Related papers: Regularity Analysis and Tensor Neural Network Meth…
Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such…
Fast and accurate solutions of time-dependent partial differential equations (PDEs) are of pivotal interest to many research fields, including physics, engineering, and biology. Generally, implicit/semi-implicit schemes are preferred over…
Quantum neural networks represent a new machine learning paradigm that has recently attracted much attention due to its potential promise. Under certain conditions, these models approximate the distribution of their dataset with a truncated…
We develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
In recent years, new regularization methods based on (deep) neural networks have shown very promising empirical performance for the numerical solution of ill-posed problems, e.g., in medical imaging and imaging science. Due to the…
It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
We apply the deep learning neural network architecture to the two-level system in quantum optics to solve the time-dependent Schrodinger equation. By carefully designing the network structure and tuning parameters, above 90 percent accuracy…
We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent…
The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to H\"{o}rmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We…
We consider the problem of solving mixed random linear equations with $k$ components. This is the noiseless setting of mixed linear regression. The goal is to estimate multiple linear models from mixed samples in the case where the labels…
This paper explores the integration of Diophantine equations into neural network (NN) architectures to improve model interpretability, stability, and efficiency. By encoding and decoding neural network parameters as integer solutions to…
Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first…
This article describes a method for constructing approximations to periodic solutions of dynamic Lorenz system with classical values of the system parameters. The author obtained a system of nonlinear algebraic equations in general form…
Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which…
The transferable neural network (TransNet) is a two-layer shallow neural network with pre-determined and uniformly distributed neurons in the hidden layer, and the least-squares solvers can be particularly used to compute the parameters of…
We study iterative finite element approximations for the numerical approximation of semilinear elliptic boundary value problems with monotone nonlinear reactions of subcritical growth. The focus of our contribution is on an optimal a priori…
This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated…
In this work, we describe a new approach that uses deep neural networks (DNN) to obtain regularization parameters for solving inverse problems. We consider a supervised learning approach, where a network is trained to approximate the…