Adaptive SIREN-PINN with Principled Initialization: A Frequency-Aware and Singularity-Robust Framework for Solver-Free Acoustic Seismic Wave Modeling

The eikonal equation plays an important role across multidisciplinary branches of science and engineering. In geophysics, the eikonal equation, and its characteristics, are used in addressing two fundamental questions pertaining to seismic waves: what paths do the seismic waves take (its spreading)? and how long do they take? There have been numerous attempts to solve the eikonal equation, which can be broadly categorized as finite-difference and physics informed neural network (PINN) based approaches. While the former has been developed and optimized over the years, it still inherits some numerical inaccuracies and also the cost scales exponentially with the velocity model size. More importantly, it requires upwind calculations to satisfy the viscosity solution. PINNs, on the other hand, have shown great promise due to several features allowing for higher accuracy and scalability than conventional approaches. In this paper, we demonstrate another unique feature of PINN solutions, specifically its flexibility resulting from the global nature of its NN functional optimization, allowing for functional gradients referred to as automatic differentiation. This feature allows us to overcome the inability of conventional methods to handle large areas of missing information (gap) in the velocity model. We find empirically that the PINNs interpolation-extrapolation inherent capability enables us to circumvent a scenario when traveltime modelling is performed on velocity models containing gaps. Such a capability is crucial when performing traveltime modelling using the global tomographic Earth velocity model.

Modeling density-driven flow in porous media by physics-informed neural networks for CO2 sequestration

The thermal-fluid coupling phenomenon of silicon melt is significant in the growth process of silicon single crystals. Complex convection affects the temperature and concentration distribution of the silicon melt. Therefore, establishing and solving the thermal-fluid coupling model of silicon melts is crucial to optimizing the crystal growth process and improving crystal quality. Traditional numerical simulation methods have limitations in regard to optimization, control, and real-time monitoring. Physics-Informed Neural Network (PINN) does not require model discretization, after training, it can make predictions quickly, showing potential for industrial applications. However, when solving practical industrial coupling models, PINN often struggles to converge due to large parameter values and significant gaps between solution variables. Moreover, solving the thermal-fluid coupling model with PINN can be treated as a multitask problem, where the gradients of different equations interfere with each other, leading to gradient confusion, slow convergence, or even divergence. Therefore, this paper proposes a gradient normalized PINN (GNPINN) for solving the thermal-fluid coupling model of silicon melt. GNPINN balances the contribution of each task, ensuring a more equitable training speed between different tasks to stabilize the training process of the coupling model. This paper considers the thermal-fluid coupling model of silicon melt under different rotation conditions. GNPINN can accurately and comprehensively capture the complex temperature, velocity, and pressure distribution of silicon melt compared with other methods. Additionally, the experimental results uncover the flow and heat transfer properties of silicon melt, validating the effectiveness and industrial applicability of GNPINN.

Physics-Informed Neural Networks (PINNs) embed governing differential equations into training, enabling solutions of ODEs and PDEs. This review consolidates theoretical foundations (expressive capacity, automatic differentiation) and core methods (loss design, constraint enforcement, sampling, optimization), while surveying applications from baseline PINNs to advanced variants such as multi-physics coupling, domain decomposition, frequency-enhanced representations, and operator-learning hybrids. Comparative synthesis links architectural and training choices to equation type, data conditions, and computational budgets. A unified benchmarking framework is proposed with standard PDE tasks, accuracy metrics, collocation budgets, and transparent reporting for fair comparison with classical solvers. Evidence positions PINNs as complementary to traditional methods-effective for inverse problems, data assimilation, irregular domains, and parametric inference-while challenges remain in scalability, spectral bias, constraint enforcement, and reproducibility. The review offers a coherent synthesis with actionable guidance for scientific and engineering applications.

Chapter 3 - Seismic wave modeling and high-resolution imaging

Partial Differential Equations (PDEs) are foundational descriptive tools in scientific and engineering disciplines, enabling the exploration of phenomena from material design to climate dynamics and the propagation of pandemics and wildfires. Establishing control of PDEs, especially when the underlying phenomena are only partially observable, holds great potential for fostering technological innovations. This ambition faces considerable obstacles, as a robust sample-based solution remains elusive. Addressing this challenge, this thesis introduces a framework leveraging physics-informed machine learning (ML) to enable control of PDEs. Central to this framework is an agent, equipped with sensors to perceive its environment, that generates a low-dimensional physics-informed representation of the environment. This representation enables the derivation of an optimal policy for sparse control by the agent’s actuators. The framework provides a novel objective function which guides the agent to learn from its environment. Two alternative implementations of the objective are shown in the particular setting of the acoustic wave equation. One implementation employs differentiable numerical integration, while the other relies on physics-informed neural networks (PINN). The superiority of these approaches are demonstrated in prediction and control of scattered acoustic wave energy compared to the state-of-the-art methods.

Based on the common equation of seismic wave fields and its Hamiltonian formulation, an implicit symplectic method (ISM) is proposed for seismic wave modelingin this paper. Combined with helix boundary conditions, the spectral factorization method is applied to give a fast way for the ISM modeling. The propagation of seismic wave can be described by a same scalar acoustic euqation for homogeneous, complex isotropic and anisotropic media. With the assumption of no dissipation, the scalar equation depicts the symplectic evolution of seismic wave fields which is an infinite dimensional Hamiltonian system. Therefore, the scalar equation can be rewritten into Hamiltonian formulation and consequently its symplectic schemes can be given. Compared with the explicit symplectic method (ESM), ISM is unconditionally stable and permits big time steps and long‐time calculations which may be impossible for ESM. But the inverse of large matrix must be calculated at each step of ISM, which is a time consuming process. Fortunately, the matrix is band‐shape with a helical boundary conditions. In addition, the nonzero units in each row of the matrix not only are located in the same relative positions but also have similar values. Therefore, spectral factorization method can be used to get the inversion of the matrix in a very quick way. Based on the spectral factorization method, an ISM scheme with second order in time is applied to model the wave fields in constant velocity, layer velocity and Marmousi velocity models. The results indicates that with the help of spectral factorization ISM is a quite good method in seismic wave modeling.

Fourier feature-enhanced multi-layer residual stacking network: A novel multiscale modeling approach for physics-informed neural networks.

WarpPINN: Cine-MR image registration with physics-informed neural networks.

T he old saying “putting an ear to the ground” may turn out to be literally true for elephants, according to a recent report by O'Connell-Rodwell and her colleagues ([1][1]). When elephants generate their low-frequency vocalizations (rumbles), acoustic sound waves traveling through the air are accompanied by seismic waves that travel through the ground. Given that a variety of creatures, from insects to rodents and even the enormous elephant seal, use the generation and detection of terrestrial vibrations to communicate, O'Connell-Rodwell and her team wondered if elephants, too, have this seismic signaling capability. Eschewing the grasslands of Africa for a residential facility in Texas, the scientists analyzed acoustic and seismic signals generated by two captive Asian elephants. Whenever the elephants “rumbled” or stomped their feet during mock charges, seismic data were collected with geophones—sensitive microphones placed 10 m and 30 m from the elephants' pen that transformed terrestrial vibrations into electrical signals. Simultaneously, acoustic data were collected with audio equipment. The seismic and acoustic waves generated by rumbles and foot stomps had similar frequencies (20 to 24 Hz) that fell within the ideal range for the long-distance transmission of low-frequency sounds. Intriguingly, the two sets of waves had different velocities (248 to 264 m/s in the ground, and 309 m/s in the air), so that they were no longer in phase as they traveled further away from their source. This hints that elephants may make their rumbles and foot stomps loud enough to produce separate acoustic and seismic waves. ![Figure][2] CREDIT: BEN AND LYNETTE HART Through mathematical modeling, the investigators estimated that the seismic waves created by their stomping elephants traveled at least 36 km. Conceivably, these long-distance seismic signals may enable elephants to communicate with other herds. The location of a distant herd could be pinpointed by assessing the time delay between the arrival of seismic and acoustic signals. Elephants are known to move toward thunderstorms that are more than 25 km away, too far for them to hear the sound of thunder. It is possible that they can detect the terrestrial vibrations associated with distant storms, presumably a major advantage in their search for new water sources. Generating seismic signals is easy, but what about detecting them? The elephant's trunk has mechanoreceptors that respond to mechanical pressure. There may also be similar receptors in the elephant's well-innervated foot pads. These mechanoreceptors may explain foot-lifting behavior during which elephants lean forward and lift up one foot, possibly to improve their sensitivity to ground vibrations. Establishing that elephants use seismic signaling for long-distance communication is no easy task. Undaunted, O'Connell-Rodwell and her team plan to train their elephants to respond to seismic waves by pulling a lever with their trunk or pressing a button with their feet—with, of course, the provision of a reward for the correct response. 1. [↵][3]1. C. E. O'Connell-Rodwell, 2. B. T. Arnason, 3. L. A. Hart , J. Acoust. Soc. Am. 108, 3066 (2000) [journal]. [OpenUrl][4][CrossRef][5][PubMed][6][Web of Science][7] [1]: #ref-1 [2]: pending:yes [3]: #xref-ref-1-1 View reference 1 in text [4]: {openurl}?query=rft.jtitle%253DJournal%2Bof%2Bthe%2BAcoustical%2BSociety%2Bof%2BAmerica%26rft.stitle%253DJournal%2Bof%2Bthe%2BAcoustical%2BSociety%2Bof%2BAmerica%26rft.aulast%253DO%2527Connell-Rodwell%26rft.auinit1%253DC.%2BE.%26rft.volume%253D108%26rft.issue%253D6%26rft.spage%253D3066%26rft.epage%253D3072%26rft.atitle%253DSeismic%2Bproperties%2Bof%2BAsian%2Belephant%2B%2528Elephas%2Bmaximus%2529%2Bvocalizations%2Band%2Blocomotion.%26rft_id%253Dinfo%253Adoi%252F10.1121%252F1.1323460%26rft_id%253Dinfo%253Apmid%252F11144599%26rft.genre%253Darticle%26rft_val_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Ajournal%26ctx_ver%253DZ39.88-2004%26url_ver%253DZ39.88-2004%26url_ctx_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Actx [5]: /lookup/external-ref?access_num=10.1121/1.1323460&link_type=DOI [6]: /lookup/external-ref?access_num=11144599&link_type=MED&atom=%2Fsci%2F291%2F5505%2F842.atom [7]: /lookup/external-ref?access_num=000165780300039&link_type=ISI

We use the Earth Simulator to model seismic wave propagation resulting from large earthquakes on a global scale. Simulations are conducted based upon the spectral-element method, a high-degree finite- element technique. We include the full complexity of the Earth, i.e., a three-dimensional seismic wave veloci- ty and density structure, a 3-D crustal model, the elliptical figure of the Earth as well as topography and bathymetry. We use 507 nodes of the Earth Simulator (4056 processors) and model the three dimensional Earth with 13.7 billion grid points. A total of 7 terabytes of memory is needed. We have reached a vectoriza- tion ratio of 99.3%, and attained a total performance of 10 teraflops (32% of the peak performance). The very high resolution of the mesh allows us to calculate theoretical seismic waves for realistic fully three-dimen- sional Earth model, which are accurate at periods of 3.5 seconds and longer. We have modeled seismic waves generated by recent large earthquakes and show the importance of including three-dimensional structure in the seismic wave simulation. These theoretical seismograms calculated for fully three-dimensional Earth models should improve our understanding of earthquake source mechanisms and the Earth dynamics.

Modeling Partial differential equations (PDEs) is a well-known challenge in the field of scientific computing. In particular, the linear acoustic wave PDE forms a significant problem in modeling due to its oscillatory behavior and multi-scale tendency. With the recently introduced class of deep neural networks, the physics-informed neural networks (PINNs), a mesh-free approach can now be utilized to model the wave PDE without the need for a previously known solution for training. Other training challenges remain such as the selection of hyperparameters and loss convergence. In this paper, we propose an enhancement that focuses on the assigned weights for the PINN loss function terms in order to more accurately model the wave PDE in a homogeneous, inhomogeneous domain, and with a higher frequency wave source function. The proposed enhancement reflected reduced residual and relative error values compared to the baseline architecture.

Applying an advanced temporal and spatial high-order finite-difference stencil to 3D seismic wave modeling

SUMMARYIn seismic wave modelling, the boundary reflections caused by the computational grid edges should be reduced to produce accurate simulation results. The perfectly matched layer (PML) method is one of the popular techniques to suppress such artificial reflections, because it can be easily applied to the first-order wave equation in many numerical methods. However, one issue of the PML method is that the stability condition might be violated in complex elastic anisotropic media. In these cases, the PML method will not attenuate the boundary reflections but rather introduce strong artefacts in the simulation results. To tackle this problem, we propose a generalized stiffness reduction method (GSRM) as a substitute for the PML method. We first derive the stability conditions of the PML method and analyse the suitable conditions for their application to time- and frequency-domain seismic wave modelling. Then, we develop a simple and effective numerical implementation of the GSRM to attenuate the boundary reflections and apply it to seismic wave modelling in elastic anisotropic media. We give some numerical experiments to demonstrate the feasibility and advantages of the GSRM compared to the PML method. Numerical examples show the GSRM is conceptually simpler, more computationally efficient and more straightforward in terms of numerical implementation than the PML method for seismic modelling using either first- or second-order time- and frequency-domain wave equations.

We present a numerical model for blast-induced seismic waves. This model is based on the transformation mechanism of the seismic wave field, and the process of the seismic wave’s generation is divided into two stages. The first stage is the generation of elastic waves. Due to the plastic deformation of the geotechnical medium in this stage, we have established the Euler model to describe it. The second stage is the prorogation of the elastic waves; we have established the Lagrange models to describe this stage. Finally, the influence of the main parameters of the explosive sources (detonation pressure and expansion index) on the seismic wave is analyzed by the established model. The results show that the proposed model in this paper can reasonably predict the evolution law of the seismic wave field based on the explosive source parameters.