Three ways to solve partial differential equations with neural networks—A review
J Blechschmidt, OG Ernst - GAMM‐Mitteilungen, 2021 - Wiley Online Library
Neural networks are increasingly used to construct numerical solution methods for partial
differential equations. In this expository review, we introduce and contrast three important …
differential equations. In this expository review, we introduce and contrast three important …
An overview on deep learning-based approximation methods for partial differential equations
It is one of the most challenging problems in applied mathematics to approximatively solve
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
high-dimensional partial differential equations (PDEs). Recently, several deep learning …
On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs
Physics informed neural networks (PINNs) are deep learning based techniques for solving
partial differential equations (PDEs) encounted in computational science and engineering …
partial differential equations (PDEs) encounted in computational science and engineering …
A physics-guided neural network framework for elastic plates: Comparison of governing equations-based and energy-based approaches
One of the obstacles hindering the scaling-up of the initial successes of machine learning in
practical engineering applications is the dependence of the accuracy on the size and quality …
practical engineering applications is the dependence of the accuracy on the size and quality …
Deepreach: A deep learning approach to high-dimensional reachability
Hamilton-Jacobi (HJ) reachability analysis is an important formal verification method for
guaranteeing performance and safety properties of dynamical control systems. Its …
guaranteeing performance and safety properties of dynamical control systems. Its …
Sliding-mode surface-based approximate optimal control for nonlinear multiplayer Stackelberg-Nash games via adaptive dynamic programming
This paper studies the sliding-mode surface (SMS)-based approximate optimal control issue
for a class of nonlinear multiplayer Stackelberg-Nash games (MSNGs). First, considering …
for a class of nonlinear multiplayer Stackelberg-Nash games (MSNGs). First, considering …
NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators
Uncertainty quantification (UQ) in machine learning is currently drawing increasing research
interest, driven by the rapid deployment of deep neural networks across different fields, such …
interest, driven by the rapid deployment of deep neural networks across different fields, such …
[HTML][HTML] Hutchinson trace estimation for high-dimensional and high-order physics-informed neural networks
Abstract Physics-Informed Neural Networks (PINNs) have proven effective in solving partial
differential equations (PDEs), especially when some data are available by seamlessly …
differential equations (PDEs), especially when some data are available by seamlessly …
Optimally weighted loss functions for solving pdes with neural networks
Recent works have shown that deep neural networks can be employed to solve partial
differential equations, giving rise to the framework of physics informed neural networks …
differential equations, giving rise to the framework of physics informed neural networks …
Adaptive deep learning for high-dimensional Hamilton--Jacobi--Bellman equations
Computing optimal feedback controls for nonlinear systems generally requires solving
Hamilton--Jacobi--Bellman (HJB) equations, which are notoriously difficult when the state …
Hamilton--Jacobi--Bellman (HJB) equations, which are notoriously difficult when the state …