Phase-Retrieval-Based PINN for Acoustic Fields

• The paper introduces PRB-PINN, which integrates independent magnitude and phase networks with a Helmholtz PDE loss to accurately reconstruct complex acoustic fields.
• The method leverages a joint optimization strategy that decouples magnitude and phase inference by embedding a phase-retrieval mechanism within the PINN framework.
• Experimental results in a 3D acoustic domain show that PRB-PINN reduces test log-spectral loss by 1–3 dB compared to baseline methods, especially under sparse sensor conditions.

Phase-retrieval-based physics-informed neural networks (PRB-PINNs) address the task of reconstructing acoustic magnitude fields when only spatially sparse magnitude measurements are available and phase information is inaccessible or unreliable. This method leverages PINNs by augmenting them with a phase-retrieval mechanism, enabling full-field magnitude estimation through explicit modeling of the phase and magnitude as independent neural submodules. The central innovation lies in the integration of a physics-based PDE loss that enables the implicit recovery of spatial phase information, producing a consistent complex field solution from magnitude data alone (Schrader et al., 27 Jan 2026).

1. Governing Equations and Physics Constraints

The problem formulation assumes a free-field acoustic environment at a fixed frequency, with the pressure field $u(x)$ governed by the homogeneous Helmholtz equation: $(\Delta + k^2)\,u(x) = 0, \quad x \in \Omega$ where $\Delta$ denotes the 3D Laplacian, $k$ is the wavenumber, and $\Omega$ is the spatial domain of interest.

To enforce physical correctness, the method defines a physics loss computed as the mean squared residual of the Helmholtz operator on $P$ randomly sampled collocation points $\{x_p^\perp\}_{p=1}^P$ in $\Omega$: $$\mathcal{L}_{\rm PDE} = \frac{1}{P}\sum_{p=1}^P \left| (\Delta + k^2)u(x_p^\perp)\right|^2$$ No explicit Neumann or nontrivial boundary term is used in the basic acoustic room setup; Dirichlet and magnitude constraints are incorporated via data loss terms.

2. Network Architecture and Field Representation

The PRB-PINN models the magnitude and phase of the acoustic field with two independent multilayer perceptrons (MLPs):

• The magnitude network $M_\theta(x)$ outputs $M_\theta(x)\in\mathbb{R}^+$.
• The phase network $\phi_\varphi(x)$ outputs $\phi_\varphi(x)\in\mathbb{R}$.

Both networks are parameterized identically with 4 hidden layers, 256 units per layer, and tanh activations. The input is projected via a random Fourier feature embedding $\gamma(x) = [\sin(Bx), \cos(Bx)]^\top$ with $B \in \mathbb{R}^{128\times 3}$ drawn from a standard normal distribution $\mathcal{N}(0, I)$.

At each spatial point, the complex-valued pressure field is reconstructed as: $$A(x) = M_\theta(x)\exp\bigl(i\,\phi_\varphi(x)\bigr) \approx u(x)$$

3. Loss Function and Data Fitting

The total loss function is a weighted combination of data-fitting and physics terms: $$\mathcal{L} = \lambda_{\rm data}\mathcal{L}_{\rm data} + \lambda_{\rm PDE}\mathcal{L}_{\rm PDE}$$ with $\lambda_{\rm data}=10^{-1}$ and $\lambda_{\rm PDE}=10^{-3}$.

• The data-fitting loss leverages only magnitude measurements $\{a_m = |s_m|\}_{m=1}^M$ at sensor positions $\{x_m^m\}$. A log-spectral loss is employed: $$\mathcal{L}_{\rm data} = \frac{1}{M}\sum_{m=1}^M \left|20\log_{10}\left(\frac{a_m}{|A(x_m^m)|}\right)\right|$$ which was found to correlate better with perceptual error than an MSE in magnitude.

4. Phase Retrieval Mechanism

Phase is not measured directly and is instead implicitly estimated via the PDE regularizer. The parameters of the phase network $\phi_\varphi(x)$ are initialized randomly. During training, minimization of the $\mathcal{L}_{\rm PDE}$ term steers the phase estimates towards physically valid configurations compatible with the Helmholtz equation.

The phase retrieval is thus embedded in the joint optimization over both magnitude and phase networks, with no explicit projection or iterative phase-unwrapping methods used. Global phase shifts are not penalized, reflecting inherent ambiguity; additional phase-smoothness terms are unnecessary in the reported scenarios.

5. Training Setup and Hyperparameter Regime

Training uses the AdamW optimizer with an initial learning rate of $10^{-3}$, decaying to $0.9\times$ every 10,000 iterations. The schedule runs for $5\times 10^5$ steps. The PDE loss is computed on batches of collocation points (typically several thousand), while all $M$ measurement points are utilized for the data loss at every iteration. Both networks are trained from scratch in a joint, one-stage procedure—no curriculum or pretraining is required (Schrader et al., 27 Jan 2026).

6. Experimental Evaluation

The methodology was evaluated via simulation in a rectangular room of size $$3\,\mathrm{m}\times 4\,\mathrm{m}\times 6\,\mathrm{m}$$ ($T_{60}\approx 200\,$ms). The reconstruction target is a $1\,\mathrm{m}^3$ cube at the center of the room, discretized on a $33^3$ spatial grid. The number of sensors $M$ takes values in {5, 10, 20, 50}, with measurement positions randomly sampled.

A total of 64 point sources are positioned outside the reconstruction domain, half of which are used for hyperparameter tuning, the remainder for evaluation. Single frequencies at 200, 400, and 600 Hz are considered.

Performance is quantified by the test-set log-spectral loss in decibels as defined above. Baselines include nearest-neighbor interpolation and a neural field without PDE regularization ($\lambda_{\rm PDE}=0$). PRB-PINN consistently outperformed both baselines across sensor counts and frequencies, achieving $1$–$3$ dB lower test losses. Visualizations display high-fidelity reconstruction of standing-wave amplitude patterns otherwise blurred or distorted by baseline methods.

A summary of the experimental setup and results is given below.

Sensor Count $M$	Frequency (Hz)	Method	Typical log-spectral loss (dB)	Qualitative Outcome
20	200	PRB-PINN	Lower by 1–3 dB	Pattern closely matches ground truth
20	200	Baseline NF	Higher	Smoothed/blurred nodes
20	200	Interpolator	Highest	Significant loss of fine features

7. Discussion, Limitations, and Extensions

The PRB-PINN approach reliably converges in $O(10^5)$ iterations. Accuracy improves sharply with increased sensor count, though at higher frequencies (e.g., 600 Hz), even relatively dense sampling may not suffice for full spatial resolution. The PDE regularizer offers substantial robustness to data sparsity and moderate magnitude noise (perturbations up to $\pm 1$ dB).

Potential extensions include multi-frequency joint optimization to resolve phase ambiguity, explicit phase-smoothness or total-variation regularization, application to domains with scattering or nontrivial boundaries, and integration of learned priors to further reduce sensor requirements. Limitations include potential non-uniqueness in extremely sparse settings and sensitivity at high frequencies or for pathological sensor placements (Schrader et al., 27 Jan 2026).

The method represents a principled architecture for incorporating physical constraints into neural field regression tasks under magnitude-only measurement regimes, with demonstrated superiority over purely data-driven approaches.