Eikonal equation solved with a novel Lattice Boltzmann method framework

Abstract Ambient noise surface wave traveltime tomography has been increasingly used to investigate shallow crustal structures. In relatively small‐scale studies, factors such as topography and the uneven distribution of ambient noise sources may have significant influence on the tomographic results. In addition, anisotropy can also affect surface wave propagation, but is usually neglected. In this study, we adopt an anisotropic eikonal equation to model Rayleigh wave phase traveltime in weak anisotropic media with topographic variation. An inversion scheme is developed to invert for both shear wave velocity (Vs) and anisotropy using adjoint‐state method. This tomography method is applied in the Huidong region, located near the southwestern segment of the Lianhuashan Fault Zone, China. Rayleigh wave phase delays caused by the uneven distribution of ambient noise sources are observed. This effect is corrected through traveltime correction which is determined by inverting for the azimuthal amplitude density of ambient noise. The tomographic results reveal low‐Vs anomalies and ENE‐oriented fast directions that are consistent with the strike of the Danshui Fault. In addition, a hidden fault is inferred from the NW‐oriented fast directions and low‐Vs anomalies.

Neural shape representation generally refers to representing 3D geometry using neural networks, e.g., computing a signed distance or occupancy value at a specific spatial position. In this paper we present a neural-network architecture suitable for accurate encoding of 3D shapes in a single forward pass. Our architecture is based on a multi-scale hybrid system incorporating graph-based and voxel-based components, as well as a continuously differentiable decoder. The hybrid system includes a novel way of voxelizing point-based features in neural networks by projecting the point "feature-field" onto a grid. This projection is insensitive to local point density, and we show that it can be used to obtain smoother and more detailed reconstructions, in particular when combined with oriented point clouds as input. Our architecture also requires only a single forward pass, instead of the latent-code optimization used in auto-decoder methods. Furthermore, our network is trained to solve the well-established eikonal equation and only requires knowledge of the zero-level set for training and inference. We additionally propose a modification to the aforementioned loss function for the case that surface normals are not well defined, e.g., in the context of non-watertight surfaces and non-manifold geometry. Overall, our method consistently outperforms other baselines on the surface reconstruction task across a wide variety of datasets, while being more computationally efficient and requiring fewer parameters.

Distance functions are crucial in robotics for representing spatial relationships between a robot and its environment. They provide an implicit, continuous, and differentiable representation that integrates seamlessly with control, optimization, and learning. While standard distance fields rely on the Euclidean metric, many robotic tasks inherently involve non-Euclidean structures. To this end, we generalize Euclidean distance fields to more general metric spaces by solving the Riemannian eikonal equation, a first-order partial differential equation whose solution defines a distance field and its associated gradient flow on the manifold, enabling the computation of geodesics and globally length-minimizing paths. We demonstrate that geodesic distance fields —the classical Riemannian distance function represented as a global, continuous, and queryable field—are effective for a broad class of robotic problems where Riemannian geometry naturally arises. To realize this, we present a neural Riemannian eikonal solver (NES) that solves the equation as a mesh-free implicit representation without grid discretization, scaling to high-dimensional robot manipulators. Training leverages a physics-informed neural network (PINN) objective that constrains spatial derivatives via the PDE residual and boundary/metric conditions, so the model is supervised by the governing equation and requires no labeled distances or geodesics. We propose two NES variants, conditioned on boundary data and on spatially varying Riemannian metrics, underscoring the flexibility of the neural parameterization. We validate the effectiveness of our approach through extensive examples, yielding minimal-length geodesics across diverse robot tasks involving Riemannian geometry. Additionally, we validate the method in a dynamics-aware motion-planning task for energy-efficient trajectory generation, with comparisons to baseline approaches. Project website: https://sites.google.com/view/geodf .

Abstract We introduce a phase-consistent framework that bridges geometric and wave descriptions of light propagation in inhomogeneous media. By reconstructing the optical phase directly from Hamiltonian optics, we establish a first-principles connection between local refractive-index variations and the emergence of diffraction patterns. The recovered Hamiltonian phase is subsequently propagated using scalar diffraction within Fourier optics, providing a unified description of ray trajectories and interference within a single physical framework. We validate this approach experimentally using thermal-lens diffraction, a paradigmatic system in which continuously distributed refractive-index gradients generate complex interference patterns beyond the scope of thin-lens or ray-optical models. The reconstructed phase quantitatively reproduces the observed diffraction patterns across different transport regimes, demonstrating that diffraction arises directly from accumulated Hamiltonian phase. Beyond thermal lensing, the framework applies broadly to optical and other wave systems governed by an eikonal equation, establishing diffraction as a direct manifestation of Hamiltonian phase accumulation.

Weak solutions m\colon\Omega\subset\mathbb{R}^{2}\to\mathbb{R}^{2} of the eikonal equation, \begin{align*}|m|=1\text{ a.e.} \quad \text{and} \quad \mathrm{div} m =0,\end{align*} arise naturally as sharp interface limits of bounded energy configurations in various physically motivated models, including the Aviles–Giga energy. The distributions \mu_{\Phi}=\mathrm{div}\Phi(m) , defined for a class of smooth vector fields \Phi called entropies, carry information about singularities and energy cost. If these entropy productions are Radon measures, a long-standing conjecture predicts that they must be concentrated on the 1-rectifiable jump set of m –as they do if m has bounded variation (BV) thanks to the chain rule. We establish this concentration property, for a large class of entropies, under the Besov regularity assumption \begin{align*}m\in B^{1/p}_{p,\infty} \quad\Longleftrightarrow\quad\sup_{h\in \mathbb R^2\setminus\lbrace 0\rbrace} \frac{\|m(\cdot +h)-m\|_{L^p}}{|h|^{1/p}} <\infty,\end{align*} for any 1\leq p<3 , thus going well beyond the BV setting ( p=1 ) and leaving only the borderline case p=3 open.

On (discounted) global Eikonal equations in metric spaces

Abstract We propose a novel variational approach for computing neural Signed Distance Fields (SDF) from unoriented point clouds. To this end, we replace the commonly used eikonal equation with the heat method, carrying over to the neural domain what has long been standard practice for computing distances on discrete surfaces. This yields two convex optimisation problems for whose solution we employ neural networks: We first compute a neural approximation of the gradients of the unsigned distance field through a small time step of heat flow with weighted point cloud densities as initial data. Then, we use it to compute a neural approximation of the SDF. We prove that the underlying variational problems are well‐posed. Through numerical experiments, we demonstrate that our method provides state‐of‐the‐art surface reconstruction and consistent SDF gradients. Furthermore, we show in a proof‐of‐concept that it is accurate enough for solving a PDE on the zero‐level set.

The Eikonal equation is a fundamental partial differential equation in geophysics to describe seismic wavefront propagation. Accurate and efficient solutions to the Eikonal equation are crucial for various geophysical applications, including seismic migration and traveltime tomography. To address the computational bottlenecks in inversion workflows, an Eikonal solver is needed that can generalize across variations in the velocity model. In this work, we propose a generalized Eikonal solver using Fourier-DeepONet, a hybrid neural operator framework that combines the strengths of DeepONet and Fourier Neural Operator (FNO). By leveraging DeepONet for encoding velocity models and background traveltime fields, and FNO for decoding and predicting traveltimes, Fourier-DeepONet achieves superior accuracy and flexibility compared to existing methods. Once trained, the neural operator can rapidly predict traveltimes for new, unseen velocity models and source locations without retraining, offering a substantial computational advantage. We validate the performance of the proposed method on the OpenFWI velocity families, demonstrating its superior generalization capability compared to vanilla DeepONet. Our approach offers a scalable and efficient solution for real-time seismic modeling and inversion tasks, with potential to significantly reduce computational costs in large-scale geophysical applications.

ABSTRACT Traveltime computation is an effective way to simulate seismic wave propagation in isotropic and anisotropic media. It often requires the phase and group velocities along a ray direction. The phase and group velocities are not functions of the ray direction, but functions of the slowness direction, which motivates the need for efficient numerical approaches for computing the slowness direction. A generalized method was proposed to achieve this goal in 3D tilted transverse isotropic media, which involves solving a nonlinear equation of two unknowns, the inclination and azimuthal angles of the slowness direction, and is computationally demanding. To overcome the difficulty, the equation was recast by projecting it onto the local coordinates that align with the axis of symmetry of the medium. Under the local coordinate system, the nonlinear equation reduces to an equation of one unknown, the inclination angle, with the azimuthal angle obtained from the given ray direction. Hence, the nonlinear equation can be solved efficiently and its solutions can be used to calculate the phase and group velocities. As an application, the proposed method was used to compute traveltimes in 3D vertical transverse isotropic (VTI) media. The calculated group velocities are incorporated into the fast sweeping method to solve the original anisotropic Eikonal equation directly on uniform meshes. The feasibility of the proposed group velocity calculation method is evaluated in three 3D homogeneous anisotropic models, and the application on traveltime computation is tested on a 3D homogeneous VTI model and the British Petroleum anisotropic VTI model.

Purpose. To justify and quantitatively evaluate the characteristic dimensions of regions L within which the linearization of the eikonal equation has a correct solution in the context of kinematic ray tomography based on the principles of geometrical optics. Methodology. The authors use the theoretical foundations for determining the correctness of solving the seismic problem of the kinematic ray tomography method based on Taylor approximation combined with regularization (the Geyko’s method). To assess the characteristic dimensions of the regions, model seismic profiles, including PANCAKE, as well as global mantle tomography data, are applied. The analysis was carried out using models of the velocity structure of the Earth’s crust and mantle in the Carpathian region, taking into account the main tectonic units. Findings. The characteristic dimensions of the linearization region, which determine the resolution of the method, vary significantly ‒ from  0.2 km in the crust to ≈100 km in the mantle. It was determined that these sizes mainly depend on the geometry of seismic rays and the velocity structure of the medium. The main factor influencing the size of region L was found to be the size of the time spline window selected when forming one-dimensional travel-time curves in the common midpoint format. It was established that the errors of the kinematic tomography method are errors in estimating the depth of penetration of refracted rays. The paper shows how to calculate these errors and use them to assess the accuracy of the kinematic tomography method in the format of seismic velocities. Originality. Quantitative criteria are proposed for the applicability of the linearized approach to solving the eikonal equation in heterogeneous media within the framework of the kinematic method using Taylor approximation. For the first time, a survey design network for regional tomography by the kinematic method with the possibility of setting the resolution was calculated. The effect of errors on the method’s results is demonstrated. Practical value. The study results have important applied significance for optimizing the configurations of seismic observation networks. They allow for a more substantiated parameterization when constructing kinematic models, ensuring a balance between resolution and stability of solutions. The methodology makes it possible to design kinematic tomography surveys using natural sources of seismic waves with predictability of results at a level comparable to deep seismic sounding methods, despite the uneven distribution of sources and receivers.

This study aims to improve the numerical solution of the two-dimensional Eikonal equation, a fundamental partial differential equation in wave propagation and geometric optics, by creating highly efficient and precise recursive schemes. These schemes enhance computational performance, reduce numerical complexity, and increase solution accuracy for essential applications in seismic imaging, computer vision, and robotic path planning. We propose two innovative recursive schemes that enhance an advanced adaptation of the modified decomposition method. These schemes break down the nonlinear Eikonal equation into manageable components, enabling robust iterative solutions. The methodology is rigorously derived and confirmed through extensive numerical simulations across various examples, featuring multiple boundary conditions and domain complexities. Results indicate that the proposed schemes surpass existing methods in terms of accuracy, stability for complex geometries, and computational efficiency, as shown by faster convergence rates, making them suitable for large-scale challenges in applied mathematics and engineering. This work introduces two novel recursive schemes, specifically designed for the Eikonal equation, which extend the modified decomposition method with innovative adaptations. These schemes offer a unique blend of simplicity, scalability, and durability, addressing the limitations of traditional methods and providing novel numerical algorithms and insights that greatly enhance the computational toolkit for tackling nonlinear partial differential equations in scientific and industrial applications contexts.

The study presents the model of dynamic urban boundary evolution using the Eikonal equation arising in the wave Partial Differential Equation (PDE) using Huygens principle. It is shown how the mathematical model may be used in analyzing urban morphology through the formulation of wave equations that can be solved to represent the development of an urban area. The technique uses the urban boundary obtained through satellite data and estimates the parameters used in the ordinary differential equations (ODE) that arise when solving the eikonal PDE using the method of characteristics. The proposed mathematical framework is implemented for a small study area to demonstrate its functionality. The boundary of the study area for three years is extracted using satellite remote sensing data. The initial year boundary is discretized into fifty subdivisions. The Huygens principle is used to determine the parameters - the rate of change in magnitude and direction of urban boundaries at these selected boundary points. Once the parameters are estimated, they are related to various potential urban boundary expansion drivers using neural networks. The calibrated model shows a high degree of accuracy and may be used to further interpolate or extrapolate the boundary data.

A parallel multiscale FIM approach in solving the Eikonal equation on GPU

Fast Sweeping Method Based on Equivalent Slowness for Solving the Eikonal Equation

We propose an embedding method to solve the eikonal equation on implicit surfaces, where implicit surfaces are defined by signed distance functions. Building upon a recently established diffeomorphic embedding method for hyperbolic conservation laws, we introduce a novel anisotropic eikonal equation within a tubular neighborhood of an implicit surface to approximate the eikonal equation defined on the implicit surface. Our primary contribution lies in a formulation to ensure that the solution of the Hamilton-Jacobi equation of anisotropic eikonal type remains constant along the normal direction of the surface. To solve the resulting equation numerically, we employ the Lax- Friedrichs based high-order weighted essentially nonoscillatory fast sweeping method, and we further introduce two effective singularity factorization approaches to address the upwind singularity near the point source. While standard singularity factorization formulations apply only to point-source singularities, our new methods accommodate line-source singularities when the boundary condition is extended along the normal direction of the surface. These new factorization strategies enable us to compute uniformly high-order accurate solutions for the eikonal equation on an implicit surface. Furthermore, our method can be readily extended to handle anisotropic eikonal equations or general static Hamilton-Jacobi equations defined on implicit surfaces. Two- and three-dimensional examples demonstrate the accuracy and efficiency of our new embedding method.

In gas reservoirs, high gas velocity causes significant inertial effects, leading to a nonlinear relationship between pressure gradient and velocity, especially near wellbores or fractures. In such cases, Darcy’s law is inadequate, and the Forchheimer equation is commonly used to model nonlinear flow behavior. Although the Forchheimer equation improves simulation accuracy for high-velocity flow in porous media, incorporating it into conventional numerical simulations greatly increases computational time, as nonlinear flow equations must be solved over the entire reservoir. This difficulty is exacerbated in heterogeneous fractured reservoirs, where complex fracture–matrix interactions and localized high-velocity flow complicate solving nonlinear equations. To address this, this work proposes a fast numerical simulation method based on diffusive time of flight (DTOF). By using DTOF as a spatial coordinate, the original three-dimensional flow equations incorporating the Forchheimer equation are reduced to a one-dimensional form, enhancing computational efficiency. DTOF represents the diffusive time for a pressure disturbance from a well to reach a specific reservoir location and can be efficiently computed by solving the Eikonal equation via the fast marching method (FMM). Once the DTOF field is obtained, the three-dimensional problem is transformed into a one-dimensional problem. This dimensionality reduction enables fast and reliable modeling of nonlinear high-velocity gas transport in complex reservoirs. The proposed method’s results show good agreement with those from COMSOL Multiphysics, confirming its accuracy in capturing nonlinear gas flow behavior.

Central-differencing quasi-Newton method for solving the Eikonal equation with application to wall distance computation

Abstract We develop a novel seismic attenuation tomography method that is formulated under a matrix‐free framework and avoids ray tracing. The key ingredient is the use of the adjoint‐state technique to compute the gradient of the misfit function, defined as the difference between observed and synthetic , to obtain an optimal attenuation model. Two advancements have been achieved over previous ray‐based methods. First, is accurately computed using a grid‐based method to solve the Eikonal equation, followed by solving the governing equation. Second, the inversion mainly requires solving the forward and adjoint equations, with a cost roughly twice that of forward modeling alone. The proposed method is benchmarked by imaging the structure of the Hikurangi subduction zone beneath northern New Zealand. The amplitude spectra of P‐waves from local earthquakes are inverted to estimate at 1 Hz, along with the seismic moment and corner frequency of each event. A total of measurements from 6,478 events at 44 stations are used. The new model shows high in the subducted, cold Pacific plate, adjacent to the low mantle wedge, matching previous results. Our findings reveal contrasting high and low , at depths of km beneath the backarc, may indicating the presence of a lithosphere‐asthenosphere boundary (LAB). The low beneath the backarc originates from depths of 150–200 km to the LAB, ascending along a nonvertical path toward the region beneath the arc, which may result from the dehydration of the deeply subducted slab.

Abstract We study the electric Helmholtz equation $$\Delta u + Vu + \lambda u =f$$ Δ u + V u + λ u = f and show that, for certain potentials, the solution u given by the limited absorption principle obeys a Sommerfeld radiation condition. We use a non-spherical approach based on the solution K of the eikonal equation $$|\nabla K|^2=1 + \frac{p}{\lambda }$$ | ∇ K | 2 = 1 + p λ to improve previous results in that area and extend them to long-range potentials which decay like $$|x|^{-2-\alpha }$$ | x | - 2 - α at infinity, with $$\alpha > 0$$ α > 0 .