Physics-informed neural operators for the in situ characterization of locally reacting sound absorbers

This paper proposes an interpretable neural operator framework, the Kernel Operator Network (KernelOnet), which incorporates kernel functions explicitly into the neural operator architecture, so that the operator structure matches the kernel-expansion form used in boundary-type kernel-expansion methods. Unlike traditional neural operators such as DeepONet, which learn basis functions implicitly through deep networks, KernelOnet replaces the trunk network with explicit kernels and offers three complementary kernels: a data-driven learnable kernel, in which a neural network parameterizes a radial basis function learned from data, and which for constant-coefficient linear problems can be regarded as a non-singular fundamental solution; a physics-informed kernel, which embeds physical information such as analytic fundamental solutions into the network structure, so that the expansion satisfies the governing equation automatically and can be trained without supervision on boundary conditions alone, with no interior solution data; and a hybrid kernel, which splits the solution, according to the linear principal part of the governing equation, into a homogeneous part spanned by analytic fundamental solutions and a source part carried by low-rank learned correction kernels, thereby balancing physical priors against data fitting on nonlinear problems lacking an analytic fundamental solution. On three benchmarks and one engineering problem in a shallow-water waveguide, KernelOnet attains high accuracy; where comparable with DeepONet, it is more accurate with fewer learnable parameters. Its unsupervised configuration needs no interior solution labels, and its per-query inference cost is far below that of per-instance solvers, offering an effective route to acoustic propagation in unbounded exterior domains that general-purpose neural operators struggle to handle.

First-order Ambisonics (FOA) is a standard spatial audio format based on spherical harmonic decomposition. Its zeroth- and first-order components capture the sound pressure and particle velocity, respectively. Recently, physics-informed neural networks have been applied to the spatial interpolation of FOA signals, regularizing the network outputs based on soft penalty terms derived from physical principles, e.g., the linearized momentum equation. In this paper, we reformulate the task so that the predicted FOA signal automatically satisfies the linearized momentum equation. Our network approximates a scalar function called velocity potential, rather than the FOA signal itself. Then, the FOA signal can be readily recovered through the partial derivatives of the velocity potential with respect to the network inputs (i.e., time and microphone position) according to physics of sound propagation. By deriving the four channels of FOA from the single-channel velocity potential, the reconstructed signal follows the physical principle at any time and position by construction. Experimental results on room impulse response reconstruction confirm the effectiveness of the proposed framework.

This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network based on the DeepONet framework. We consider a 2D square domain with an inclusion of arbitrary boundary geometry at its center. This inclusion acts as a scatterer for an incoming harmonic wave. The aim is to learn the operator linking the geometry of the scatterer to the resulting scattered field. A signed distance function to the boundary of the inner inclusion, evaluated at several points in the domain, is used to encode its geometry. It serves as input for the branch part of the DeepONet architecture, while local information is used as input for the trunk part. This approach enables the encoding of arbitrary geometries, whether they are parameterized or not. The evaluation of the model on unseen geometries is compared with its finite element method (FEM) equivalent to test its generalization capabilities. The trained network weights implicitly embed the local physics and their interaction with the domain geometry. If the training space sufficiently covers the target evaluation space, the model can generalize accordingly. Furthermore, it can be refined to extend to another region of interest without retraining from scratch. This framework also avoids the need to remesh the domain for each geometry. The proposed approach delivers a computationally lighter surrogate model than FEM alternatives and avoids relying on FEM-generated training data.