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We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving optimization problem on the finest finite element space is…
Deep unfolding methods---for example, the learned iterative shrinkage thresholding algorithm (LISTA)---design deep neural networks as learned variations of optimization methods. These networks have been shown to achieve faster convergence…
In this paper, we study adaptive neuron enhancement (ANE) method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The ANE method is a self-adaptive method generating a two-layer spline NN and a numerical…
This paper studies a low-communication algorithm for solving elliptic partial differential equations (PDE's) on high-performance machines, the nested iteration with range decomposition algorithm (NIRD). Previous work has shown that NIRD…
The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson-Neumann partial differential equations (PDEs) with Neumann boundary conditions.…
Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel alternative to calculating…
Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite…
By employing non-equispaced grid points near boundaries, boundary-optimized upwind finite-difference operators of orders up to nine are developed. The boundary closures are constructed within a diagonal-norm summation-by-parts (SBP)…
Automated mathematical reasoning is a challenging problem that requires an agent to learn algebraic patterns that contain long-range dependencies. Two particular tasks that test this type of reasoning are (1) mathematical equation…
We propose a randomized physics-informed neural network (PINN) or rPINN method for uncertainty quantification in inverse partial differential equation (PDE) problems with noisy data. This method is used to quantify uncertainty in the…
In many applications, we need algorithms which can align partially overlapping point sets and are invariant to the corresponding transformations. In this work, a method possessing such properties is realized by minimizing the objective of…
This paper describes the application of the method of probabilistic solutions (MPS) to numerically solve the Dirichlet generalized and classical harmonic problems for irregular n sided pyramidal domains. Here, generalized means that the…
The purpose of this article is to study extrapolation of solvability for boundary value problems of elliptic systems in divergence form on the upper half-space assuming De Giorgi type conditions. We develop a method allowing to treat each…
In this paper, we consider approximating the parameter-to-solution maps of parametric partial differential equations (PPDEs) using deep neural networks (DNNs). We propose an efficient approach combining reduced collocation methods (RCMs)…
Physics-informed neural networks offered an alternate way to solve several differential equations that govern complicated physics. However, their success in predicting the acoustic field is limited by the vanishing-gradient problem that…
Computing the regularized solution of Bayesian linear inverse problems as well as the corresponding regularization parameter is highly desirable in many applications. This paper proposes a novel iterative method, termed the Projected Newton…
This paper analyzes local convergence of the block Newton (BN) method introduced in [5, 6] for one-dimensional shallow neural network approximation to functions and diffusion-reaction problems. The BN method consists of the 2x2 block…
In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential…
We propose a method combining boundary integral equations and neural networks (BINet) to solve partial differential equations (PDEs) in both bounded and unbounded domains. Unlike existing solutions that directly operate over original PDEs,…
In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for…