相关论文: Learning the Helmholtz equation operator with Deep…
We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the subdomain problems satisfy first-order absorbing (impedance) transmission conditions, and exchange of information between subdomains…
Deep learning methods have proven capable of recovering operators between high-dimensional spaces, such as solution maps of PDEs and similar objects in mathematical physics, from very few training samples. This phenomenon of data-efficiency…
The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast…
Operator learning provides methods to approximate mappings between infinite-dimensional function spaces. Deep operator networks (DeepONets) are a notable architecture in this field. Recently, an extension of DeepONet based on model…
We consider the numerical solution of high-frequency scattering problems modeled by the Helmholtz equation with a bounded obstacle. Although the analysis of this problem dates back at least 50 years, over the past decade or so, tools and…
In this paper, we propose a deep learning-enhanced multigrid solver for high-frequency and heterogeneous Helmholtz equations. By applying spectral analysis, we categorize the iteration error into characteristic and non-characteristic…
Neural operators are neural network-based surrogate models for approximating solution operators of parametric partial differential equations, enabling efficient many-query computations in science and engineering. Many applications,…
We present a deep learning-based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning,…
In this work, we propose a method to learn the solution operators of PDEs defined on varying domains via MIONet, and theoretically justify this method. We first extend the approximation theory of MIONet to further deal with metric spaces,…
In this paper, a two-dimensional Dirichlet-to-Neumann (DtN) finite element method (FEM) is developed to analyze the scattering of SH guided waves due to an interface delamination in a bi-material plate. During the finite element analysis,…
Burn injuries present a significant global health challenge. Among the most severe long-term consequences are contractures, which can lead to functional impairments and disfigurement. Understanding and predicting the evolution of post-burn…
Nonlinearity parameter tomography leads to the problem of identifying a coefficient in a nonlinear wave equation (such as the Westervelt equation) modeling ultrasound propagation. In this paper we transfer this into frequency domain, where…
In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as…
Recently, deep neural networks (DNNs) have become powerful tools for solving inverse scattering problems. However, the approximation and generalization rates of DNNs for solving these problems remain largely under-explored. In this work, we…
The accurate estimation of the state of complex uncertain physical systems requires reconciling theoretical models, with inherent imperfections, with noisy experimental data. In this work, we propose an effective hybrid approach that…
We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial…
Poroelasticity -- coupled fluid flow and elastic deformation in porous media -- often involves spatially variable permeability, especially in subsurface systems. In such cases, simulations with random permeability fields are widely used for…
Deep operator networks (DeepONet) and neural operators have gained significant attention for their ability to map infinite-dimensional function spaces and perform zero-shot super-resolution. However, these models often require large…
Deep Operator Networks (DeepONets) are among the most prominent frameworks for operator learning, grounded in the universal approximation theorem for operators. However, training DeepONets typically requires significant computational…
Multi-task learning (MTL) is an inductive transfer mechanism designed to leverage useful information from multiple tasks to improve generalization performance compared to single-task learning. It has been extensively explored in traditional…