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We consider the use of Gaussian Processes (GPs) or Neural Networks (NNs) to numerically approximate the solutions to nonlinear partial differential equations (PDEs) with rough forcing or source terms, which commonly arise as pathwise…
In recent years, deep learning-based methods have been proposed for solving inverse scattering problems (ISPs), but most of them heavily rely on data and suffer from limited generalization capabilities. In this paper, a new solving scheme…
The computational complexity of classical numerical methods for solving Partial Differential Equations (PDE) scales significantly as the resolution increases. As an important example, climate predictions require fine spatio-temporal…
The use of neural networks to solve differential equations, as an alternative to traditional numerical solvers, has increased recently. However, error bounds for the obtained solutions have only been developed for certain equations. In this…
Partial differential equations (PDEs) are widely used to describe relevant phenomena in dynamical systems. In real-world applications, we commonly need to combine formal PDE models with (potentially noisy) observations. This is especially…
DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with…
Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential…
In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…
Physics-informed neural networks (PINN) have recently become attractive for solving partial differential equations (PDEs) that describe physics laws. By including PDE-based loss functions, physics laws such as mass balance are enforced…
In this paper, local H\"older regularization is incorporated into a physics-informed neural networks (PINNs) framework for solving elliptic partial differential equations (PDEs). Motivated by the interior regularity properties of linear…
Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability.…
Recently, the advent of deep learning has spurred interest in the development of physics-informed neural networks (PINN) for efficiently solving partial differential equations (PDEs), particularly in a parametric setting. Among all…
It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that…
Several recent works in scientific machine learning have revived interest in the application of neural networks to partial differential equations (PDEs). A popular approach is to aggregate the residual form of the governing PDE and its…
This paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear…
We propose a data-driven method for recovering miss-ing parts of 3D shapes. Our method is based on a new deep learning architecture consisting of two sub-networks: a global structure inference network and a local geometry refinement…
Solving inverse problems in dynamical systems governed by high-dimensional coupled ordinary differential equations (ODEs) is a ubiquitous challenge in scientific machine learning. In many real-world applications, researchers seek to uncover…
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…
The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging…
This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear…