Related papers: Solving Partial Differential Equations with Random…

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem…

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…

In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs). The main idea is to use a neural network to learn the…

The physics informed neural network (PINN) is a promising method for solving time-evolution partial differential equations (PDEs). However, the standard PINN method may fail to solve the PDEs with strongly nonlinear characteristics or those…

Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity…

In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies…

Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural…

In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained increasing popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning…

We propose several approaches for solving differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are…

Partial differential equations (PDEs) form the backbone of simulations of many natural phenomena, for example in climate modeling, material science, and even financial markets. The application of physics-informed neural networks to…

Neural networks can be trained to solve partial differential equations (PDEs) by using the PDE residual as the loss function. This strategy is called "physics-informed neural networks" (PINNs), but it currently cannot produce high-accuracy…

This paper focuses on solving large-scale, ill-conditioned, and overdetermined sparse least squares problems that arise from numerical partial differential equations (PDEs), mainly from the random feature method. To address these…

Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error,…

In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…

Physics-informed neural networks (PINNs) have recently emerged as a prominent paradigm for solving partial differential equations (PDEs), yet their training strategies remain underexplored. While hard prioritization methods inspired by…

Physics-informed deep learning has emerged as a promising framework for solving partial differential equations (PDEs). Nevertheless, training these models on complex problems remains challenging, often leading to limited accuracy and…

Solving partial differential equations (PDEs) with machine learning has recently attracted great attention, as PDEs are fundamental tools for modeling real-world systems that range from fundamental physical science to advanced engineering…

Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from…

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…

Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient…