Helmholtz Equation Research Articles (Page 1)

Abstract Physics-informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws within neural architectures. However, these classical approaches often require a large number of parameters to achieve reasonable accuracy, particularly for complex PDEs. In this paper, we present a quantum-classical physics-informed neural network (QCPINN) that combines quantum and classical components, allowing us to solve PDEs with significantly fewer parameters while maintaining comparable accuracy and convergence to classical PINNs. We systematically evaluated two quantum circuit architectures across various configurations on five benchmark PDEs to identify optimal QCPINN designs. Our results demonstrate that the QCPINN achieves stable convergence and comparable accuracy while using only 10-30% of the trainable parameters required by classical PINNs. This approach also results in a significant reduction in the relative L2 error for Helmholtz, Klein-Gordon, and Convection-diffusion equations, with a reduction ranging from 4% to 64% across various fields. These findings demonstrate the potential of parameter efficiency and solution accuracy in physics-informed machine learning, allowing for a substantial decrease in model complexity without compromising solution quality. QCPINN presents a promising pathway to address the computational challenges associated with solving PDEs.

In this study, we consider new optical soliton solutions of one of the most important non-integrable model arising in optical fibres, namely Nonlinear Helmholtz equations (NHEs) that describes transverse interactions, transmission of coupled waves and optical solitons’ propagation in the field of fiber optics. We apply an adapted method to obtain some novel plethora of optical quasi-periodic soliton solutions. These solutions are presented in the shape of exponential, hyperbolic, trigonometric and rational functions. A set of 3D visualization, contour plots and 2D curves of these solutions physical relevance are presented with implications for the nonlinear optics. These figures also reveal that the established optical solitons exhibit quasi-periodicity due to the combination of linear periodic and axial perturbations, and that the presence of quasi-periodical perturbations of the solitons leads to the formation of the fractal-like structures. We also study the chaotic/periodic and bifurcation behavior, associated with the model, in the light of Hamiltonian analysis, as a consequence, we find positive results of the quasi-periodicity and fractal-like structures in the systems under consideration. Apart from offering novel analytical perspectives for dealing with the coupled NHEs, the present results would also be a concrete contribution to the understanding the soliton wave dynamics in complicated nonlinear media.

Passive Inverse Obstacle Scattering Problems for the Helmholtz Equation

Two-Level Hybrid Schwarz Preconditioners for the Helmholtz Equation with High Wave Number

Agent-Physics-Informed Neural Network solving frequency-domain Helmholtz equation related forward and inverse problems

Accurate and efficient numerical methods are vital for solving the wave equation in applications such as acoustics, electromagnetics, and seismology. However, large-scale and strongly heterogeneous models give rise to prohibitive computational costs because of large dimensions of the discrete wave equation systems. We develop a multiscale finite-difference method to solve the frequency-domain counterpart of the wave equation in elastic media, that is the elastic wave Helmholtz equation. This multiscale method reduces the dimension of the discrete wave equation system that is assembled using a conventional single-scale finite-difference frequency-domain method. The model reduction is achieved by multiscale basis functions constructed from local elastic Helmholtz problems, and can incorporate the medium heterogeneity at the fine scale into the dimension-reduced discrete wave equation system at the coarse scale. To capture the medium property variations at the fine scale accurately, we employ a multi-node coarse element scheme, which involves more than four coarse nodes in a coarse element, to replace the four-node coarse element designed for the first-order multiscale basis functions. The proposed method contributes to a significant computational cost reduction of solving the elastic wave Helmholtz equation without reducing its accuracy. Numerical tests show that our method can implement finite-difference frequency-domain elastic wave modeling with higher efficiency and lower memory cost than the conventional finite-difference frequency-domain methods.

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 .

We consider a bounded open subset Ω of R n of class C 1 , α for some α ∈ ] 0 , 1 [ , and we define a distributional outward unit normal derivative for α-Hölder continuous solutions of the Helmholtz equation in the exterior of Ω that may not have a classical outward unit normal derivative at the boundary points of Ω and that may have an infinite Dirichlet integral around the boundary of Ω. Namely for solutions that do not belong to the classical variational setting. Then we show a Schauder boundary regularity result for α-Hölder continuous functions that have the Laplace operator in a Schauder space of negative exponent and we prove a uniqueness theorem for α-Hölder continuous solutions of the exterior Dirichlet and impedance boundary value problems for the Helmholtz equation that satisfy the Sommerfeld radiation condition at infinity in the above mentioned nonvariational setting.

Interpreting material anisotropy through the fractional wave equation.

A study of Cauchy problem of the Helmholtz equation based on a relaxation model: Regularization and analysis

Introduction. In recent years, the field of mathematics specializing in the application of artificial neural networks has been rapidly developing. In this work, a new method for constructing a neural network for solving wave differential equations is proposed. This method is particularly effective in solving boundary value problems for domains of complex geometric shapes.Materials and Methods. A method is proposed for constructing a neural network designed to solve the wave equation in a planar domain G bounded by an arbitrary closed curve. It is assumed that the boundary conditions are periodic functions of time t, and the steady-state regime is considered. When constructing the neural network, the activation functions are taken as derivatives of singular solutions of the Helmholtz equation. The singular points of these solutions are uniformly distributed along closed curves surrounding the domain boundary. The training set consists of a set of particular solutions of the Helmholtz equation.Results. Results were obtained for the solution of the first boundary value problem in various domains of complex geometric shape and under different boundary conditions. The results are presented in tables containing both the exact solutions of the problem and the solutions obtained using the neural network. A graphical comparison is also provided between the exact solution and the solution obtained with the constructed neural network.Discussion. The presented computational results demonstrate the efficiency of the proposed method for constructing neural networks that solve boundary value problems of partial differential equations in domains of complex geometry.Conclusion. The further development of the proposed method may be applied to solving boundary value problems for the wave equation in exterior domains. Of particular interest is the application of this method to diffraction problems.

A Final Value Problem for the Nonlinear Modified Helmholtz Equation Associated with the Nonlinear Wave Velocity

The phase-only generation of self-accelerating Weber beams.

This study explores the acoustic behavior of flexible cylindrical shells incorporating membrane discs at structural interfaces, focusing on their influence on wave propagation characteristics. The dynamics of the embedded membrane discs are modeled at the junctions between different shell segments, and the resulting boundary value problem is addressed using a combination of the Mode-Matching (MM) and Galerkin methods. The governing equations comprise the Helmholtz equation in the fluid domain and the Donnell–Mushtari shell equations in the elastic guiding regions. To ensure the accuracy and convergence of the semi-analytical solution, generalized orthogonality conditions are employed. A truncated modal expansion is used to reconstruct the matching conditions and enforce physical conservation laws at the interfaces. Numerical simulations are conducted to examine the effects of geometric parameters—such as the radii of adjacent shell segments, the size of the membrane discs, and the excitation frequency—providing valuable insights for the design and optimization of waveguide-based acoustic attenuation systems.

A sparse fundamental solution neural network (SFSNN) for solving the Helmholtz equation with constant coefficients and relatively large wave numbers $k$ is proposed. The method combines the strengths of fundamental solution techniques and neural networks by employing a radial basis function neural network, where fundamental solution functions serve as activation functions. Since these functions inherently satisfy the homogeneous Helmholtz equation, SFSNN only requires boundary sampling, significantly accelerating training. To enhance sparsity and generalization, an $ℓ_1$ regularization term of the weights is introduced into the loss function, reformulating the weight optimization as a least absolute shrinkage and selection operator (Lasso) problem. This not only reduces the number of basis functions but also improves the network’s generalization capability. Numerical experiments validate the method’s effectiveness for high-wavenumber isotropic Helmholtz equations in two dimensions and three dimensions. The results reveal that when the analytical solution is a linear combination of fundamental solutions, SFSNN accurately identifies their centers. Otherwise, the number of required basis functions scales as $N =\mathcal{O}(k^{(τ(d−1))}),$ where $τ < 1$ and $d$ is the problem dimension. Moreover, SFSNN has been successfully extended to non-homogeneous and semi-infinite Helmholtz equations, achieving high accuracy. Codes of the examples in this paper are available at https://github.com/wangzhiwensuda/SFSNN-Helmholtz-problem.

In this paper, the dual boundary integral formulation of the two-dimensional Helmholtz equation with complex wave number is derived. The presence of damping in the medium results in the Helmholtz equation incorporating complex wave numbers in mathematical models. To address the singular and hypersingular integrals, the addition theorem is used to expand the four kernel functions, originally expressed with complex variables in the dual formulation, into purely real-variable functions in a series form. Consequently, the singular and hypersingular integrals are successfully transformed into the summation of regular integrals in an infinite series through the proposed regularization technique. The regular integrals are then computed using the Gaussian quadrature rule. This paper examines the occurrence of eigenvalues in both interior and exterior Helmholtz problems to understand how damping influences resonances. To validate the proposed formulation, three cases with exact solutions are used as standard benchmarks to evaluate the convergence and accuracy of the developed dual boundary element method program. Finally, two more general cases with amoeba-shaped geometry, which lack an exact solution and pose challenges in obtaining a convergent solution due to their irregular shape, are considered to evaluate the applicability and effectiveness of the proposed formulation.

Solving partial differential equations (PDEs) is essential in scientific forecasting and fluid dynamics. Traditional approaches often incur expensive computational costs and tradeoffs in efficiency and accuracy. Recent deep neural networks have improved the accuracy but require high-quality training data. Physics-informed neural networks effectively integrate physical laws to reduce the data reliance in limited sample scenarios. A novel machine-learning framework, Chebyshev physics-informed Kolmogorov–Arnold network (ChebPIKAN), is proposed to integrate the robust architectures of KAN with physical constraints to enhance the calculation accuracy of PDEs for fluid mechanics. We study the fundamentals of KAN, take advantage of the orthogonality of Chebyshev polynomial basis functions in spline fitting, and integrate physics-informed loss functions that are tailored to specific PDEs in fluid dynamics, including Allen–Cahn equation, nonlinear Burgers equation, Helmholtz equations, Kovasznay flow, cylinder wake flow, and cavity flow. Extensive experiments demonstrate that the proposed ChebPIKAN model significantly outperforms the standard KAN architecture in solving various PDEs by effectively embedding essential physical information. These results indicate that augmenting KAN with physical constraints can alleviate the overfitting issues of KAN and improve the extrapolation performance. Consequently, this study highlights the potential of ChebPIKAN as a powerful tool in computational fluid dynamics and proposes a path toward fast and reliable predictions in fluid mechanics and beyond.

This paper presents relatively simple formulations of the problem of acoustic scattering by flooded and hollow elastic shells immersed in fluids, which can serve as a basis for efficient numerical models. The full rigorous formulation of the problem, which involves the Helmholtz equations for acoustic pressures in the fluids and the Navier equation for three-dimensional displacements in the elastic material, is reduced to a boundary value problem only for the Helmholtz equations with effective boundary conditions relating the boundary pressures and normal displacements on both sides of the shell. To that end, the thin elastic shell is regarded as a neighborhood of its midsurface, and the boundary values of the elastic quantities (displacements and stresses) are expressed via their expansions about the midsurface, considering the shell thickness as a small parameter. In this paper, the expansion is restricted to the first order. Despite relative simplicity, the first-order models can describe elastic effects rather well, which is demonstrated by the comparison with the exact solutions for the case of spherical elastic shells. In particular, the boundary element method numerical solutions reproduce the low-frequency resonant peaks and dips of the exact solutions.

Fourier beyond dispersion: Wavenumber explicit and precise accuracy of FDMs for the Helmholtz equation

The mathematical model of water infiltration in a furrow irrigation channel with an impermeable layer in homogeneous soil is formulated as a Boundary Value Problem (BVP) with the Modified Helmholtz Equation as the governing equation and mixed boundary conditions. The purpose of this study is to solve the infiltration problem using the Dual Reciprocity Boundary Element Method (DRBEM). The results show that the highest values of suction potential and water content are located beneath the permeable channel, while the lowest values are found at the soil surface outside the channel and beneath the impermeable layer. The values of suction potential and water content increase over time t and converge, indicating stability in the infiltration process. These findings align well with real-world scenarios, demonstrating that the developed mathematical model and its numerical solution using DRBEM accurately illustrate the time-dependent water infiltration process in impermeable furrow irrigation channels.