Region-to-Region Sound Field Reconstruction

• Region-to-region sound field reconstruction is a method to interpolate acoustic transfer functions across continuously varying source and receiver regions while satisfying the Helmholtz equation and reciprocity.
• It leverages advanced deep learning architectures, kernel interpolation, and Bayesian models to accurately capture spatial field structures, especially at high frequencies.
• Techniques enforce physical regularity and permutation invariance, ensuring that the reconstructed fields adhere to inherent acoustic symmetry and domain-specific constraints.

Region-to-region sound field reconstruction concerns the estimation or interpolation of the sound field—typically the Acoustic Transfer Function (ATF), room transfer function (RTF), or pressure field—across all combinations of continuously varying source and receiver regions within a domain, subject to physical and often data-driven constraints. Whereas traditional point-to-region techniques focus on varying receiver positions with a fixed source, region-to-region methods interpolate over both source and receiver locations, addressing the geometry, symmetry, and physical regularity inherent in real-world acoustic propagation (Chen et al., 27 Jan 2026). A variety of approaches have emerged, including deep learning architectures that encode permutation invariance for source–receiver symmetry, physics-informed neural networks, variational and kernel-based methods, Bayesian models leveraging partial boundary priors, and explicit integral-equation or basis-function decompositions. This article systematically reviews region-to-region sound field reconstruction, drawing on recent contributions in the field.

1. Mathematical Formulation and Theoretical Framework

Region-to-region reconstruction is formulated as learning or recovering a function $P(r,s,f)$ describing the complex-valued ATF at position $r$ (receiver) due to a source at $s$ (source position) for frequency $f$; $r\in\Omega_R$, $s\in\Omega_S$, with both $\Omega_R$ and $\Omega_S$ being spatial regions of interest. The governing equation is typically the homogeneous Helmholtz equation:

$\nabla^2 P(r, s, f) + k^2 P(r, s, f) = 0,$

where $k=2\pi f/c$ is the wavenumber and $c$ is the speed of sound. Fundamental physical constraints such as acoustic reciprocity must be satisfied: $P(r,s,f) = P(s,r,f)$. Given $N$ measured ATFs $\{P(r_i,s_i,f_i)\}_{i=1}^N$ in $\Omega_R\times\Omega_S$, the objective is to infer $\hat{P}(r,s,f)$ for all $(r,s)\in\Omega_R\times\Omega_S$ such that the interpolated field matches observed data, satisfies the Helmholtz equation everywhere, and encodes symmetry constraints (Chen et al., 27 Jan 2026).

2. Permutation-Invariant Physics-Informed Neural Networks (PI-PINN)

A significant advance is the permutation-invariant physics-informed neural network for region-to-region ATF reconstruction (Chen et al., 27 Jan 2026). The PI-PINN adopts a Deep Set architecture to encode the unordered pair $(r, s)$, ensuring reciprocity by construction. Specifically, it defines

$$\hat{P}(r, s, f) = \rho \left( \phi(r) + \phi(s) \right),$$

where $\phi$ and $\rho$ are multi-layer perceptrons (MLPs) with tanh activations and hidden dimension 128. This structure guarantees $\hat{P}(r,s,f) = \hat{P}(s,r,f)$, obviating the need for an explicit loss term to enforce symmetry.

The training objective combines a data-fitting loss and a PDE residual:

• Data loss: $$L_\mathrm{data} = \frac{1}{N_\mathrm{data}} \sum_i \|\hat{P}(r_i, s_i, f_i) - P(r_i, s_i, f_i)\|^2$$,
• PDE loss: $$L_\mathrm{PDE} = \frac{1}{N_\mathrm{PDE}} \sum_j \|\nabla^2 \hat{P}(r_j, s_j, f_j) + k_j^2 \hat{P}(r_j, s_j, f_j)\|^2$$,
• Total loss: $$L_\mathrm{total} = L_\mathrm{data} + \lambda L_\mathrm{PDE}$$ (with $\lambda=1$).

Ground truth and interpolated ATFs are compared using frequency-resolved normalized MSE (NMSE):

$$\mathrm{NMSE}(f) = 10 \log_{10} \frac{\sum_n \|P(q_n, f) - \hat{P}(q_n, f)\|^2}{\sum_n \|P(q_n, f)\|^2},$$

with $q_n = (r, s)$ pairs over the test region. The PI-PINN, trained on experimental ATF data, demonstrates superior generalization over kernel methods especially at high frequencies and captures spatial field structure with higher fidelity (Chen et al., 27 Jan 2026).

3. Alternative Approaches and Comparative Methodologies

Several methodologies complement or compete with the PI-PINN strategy:

3.1 Kernel Interpolation and Weighted Pressure Matching

Kernel ridge regression (KRR) interpolates the sound field over space using reproducing kernels derived from the physical Green’s functions. Weighted pressure matching (WPM) further extends discrete control-point-based synthesis to optimize the regionally integrated error, resulting in a weighted least squares solution:

$$d^* = (G^H W G + \eta I)^{-1} G^H W p_\mathrm{des},$$

where $G$ encodes transfer functions from loudspeaker sources to control points; $W$ is a region-dependent weighting matrix constructed via kernel interpolation, typically using Helmholtz or von-Mises–Fisher kernels (Koyama et al., 2022). WPM provides more uniform field synthesis and higher accuracy, especially at higher frequencies or with sparse control points.

3.2 Physics-Informed Neural Architectures

Physics-informed neural networks (PINNs) and compact acoustics-informed neural networks (AINN) enforce the governing PDE—usually via a volumetric loss (PDE residual) collocated at points throughout the region, in addition to data-fitting at receivers or boundaries (Karakonstantis et al., 2024, Ma et al., 2024). Boundary-integral networks (PIBI-Net) restrict neural inference to the boundary, reconstructing the field in the interior by enforcing the Kirchhoff–Helmholtz boundary integral, thereby guaranteeing exact physics compliance inside the region without volumetric PDE losses (Damiano et al., 4 Jun 2025).

3.3 Bayesian and Variational Models

Boundary-informed Gaussian process priors incorporate partial (possibly uncertain) point-cloud knowledge of the region’s boundary through a regularization of plane-wave expansion coefficients, yielding a Bayesian posterior for field values at arbitrary locations in the target region. This approach bridges the gap between unconditional kernel interpolation (Tikhonov) and simulation with known boundaries, affording gains in region-to-region accuracy, particularly as the amount and quality of boundary data increases (Sundström et al., 16 Jun 2025).

3.4 Generative and Data-driven Paradigms

Deep learning approaches based on U-Net architectures (Kristoffersen et al., 2021), complex-valued networks (Ronchini et al., 2024), GANs with physical priors (Karakonstantis et al., 2023), diffusion models (Miotello et al., 2023), and neural processes with dynamic kernels (Liang et al., 2023) have been successfully used for region-to-region RTF/ATF interpolation, especially in highly underdetermined regimes. Physics-informed generative models and complex-valued neural networks can reconstruct both magnitude and phase, outperforming kernel or purely magnitude-based approaches in low-data settings and providing robust extrapolation to regions beyond the sensor array.

4. Basis Function Decompositions and Integral Representations

Explicit basis-function expansions—e.g., plane-wave, spherical/cylindrical harmonics (for interior or exterior domains), and local/divided subspace approaches—have long underpinned region-to-region sound field analysis. Discrete variable representation (DVR), for instance, enables full wavefield recovery in waveguides from vertical array measurements by directly linking mode coefficients to samples at prescribed depths, with regularization effected by modal bandwidth truncation (Makarov et al., 2021). In acousto-optic sensing, concentric-circle scanning with a circular-harmonic external expansion enables robust exterior-region reconstruction from tomographic projections, far outperforming filtered back-projection when sources lie within the reconstruction disk (Nguyen et al., 2023).

Convolutional plane-wave models hybridize local and global regularity: the field is partitioned into overlapping regions, each modeled as a sum of local plane waves, but with smoothness constraints enforced on the spatial variation of wave coefficients to preserve global coherence while remaining data efficient (Hahmann et al., 2022).

5. Evaluation Metrics and Practical Benchmarks

Quantitative benchmarking of region-to-region reconstruction methods employs metrics such as frequency-dependent normalized mean-square error (NMSE), spatial similarity (Modal Assurance Criterion), and regionally integrated signal-to-distortion ratio (SDR). These metrics assess both interpolation and extrapolation performance across the receiver and source manifolds.

Experiments indicate that physics-informed and permutation-invariant models (e.g., PI-PINN) provide robust performance in the high-frequency and extrapolation regimes, whereas boundary-based Bayesian regularization is especially advantageous where only coarse or uncertain geometric priors exist (Chen et al., 27 Jan 2026, Sundström et al., 16 Jun 2025). Generative adversarial networks with physical priors excel in retaining high-frequency spatial energy, and compact AINN implementations offer grid-free, low-parametric complexity solutions with robust pressure and gradient reconstruction in both synthetic and reverberant environments (Ma et al., 2024, Karakonstantis et al., 2023).

6. Practical Considerations, Limitations, and Future Directions

Region-to-region sound-field reconstruction is sensitive to measurement configuration (microphone placement, density), physical model accuracy (e.g., boundary conditions, medium homogeneity), and architectural or computational constraints (network depth, regularization parameter selection). Many algorithms, especially deep neural or kernel-based models, demand careful hyperparameter tuning, phase-aware loss balancing, and robust handling of sensor or boundary uncertainties.

Current limitations include:

• The dependence on domain geometry for basis or kernel selection (e.g., spherical harmonics for bounded domains, plane waves for open spaces);
• The general restriction to stationary or time-harmonic fields in most deep learning and kernel/integral approaches;
• Scalability to large-scale or three-dimensional volumetric domains;
• Robustness to sensor misplacement or measurement noise.

Emerging directions include hybrid diffusion/generative models incorporating explicit PDE constraints, learning-based adaptive basis selection, integration with differentiable numerical solvers for imposing physical laws “hard,” and expansion to real-time, region-to-region acoustic rendering in moving, complex environments (Verburg et al., 6 Oct 2025, Miotello et al., 2023, Chen et al., 27 Jan 2026).

7. Summary Table: Representative Region-to-Region Reconstruction Approaches

Approach	Key Mathematical Principle	Notable Features
PI-PINN (Chen et al., 27 Jan 2026)	Deep sets, physics-informed loss (PDE)	Reciprocally symmetric ATF, PDE-regularized, high-frequency accurate
Weighted PM (Koyama et al., 2022)	Kernel interpolation, weighted LS	Region-integrated error, flexible control-point placement
Boundary-informed GP (Sundström et al., 16 Jun 2025)	Plane-wave prior + boundary constraint	Leverages partial geometry, Bayesian hyperparameter estimation
PIBI-Net (Damiano et al., 4 Jun 2025)	Boundary integral, shallow network	Physics by construction, reduced interior parameterization
AINN (Ma et al., 2024)	Compact MLP, Helmholtz residual	Sparse measurements, pressure gradient support
Convolutional PW model (Hahmann et al., 2022)	Local-global plane-wave, smoothness	Data-efficient, flexible for wide apertures
Complex U-Net (Ronchini et al., 2024)	Complex-valued convolutions	Phase/magnitude recovery, irregular sensors
GAN with physical prior (Karakonstantis et al., 2023)	Plane-wave generator + adversarial	Extrapolation, high-frequency robustness

This synthesis reveals that region-to-region sound field reconstruction encompasses a diverse, rapidly advancing set of techniques, now centered on model-informed deep learning and hybrid variational-physical approaches. The ongoing integration of symmetry, physics, data adaptation, and robust optimization defines the leading edge of the field.