Physics and geometry informed neural operator network with application to acoustic scattering

TL;DR

The paper develops a physics- and geometry-informed neural operator, PGI-DeepONet, to learn the forward solution operator of 2D acoustic scattering across arbitrarily shaped rigid scatterers. By integrating a NURBS-based geometry parameterization with a DeepONet framework and enforcing the Helmholtz PDE and Robin/Neumann boundary conditions through a physics-driven loss, the method achieves fast, physically consistent predictions without labeled data and with strong generalization over shapes. Numerical results demonstrate accurate scattered-field predictions for both circular and arbitrary shapes, with substantial speed-ups (approximately 17×) over traditional FEM solvers, highlighting the practical impact for tasks requiring many forward evaluations. The approach generalizes to other PDEs and variable domains, offering a scalable path toward real-time, geometry-aware operator learning in engineering applications.

Abstract

In this paper, we introduce a physics and geometry informed neural operator network with application to the forward simulation of acoustic scattering. The development of geometry informed deep learning models capable of learning a solution operator for different computational domains is a problem of general importance for a variety of engineering applications. To this end, we propose a physics-informed deep operator network (DeepONet) capable of predicting the scattered pressure field for arbitrarily shaped scatterers using a geometric parameterization approach based on non-uniform rational B-splines (NURBS). This approach also results in parsimonious representations of non-trivial scatterer geometries. In contrast to existing physics-based approaches that require model re-evaluation when changing the computational domains, our trained model is capable of learning solution operator that can approximate physically-consistent scattered pressure field in just a few seconds for arbitrary rigid scatterer shapes; it follows that the computational time for forward simulations can improve (i.e. be reduced) by orders of magnitude in comparison to the traditional forward solvers. In addition, this approach can evaluate the scattered pressure field without the need for labeled training data. After presenting the theoretical approach, a comprehensive numerical study is also provided to illustrate the remarkable ability of this approach to simulate the acoustic pressure fields resulting from arbitrary combinations of arbitrary scatterer geometries. These results highlight the unique generalization capability of the proposed operator learning approach.

Figures (9)

• Figure 1: Schematic illustrates a sample (a) DeepONet architecture with branch and trunk networks. Here, $u$ represents an input parameterization function evaluated at $m$ discrete points and $\textbf{x}$ describes the spatial coordinates in a 2D domain, while $\mathcal{G}_\theta$ represents the learned operator that maps the inputs $u$ and $\textbf{x}$ to the corresponding output $\mathcal{G}_{\theta}(u)(\textbf{x})$. (b) 2D square acoustic domain $\Omega$ of size $[0,1]~m \times [0,1]~m$ with internal boundary $\Gamma_i$ and external boundary $\Gamma_e$. This also highlights the incident pressure ($p_i$) and scattered pressure ($p_s$) inside the domain.
• Figure 2: Schematics of the neural network architecture of the PGI-DeepONet. $\textbf{C}$ represents an input NURBS parameterization function evaluated for $m$ discrete shapes $\Gamma_i$, and $\textbf{x}$ describes the spatial coordinates forming the corresponding acoustic domain. In addition, $\beta(\textbf{C}; \theta_b)$ are the coefficients of the branch network, $\tau(\textbf{C}; \theta_t)$ are the coefficients of the trunk network, and $\mathcal{G}_\theta$ is the learned operator that maps the inputs $\textbf{C}$ and $\textbf{x}$ to the corresponding output $\mathcal{G}_{\theta}(u)(\textbf{x})$.
• Figure 3: Schematic of the deep ResNet architecture with $N_R$ residual blocks, where each residual block contains $N_L$ linear layers with Sine activation function and $n_w$ neurons (see dotted box). Note that the input and output of the ResNet architecture signifies the input and output of the branch and trunk networks.
• Figure 4: Schematic of (a) the 2D acoustic domain showing also the possible size range of internal boundaries ($\Gamma_i$). The size of the internal boundary can vary between a minimum and maximum value (dotted squares), i.e. $\Gamma_i \in [\Gamma^{min}_i, \Gamma^{max}_i]$. For example, $\Gamma_i$ could be a circle (in blue) with a radius between $\Gamma^{min}_i$ and $\Gamma^{max}_i$. In addition, the use of NURBS allows capturing many different arbitrary shapes contained within the size limits. (b) Coordinates used for training the physics-informed operator network on the external boundary ($\Gamma_e$), in the acoustic domain ($\Omega$), and on a sample circular internal boundary ($\Gamma_i$).
• Figure 5: Schematic of (a) the convergence plot of the loss function $\mathcal{L}$ with training epochs. Note that this plot also records the variation in the individual loss components $\mathcal{L}_{\mathcal{N}}$, $\mathcal{L}_{\mathcal{B}_i}$, and $\mathcal{L}_{\mathcal{B}_e}$. (b) The meshed geometry for a sample circular $\Gamma_i$ with PML boundaries for finite element (FE) analysis.