic spline interpolation for accurate and physically consistent sound field estimation from sparse microphone measurements. By integrating Helmholtz equation constraints via a loss function evaluated on interpolated pressure fields, the method achieves superior physical feasibility compared to standard CNNs while maintaining high estimation accuracy, especially with limited data.

ABSTRACT

A sound field estimation method based on a physics-informed convolutional neural network (PICNN) using spline interpolation is proposed. Most of the sound field estimation methods are based on wavefunction expansion, making the estimated function satisfy the Helmholtz equation. However, these methods rely only on physical properties; thus, they suffer from a significant deterioration of accuracy when the number of measurements is small. Recent learning-based methods based on neural networks have advantages in estimating from sparse measurements when training data are available. However, since physical properties are not taken into consideration, the estimated function can be a physically infeasible solution. We propose the application of PICNN to the sound field estimation problem by using a loss function that penalizes deviation from the Helmholtz equation. Since the output of CNN is a spatially discretized pressure distribution, it is difficult to directly evaluate the Helmholtz-equation loss function. Therefore, we incorporate bicubic spline interpolation in the PICNN framework. Experimental results indicated that accurate and physically feasible estimation from sparse measurements can be achieved with the proposed method.

Motivation & Objective

• To address the limitation of existing learning-based sound field estimation methods that produce physically infeasible solutions due to lack of physical constraints.
• To improve estimation accuracy under sparse measurement conditions where traditional wavefunction expansion methods degrade significantly.
• To integrate the physics-informed neural network (PICNN) framework into sound field estimation, ensuring solutions satisfy the Helmholtz equation.
• To overcome the challenge of evaluating the Helmholtz equation loss on discretized CNN outputs by introducing bicubic spline interpolation for continuous field evaluation.
• To demonstrate that the proposed method achieves both high accuracy and strong physical consistency in sound field reconstruction.

Proposed method

• The method employs a physics-informed CNN (PICNN) that minimizes a loss function penalizing deviation from the Helmholtz equation.
• Bicubic spline interpolation is applied to the CNN’s discrete pressure output to reconstruct a continuous pressure field for accurate Helmholtz equation evaluation.
• The Helmholtz-equation loss is computed as the integral of the squared residual of the Laplacian plus k² term over the domain.
• The network is trained using Adam optimizer with a learning rate of 0.01 and input standardization for scale independence.
• Phase randomization is applied during training to achieve phase-invariant generalization.
• The loss function includes a regularization term with λ = 10⁻⁵ to balance data fidelity and physical consistency.

Fig. 1 : Experimental setup. The sound field in the rectangular target region $\Omega$ is estimated. The positions of the vertices of the pentagonal room are indicated.

Experimental results

Research questions

• RQ1 Can a physics-informed CNN with bicubic spline interpolation improve sound field estimation accuracy under sparse microphone configurations compared to standard deep learning methods?
• RQ2 How effectively can the proposed method enforce physical consistency by satisfying the Helmholtz equation compared to baseline CNNs?
• RQ3 Does the integration of bicubic spline interpolation enable accurate evaluation of the Helmholtz equation loss on discretized CNN outputs?
• RQ4 How does the proposed method compare to kernel interpolation methods in terms of estimation accuracy and physical feasibility?
• RQ5 Can the method generalize well across different acoustic environments with limited training data?

Key findings

• For M=5 measurements, the proposed method achieved an NMSE of -2.44 dB, outperforming the kernel method (-1.36 dB) and showing a significant improvement over baseline CNN.
• At M=10, the proposed method achieved an NMSE of -4.88 dB and an HE of 2.35, indicating both high accuracy and strong physical consistency.
• The proposed method reduced Helmholtz-equation error (HE) by 23% on average compared to the baseline CNN, demonstrating superior physical feasibility.
• Visual results showed that the proposed method produced smoother, more natural pressure distributions without the unnatural variations seen in the baseline CNN.
• The proposed method maintained low HE values (2.06 on average at M=20) while achieving high accuracy, confirming that physical constraints are effectively enforced.
• The method demonstrated robustness across multiple random seeds, with low standard deviations in both NMSE and HE, indicating stable performance.