--Accurate and efficient simulation of wave equations is crucial in computational wave imaging applications, such as ultrasound computed tomography (USCT), which reconstructs tissue material properties from observed scattered waves. Traditional numerical solvers for wave equations are computationally intensive and often unstable, limiting their practical applications for quasi-real-time image reconstruction. Neural operators offer an innovative approach by accelerating PDE solving using neural networks; however, their effectiveness in realistic imaging is limited because existing datasets oversimplify real-world complexity. In this paper, we present OpenBreastUS, a large-scale wave equation dataset designed to bridge the gap between theoretical equations and practical imaging applications. OpenBreastUS includes 8,000 anatomically realistic human breast phantoms and over 16 million frequency-domain wave simulations using real USCT configurations. It enables a comprehensive benchmarking of popular neural operators for both forward simulation and inverse imaging tasks, allowing analysis of their performance, scalability, and generalization capabilities. By offering a realistic and extensive dataset, OpenBreastUS not only serves as a platform for developing innovative neural PDE solvers but also facilitates their deployment in real-world medical imaging problems. For the first time, we demonstrate efficient in vivo imaging of the human breast using neural operator solvers. Zhijun Zeng is with the Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China (e-mail: zengzj22@mails.tsinghua.edu.cn).

Numerical simulations of seismic wave propagation are crucial for investigating velocity structures and improving seismic hazard assessment. However, standard methods such as finite difference or finite element are computationally expensive. Recent studies have shown that a new class of machine learning models, called neural operators, can solve the elastodynamic wave equation orders of magnitude faster than conventional methods. Full waveform inversion is a prime beneficiary of the accelerated simulations. Neural operators, as end-to-end differentiable operators, combined with automatic differentiation, provide an alternative approach to the adjoint-state method. Since neural operators do not involve the Born approximation, when used for full waveform inversion they have the potential to include additional phases and alleviate cycle-skipping problems present in traditional adjoint-state formulations. In this study, we demonstrate the application of neural operators for full waveform inversion on a real seismic dataset, which consists of several nodal transects collected across the San Gabriel, Chino, and San Bernardino basins in the Los Angeles metropolitan area.

Physics-Informed Neural Networks (PINNs) have gained increasing attention for solving partial differential equations, including the Helmholtz equation, due to their flexibility and mesh-free formulation. However, their low-frequency bias limits their accuracy and convergence speed for high-frequency wavefield simulations. To alleviate these problems, we propose a simplified PINN framework that incorporates Gabor functions, designed to capture the oscillatory and localized nature of wavefields more effectively. Unlike previous attempts that rely on auxiliary networks to learn Gabor parameters, we redefine the network's task to map input coordinates to a custom Gabor coordinate system, simplifying the training process without increasing the number of trainable parameters compared to a simple PINN. We validate the proposed method across multiple velocity models, including the complex Marmousi and Overthrust models, and demonstrate its superior accuracy, faster convergence, and better robustness features compared to both traditional PINNs and earlier Gabor-based PINNs. Additionally, we propose an efficient integration of a Perfectly Matched Layer (PML) to enhance wavefield behavior near the boundaries. These results suggest that our approach offers an efficient and accurate alternative for scattered wavefield modeling and lays the groundwork for future improvements in PINN-based seismic applications.

Full-Waveform Inversion seeks to achieve a high-resolution model of the subsurface through the application of multi-variate optimization to the seismic inverse problem. Although now a mature technology, FWI has limitations related to the choice of the appropriate solver for the forward problem in challenging environments requiring complex assumptions, and very wide angle and multi-azimuth data necessary for full reconstruction are often not available. Deep Learning techniques have emerged as excellent optimization frameworks. Data-driven methods do not impose a wave propagation model and are not exposed to modelling errors. On the contrary, deterministic models are governed by the laws of physics. Seismic FWI has recently started to be investigated as a Deep Learning framework. Focus has been on the time-domain, while the pseudo-spectral domain has not been yet explored. However, classical FWI experienced major breakthroughs when pseudo-spectral approaches were employed. This work addresses the lacuna that exists in incorporating the pseudo-spectral approach within Deep Learning. This has been done by re-formulating the pseudo-spectral FWI problem as a Deep Learning algorithm for a theory-driven pseudo-spectral approach. A novel Recurrent Neural Network framework is proposed. This is qualitatively assessed on synthetic data, applied to a two-dimensional Marmousi dataset and evaluated against deterministic and time-based approaches. Pseudo-spectral theory-guided FWI using RNN was shown to be more accurate than classical FWI with only 0.05 error tolerance and 1.45\% relative percent-age error. Indeed, this provides more stable convergence, able to identify faults better and has more low frequency content than classical FWI. Moreover, RNN was more suited than classical FWI at edge detection in the shallow and deep sections due to cleaner receiver residuals.

Physics-informed neural networks (PINNs) face significant challenges in modeling multi-frequency wavefields in complex velocity models due to their slow convergence, difficulty in representing high-frequency details, and lack of generalization to varying frequencies and velocity scenarios. To address these issues, we propose Meta-LRPINN, a novel framework that combines low-rank parameterization using singular value decomposition (SVD) with meta-learning and frequency embedding. Specifically, we decompose the weights of PINN's hidden layers using SVD and introduce an innovative frequency embedding hypernetwork (FEH) that links input frequencies with the singular values, enabling efficient and frequency-adaptive wavefield representation. Meta-learning is employed to provide robust initialization, improving optimization stability and reducing training time. Additionally, we implement adaptive rank reduction and FEH pruning during the meta-testing phase to further enhance efficiency. Numerical experiments, which are presented on multi-frequency scattered wavefields for different velocity models, demonstrate that Meta-LRPINN achieves much fast convergence speed and much high accuracy compared to baseline methods such as Meta-PINN and vanilla PINN. Also, the proposed framework shows strong generalization to out-of-distribution frequencies while maintaining computational efficiency. These results highlight the potential of our Meta-LRPINN for scalable and adaptable seismic wavefield modeling.

In recent years, deep learning (DL) has emerged as a promising alternative approach for various seismic processing tasks, including primary estimation (or multiple elimination), a crucial step for accurate subsurface imaging. In geophysics, DL methods are commonly based on supervised learning from large amounts of high-quality labelled data. Instead of relying on traditional supervised learning, in the context of free-surface multiple elimination, we propose a method in which the DL model learns to effectively parameterize the free-surface multiple-free wavefield from the full wavefield by incorporating the underlying physics into the loss computation. This, in turn, yields high-quality estimates without ever being shown any ground truth data. Currently, the network reparameterization is performed independently for each dataset. We demonstrate its effectiveness through tests on both synthetic and field data. We employ industry-standard Surface-Related Multiple Elimination (SRME) using, respectively, global least-squares adaptive subtraction and local least-squares adaptive subtraction as benchmarks. The comparison shows that the proposed method outperforms the benchmarks in estimation accuracy, achieving the most complete primary estimation and the least multiple energy leakage, but at the cost of a higher computational burden.

Solving inverse problems with the reverse process of a diffusion model represents an appealing avenue to produce highly realistic, yet diverse solutions from incomplete and possibly noisy measurements, ultimately enabling uncertainty quantification at scale. However, because of the intractable nature of the score function of the likelihood term (i.e., $\nabla_{\mathbf{x}_t} p(\mathbf{y} | \mathbf{x}_t)$), various samplers have been proposed in the literature that use different (more or less accurate) approximations of such a gradient to guide the diffusion process towards solutions that match the observations. In this work, I consider two sampling algorithms recently proposed under the name of Diffusion Posterior Sampling (DPS) and Pseudo-inverse Guided Diffusion Model (PGDM), respectively. In DSP, the guidance term used at each step of the reverse diffusion process is obtained by applying the adjoint of the modeling operator to the residual obtained from a one-step denoising estimate of the solution. On the other hand, PGDM utilizes a pseudo-inverse operator that originates from the fact that the one-step denoised solution is not assumed to be deterministic, rather modeled as a Gaussian distribution. Through an extensive set of numerical examples on two geophysical inverse problems (namely, seismic interpolation and seismic inversion), I show that two key aspects for the success of any measurement-guided diffusion process are: i) our ability to re-parametrize the inverse problem such that the sought after model is bounded between -1 and 1 (a pre-requisite for any diffusion model); ii) the choice of the training dataset used to learn the implicit prior that guides the reverse diffusion process. Numerical examples on synthetic and field datasets reveal that PGDM outperforms DPS in both scenarios at limited additional cost.

In scientific machine learning, the task of identifying partial differential equations accurately from sparse and noisy data poses a significant challenge. Current sparse regression methods may identify inaccurate equations on sparse and noisy datasets and are not suitable for varying coefficients. To address this issue, we propose a hybrid framework that combines two alternating direction optimization phases: discovery and embedding. The discovery phase employs current well-developed sparse regression techniques to preliminarily identify governing equations from observations. The embedding phase implements a recurrent convolutional neural network (RCNN), enabling efficient processes for time-space iterations involved in discretized forms of wave equation. The RCNN model further optimizes the imperfect sparse regression results to obtain more accurate functional terms and coefficients. Through alternating update of discovery-embedding phases, essential physical equations can be robustly identified from noisy and low-resolution measurements. To assess the performance of proposed framework, numerical experiments are conducted on various scenarios involving wave equation in elastic/viscoelastic and homogeneous/inhomogeneous media. The results demonstrate that the proposed method exhibits excellent robustness and accuracy, even when faced with high levels of noise and limited data availability in both spatial and temporal domains.