Related papers: Multiwavelet-based Operator Learning for Different…
This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations…
Nonlocal operators with integral kernels have become a popular tool for designing solution maps between function spaces, due to their efficiency in representing long-range dependence and the attractive feature of being resolution-invariant.…
We establish a general form of explicit, input-dependent, measure-valued warpings for learning nonstationary kernels. While stationary kernels are ubiquitous and simple to use, they struggle to adapt to functions that vary in smoothness…
Performing analog computations with metastructures is an emerging wave-based paradigm for solving mathematical problems. For such devices, one major challenge is their reconfigurability, especially without the need for a priori mathematical…
Pairwise learning usually refers to a learning task which involves a loss function depending on pairs of examples, among which most notable ones include ranking, metric learning and AUC maximization. In this paper, we study an online…
We study the problem of structured output learning from a regression perspective. We first provide a general formulation of the kernel dependency estimation (KDE) problem using operator-valued kernels. We show that some of the existing…
We propose a partially learned approach for the solution of ill posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularization theory and recent advances in deep learning to…
This paper developed a systematic strategy establishing RBF on the wavelet analysis, which includes continuous and discrete RBF orthonormal wavelet transforms respectively in terms of singular fundamental solutions and nonsingular general…
Computing many eigenpairs of the Schr{\"o}dinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a…
In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [10] for convection-diffusion equations, which relies on a…
When neural networks (NNs) are used as a type of nonlinear parametric representation to solve partial differential equations (PDEs), they often display frequency-dependent learning dynamics that can differ from those seen in direct function…
Maxwell's equations, a system of linear partial differential equations (PDEs), describe the behavior of electric and magnetic fields in time and space and are essential for many important electromagnetic applications. Although numerical…
We consider the problem of operator-valued kernel learning and investigate the possibility of going beyond the well-known separable kernels. Borrowing tools and concepts from the field of quantum computing, such as partial trace and…
We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where…
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, we evaluate the effectiveness of deep operator networks (DeepONets) in solving both forward and inverse problems of partial differential equations (PDEs) on unknown manifolds. By unknown manifolds, we identify the manifold by…
The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…
A novel method for learning optimal, orthonormal wavelet bases for representing 1- and 2D signals, based on parallels between the wavelet transform and fully connected artificial neural networks, is described. The structural similarities…
In the design of engineered components, rigorous vibration testing is essential for performance validation and identification of resonant frequencies and amplitudes encountered during operation. Performing this evaluation numerically via…
We present a data-driven control framework for partial differential equations (PDEs). Our approach integrates Time-Integrated Deep Operator Networks (TI-DeepONets) as differentiable PDE surrogate models within the Differentiable Predictive…