Differentiable physics for sound field reconstruction

TL;DR

The paper tackles reconstructing a time-varying sound field $p(oldsymbol{r},t)$ from sparse measurements by learning a neural representation of the initial pressure and solving the wave equation with a differentiable finite-difference solver, ensuring hard physical constraints via automatic differentiation. The differentiable physics (DP) approach yields stable, data-efficient training and demonstrates superior accuracy and convergence compared with physics-informed neural networks (PINNs), especially under extreme undersampling and when extrapolating beyond observed regions. A sparsity-promoting constraint on the initial condition further enhances recovery with limited data, enabling accurate reconstructions across single-pulse, reverberant, and complex-source scenarios. The work highlights the potential of DP to scale to higher resolutions and to incorporate additional physics or domain knowledge, such as boundary impedance or heterogeneous media, with practical impact for acoustics in rooms and complex environments.

Abstract

Sound field reconstruction involves estimating sound fields from a limited number of spatially distributed observations. This work introduces a differentiable physics approach for sound field reconstruction, where the initial conditions of the wave equation are approximated with a neural network, and the differential operator is computed with a differentiable numerical solver. The use of a numerical solver enables a stable network training while enforcing the physics as a strong constraint, in contrast to conventional physics-informed neural networks, which include the physics as a constraint in the loss function. We introduce an additional sparsity-promoting constraint to achieve meaningful solutions even under severe undersampling conditions. Experiments demonstrate that the proposed approach can reconstruct sound fields under extreme data scarcity, achieving higher accuracy and better convergence compared to physics-informed neural networks.

Figures (8)

• Figure 1: (a): Diagram of the DP model. The inputs to the differentiable physics neural network (DPNN) are spatial coordinates $x,y$ and the output is the estimated initial pressure $g_\text{dp}(\mathbf{r};\boldsymbol{\theta})$. During training, $x,y$ are taken on a grid so that the output is $\mathbf{p}^0$, and a numerical solver is used to compute the pressure field at later times, $\mathbf{p}^1,\dots,\mathbf{p}^{n-1}$. The total loss is composed by a data fitting term $\mathcal{L}_\mathrm{data}^{\text{(d)}}$ and a regularization term $\mathcal{L}_\mathrm{reg}$. AD is used to back-propagate the loss gradient through the PDE solver and train the network. (b): Diagram of a PINN. The inputs to the network are spatio-temporal coordinates $x,y,t$ and the output is the acoustic pressure $p_\text{pinn}(\mathbf{r},t;\boldsymbol{\theta})$. AD is used for evaluating the partial derivatives in the PDE and the boundary conditions. The total loss is composed of the differential operator terms $\mathcal{L}_\mathrm{pde}$ and $\mathcal{L}_\mathrm{bcs}$, a data fitting term $\mathcal{L}_\mathrm{data},$ and a regularization term $\mathcal{L}_\mathrm{reg}^\text{(e)}$.
• Figure 2: Left: Measurement positions used for estimating the sound field. Right: simulated noisy data at each of the measurement positions over time for a sound field consisting of a single pulse at the center of the domain.
• Figure 3: Sound field consisting on a single pulse at the domain center. Each column corresponds to a time frame. Row (a): reference solution. Row (b): DP model estimation. Row (c): PINN estimation. Row (d): difference between reference and DP. Row (e): difference between reference and PINN.
• Figure 4: Dynamics of the DP neural network and PINN trained to learn the single pulse sound field. (a): Weighted loss terms, 'data' stands for $\lambda_\text{data} \mathcal{L}_\text{data}^{\text{(d)}}$, and 'sparsity' for $\lambda_\text{reg} \mathcal{L}_\text{reg}$. (b): Weighting parameters $\lambda_\text{data}$ and $\lambda_\text{reg}$. (c): Weighted loss terms, 'data' stands for $\lambda_\text{data} \mathcal{L}_\text{data}$, 'pde' for $\lambda_\text{pde} \mathcal{L}_\text{pde}$, 'sparsity' for $\lambda_\text{reg} \mathcal{L}_\text{reg}$, and 'boundary' for $\lambda_\text{bcs} \mathcal{L}_\text{bcs}$. (d): Weighting parameters $\lambda_\text{data}, \lambda_\text{pde}$, $\lambda_\text{reg}$, and $\lambda_\text{bcs}$. (e): Total weighted loss.
• Figure 5: Normalized Mean squared error as a function of (a) the SNR, (b) the source width, (c) the pulse distance to the array center, and (d) the downsample factor between the evaluation and training grids.