Related papers: A Priori Estimation of the Approximation, Optimiza…

A significant advancement in Neural Network (NN) research is the integration of domain-specific knowledge through custom loss functions. This approach addresses a crucial challenge: how can models utilize physics or mathematical principles…

Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…

Bayesian Neural Networks (BNN) have recently emerged in the Deep Learning world for dealing with uncertainty estimation in classification tasks, and are used in many application domains such as astrophysics, autonomous driving...BNN assume…

Physics-informed neural networks (PINNs) have shown promising potential for solving partial differential equations (PDEs) using deep learning. However, PINNs face training difficulties for evolutionary PDEs, particularly for dynamical…

Machine learning has been successfully applied to various fields of scientific computing in recent years. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs)…

Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at…

Recently proposed numerical algorithms for solving high-dimensional nonlinear partial differential equations (PDEs) based on neural networks have shown their remarkable performance. We review some of them and study their convergence…

Random ordinary differential equations (RODEs), i.e. ODEs with random parameters, are often used to model complex dynamics. Most existing methods to identify unknown governing RODEs from observed data often rely on strong prior knowledge.…

Neural networks with binary weights are computation-efficient and hardware-friendly, but their training is challenging because it involves a discrete optimization problem. Surprisingly, ignoring the discrete nature of the problem and using…

This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams…

Recent development of Deep Reinforcement Learning (DRL) has demonstrated superior performance of neural networks in solving challenging problems with large or even continuous state spaces. One specific approach is to deploy neural networks…

Deep neural networks are powerful tools for approximating functions, and they are applied to successfully solve various problems in many fields. In this paper, we propose a neural network-based numerical method to solve partial differential…

Compared with conventional numerical approaches to solving partial differential equations (PDEs), physics-informed neural networks (PINN) have manifested the capability to save development effort and computational cost, especially in…

Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations~(PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized…

In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…

Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we…

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on…

Deep Neural Networks have achieved remarkable success relying on the developing availability of GPUs and large-scale datasets with increasing network depth and width. However, due to the expensive computation and intensive memory,…

Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations (PDEs). In the latter the variational formulation of the Poisson problem is used in order to obtain an…