Deep learning algorithms for solving differential equations: a survey

Physics-informed neural networks (PINNs) for fluid mechanics: a review

(2022) Shengze Cai et al.   ACTA MECHANICA SINICA

Three types of periodic solutions of new (3 + 1)‐dimensional Boiti–Leon–Manna–Pempinelli equation via bilinear neural network method

(2022) Ji‐Mei Qiao et al.   MATHEMATICAL METHODS IN THE APPLIED SCIENCES

Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations

(2022) Run-Fa Zhang et al.   NONLINEAR DYNAMICS

Convergence Rate Analysis for Deep Ritz Method

(2022) Chenguang Duan et al.   Communications in Computational Physics

A mesh-free method using piecewise deep neural network for elliptic interface problems

(2022) Cuiyu He et al.   JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

Numerical solving of the generalized Black-Scholes differential equation using Laguerre neural network

(2021) Yinghao Chen et al.   DIGITAL SIGNAL PROCESSING

Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

(2021) Yulei Liao & Pingbing Ming   Communications in Computational Physics

A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics

(2021) Ehsan Haghighat et al.   COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING

Solving the Kolmogorov PDE by Means of Deep Learning

(2021) Christian Beck et al.   JOURNAL OF SCIENTIFIC COMPUTING

Periodic-soliton and periodic-type solutions of the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation by using BNNM

(2021) Jiang-Long Shen et al.   NONLINEAR DYNAMICS

Three ways to solve partial differential equations with neural networks — A review

(2021) Jan Blechschmidt et al.   GAMM Mitteilungen

Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method

(2021) Run-Fa Zhang et al.   CHAOS SOLITONS & FRACTALS

A nonlocal physics-informed deep learning framework using the peridynamic differential operator

(2021) Ehsan Haghighat et al.   COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING

When and why PINNs fail to train: A neural tangent kernel perspective

(2021) Sifan Wang et al.   JOURNAL OF COMPUTATIONAL PHYSICS

DeLISA: Deep learning based iteration scheme approximation for solving PDEs

(2021) Ying Li et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Hybrid FEM-NN models: Combining artificial neural networks with the finite element method

(2021) Sebastian K. Mitusch et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Physics-informed machine learning for reduced-order modeling of nonlinear problems

(2021) Wenqian Chen et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Extreme theory of functional connections: A fast physics-informed neural network method for solving ordinary and partial differential equations

(2021) Enrico Schiassi et al.   NEUROCOMPUTING

Optimally weighted loss functions for solving PDEs with Neural Networks

(2021) Remco van der Meer et al.   JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations

(2020) Maziar Raissi et al.   SCIENCE

Data-driven deep learning of partial differential equations in modal space

(2020) Kailiang Wu et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Extraction of mechanical properties of materials through deep learning from instrumented indentation

(2020) Lu Lu et al.   PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA

Physics‐Informed Deep Neural Networks for Learning Parameters and Constitutive Relationships in Subsurface Flow Problems

(2020) A. M. Tartakovsky et al.   WATER RESOURCES RESEARCH

Solving parametric PDE problems with artificial neural networks

(2020) YUEHAW KHOO et al.   EUROPEAN JOURNAL OF APPLIED MATHEMATICS

Relu Deep Neural Networks and Linear Finite Elements

(2020) Juncai He sci   JOURNAL OF COMPUTATIONAL MATHEMATICS

Weak adversarial networks for high-dimensional partial differential equations

(2020) Yaohua Zang et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Novel design of Morlet wavelet neural network for solving second order Lane–Emden equation

(2020) Zulqurnain Sabir et al.   MATHEMATICS AND COMPUTERS IN SIMULATION

Deep learning-based method coupled with small sample learning for solving partial differential equations

(2020) Ying Li et al.   MULTIMEDIA TOOLS AND APPLICATIONS

A kernel least mean square algorithm for fuzzy differential equations and its application in earth’s energy balance model and climate

(2020) M. Pakdaman et al.   Alexandria Engineering Journal

Wavelet neural networks based solutions for elliptic partial differential equations with improved butterfly optimization algorithm training

(2020) Lee Sen Tan et al.   APPLIED SOFT COMPUTING

A derivative-free method for solving elliptic partial differential equations with deep neural networks

(2020) Jihun Han et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Int-Deep: A deep learning initialized iterative method for nonlinear problems

(2020) Jianguo Huang et al.   JOURNAL OF COMPUTATIONAL PHYSICS

The neural network collocation method for solving partial differential equations

(2020) Adam R. Brink et al.   NEURAL COMPUTING & APPLICATIONS

Automatically discriminating and localizing COVID-19 from community-acquired pneumonia on chest X-rays

(2020) Zheng Wang et al.   PATTERN RECOGNITION

SciANN: A Keras/TensorFlow wrapper for scientific computations and physics-informed deep learning using artificial neural networks

(2020) Ehsan Haghighat et al.   COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING

B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data

(2020) Liu Yang et al.   JOURNAL OF COMPUTATIONAL PHYSICS

hp-VPINNs: Variational physics-informed neural networks with domain decomposition

(2020) Ehsan Kharazmi et al.   COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING

NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations

(2020) Xiaowei Jin et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

(2019) Christian Beck et al.   JOURNAL OF NONLINEAR SCIENCE

Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation

(2019) Run-Fa Zhang et al.   NONLINEAR DYNAMICS

ConvPDE-UQ: Convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains

(2019) Nick Winovich et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

(2019) Dongkun Zhang et al.   JOURNAL OF COMPUTATIONAL PHYSICS

A deep learning solution approach for high-dimensional random differential equations

(2019) Mohammad Amin Nabian et al.   PROBABILISTIC ENGINEERING MECHANICS

Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

(2019) Vikas Dwivedi et al.   NEUROCOMPUTING

A single layer fractional orthogonal neural network for solving various types of Lane–Emden equation

(2019) A.H. Hadian-Rasanan et al.   NEW ASTRONOMY

Forecasting stock index with multi-objective optimization model based on optimized neural network architecture avoiding overfitting

(2018) Zhou Tao et al.   Computer Science and Information Systems

DGM: A deep learning algorithm for solving partial differential equations

(2018) Justin Sirignano et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Solving Partial Differential Equation Based on Bernstein Neural Network and Extreme Learning Machine Algorithm

(2018) Hongli Sun et al.   NEURAL PROCESSING LETTERS

A unified deep artificial neural network approach to partial differential equations in complex geometries

(2018) Jens Berg et al.   NEUROCOMPUTING

Solving high-dimensional partial differential equations using deep learning

(2018) Jiequn Han et al.   PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

(2018) Yunlei Yang et al.   Advances in Difference Equations

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

(2018) M. Raissi et al.   JOURNAL OF COMPUTATIONAL PHYSICS

Application of Legendre Neural Network for solving ordinary differential equations

(2016) Susmita Mall et al.   APPLIED SOFT COMPUTING

Artificial neural network approach for a class of fractional ordinary differential equation

(2016) Ahmad Jafarian et al.   NEURAL COMPUTING & APPLICATIONS

Single Layer Chebyshev Neural Network Model for Solving Elliptic Partial Differential Equations

(2016) Susmita Mall et al.   NEURAL PROCESSING LETTERS

Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation

(2015) Muhammad Asif Zahoor Raja et al.   APPLIED SOFT COMPUTING

A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks

(2015) Keith Rudd et al.   NEUROCOMPUTING

Numerical solution of nonlinear singular initial value problems of Emden–Fowler type using Chebyshev Neural Network method

(2015) Susmita Mall et al.   NEUROCOMPUTING

A Numerical Method for Solving Fractional Differential Equations by Using Neural Network

(2015) Haidong Qu et al.   Advances in Mathematical Physics

Chebyshev Neural Network based model for solving Lane–Emden type equations

(2014) Susmita Mall et al.   APPLIED MATHEMATICS AND COMPUTATION

Buckling analysis of a beam–column using multilayer perceptron neural network technique

(2013) Manoj Kumar et al.   JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS

A Constrained Backpropagation Approach for the Adaptive Solution of Partial Differential Equations

(2013) Keith Rudd et al.   IEEE Transactions on Neural Networks and Learning Systems

Integration modified wavelet neural networks for solving thin plate bending problem

(2012) Xuejuan Li et al.   APPLIED MATHEMATICAL MODELLING

Deep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups

(2012) Geoffrey Hinton et al.   IEEE SIGNAL PROCESSING MAGAZINE

Finite-Element Neural Network-Based Solving 3-D Differential Equations in MFL

(2012) Chao Xu et al.   IEEE TRANSACTIONS ON MAGNETICS

Automated design parameter selection for neural networks solving coupled partial differential equations with discontinuities

(2012) Kevin Stanley McFall   JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS

Numerical treatment for solving one-dimensional Bratu problem using neural networks

(2012) Muhammad Asif Zahoor Raja et al.   NEURAL COMPUTING & APPLICATIONS

Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey

(2011) Manoj Kumar et al.   COMPUTERS & MATHEMATICS WITH APPLICATIONS

Unsupervised kernel least mean square algorithm for solving ordinary differential equations

(2011) Hadi Sadoghi Yazdi et al.   NEUROCOMPUTING

Two coupled neural-networks-based solution of the Hamilton–Jacobi–Bellman equation

(2010) Najla Krichen Masmoudi et al.   APPLIED SOFT COMPUTING

Numerical solution of PDEs via integrated radial basis function networks with adaptive training algorithm

(2010) Hong Chen et al.   APPLIED SOFT COMPUTING

Artificial neural network approach for solving fuzzy differential equations

(2010) Sohrab Effati et al.   INFORMATION SCIENCES