ANNs-Based Method for Solving Partial Differential Equations : A Survey

Conventionally, partial differential equations (PDE) problems are solved numerically through discretization process by using finite difference approximations. The algebraic systems generated by this process are then finalized by using an iterative method. Recently, scientists invented a short cut approach, without discretization process, to solve the PDE problems, namely by using machine learning (ML). This is potential to make scientific machine learning as a new sub-field of research. Thus, given the interest in developing ML for solving PDEs, it makes an abundance of an easy-to-use methods that allows researchers to quickly set up and solve problems. In this review paper, we discussed at least three methods for solving high dimensional of PDEs, namely PyDEns, NeuroDiffEq, and Nangs, which are all based on artificial neural networks (ANNs). ANN is one of the methods under ML which proven to be a universal estimator function. Comparison of numerical results presented in solving the classical PDEs such as heat, wave, and Poisson equations, to look at the accuracy and efficiency of the methods. The results showed that the NeuroDiffEq and Nangs algorithms performed better to solve higher dimensional of PDEs than the PyDEns.

[1]  Alaeddin Malek,et al.  Numerical solution for high order differential equations using a hybrid neural network - Optimization method , 2006, Appl. Math. Comput..

[2]  Aurélien Géron,et al.  Hands-On Machine Learning with Scikit-Learn and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems , 2017 .

[3]  William H. Press,et al.  Book-Review - Numerical Recipes in Pascal - the Art of Scientific Computing , 1989 .

[4]  Xiaoqun Cao,et al.  Solving Partial Differential Equations Using Deep Learning and Physical Constraints , 2020, Applied Sciences.

[5]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[6]  Bedy Purnama.  Pengantar Machine Learning: Konsep dan Praktikum Dengan Contoh Latihan Berbasis R dan Python , 2020 .

[7]  Eman A.Hussian,et al.  Numerical Solution of Partial Differential Equations by using Modified Artificial Neural Network , 2015 .

[8]  Marcel Bauer,et al.  Numerical Methods for Partial Differential Equations , 1994 .

[9]  Andrew J. Meade,et al.  Solution of nonlinear ordinary differential equations by feedforward neural networks , 1994 .

[10]  Justin A. Sirignano,et al.  DGM: A deep learning algorithm for solving partial differential equations , 2017, J. Comput. Phys..

[11]  R. Luebbers,et al.  The Finite Difference Time Domain Method for Electromagnetics , 1993 .

[12]  Dimitrios I. Fotiadis,et al.  Artificial neural networks for solving ordinary and partial differential equations , 1997, IEEE Trans. Neural Networks.

[13]  Zhiping Mao,et al.  DeepXDE: A Deep Learning Library for Solving Differential Equations , 2019, AAAI Spring Symposium: MLPS.

[14]  Philip Picton,et al.  Review of Artificial Neural Networks (ANN) applied to corrosion monitoring , 2012 .

[15]  Mohsen Hayati,et al.  Feedforward Neural Network for Solving Partial Differential Equations , 2007 .

[16]  Mark S. Gockenbach,et al.  Partial Differential Equations - Analytical and Numerical Methods (2. ed.) , 2011 .

[17]  Yoshua Bengio,et al.  On the Spectral Bias of Neural Networks , 2018, ICML.

[18]  Hyuk Lee,et al.  Neural algorithm for solving differential equations , 1990 .

[19]  Lorenzo Rosasco,et al.  Why and when can deep-but not shallow-networks avoid the curse of dimensionality: A review , 2016, International Journal of Automation and Computing.

[20]  Ali Al-Aradi,et al.  Solving Nonlinear and High-Dimensional Partial Differential Equations via Deep Learning , 2018, 1811.08782.

[21]  Pavlos Protopapas,et al.  NeuroDiffEq: A Python package for solving differential equations with neural networks , 2020, J. Open Source Softw..

[22]  Alexander Koryagin,et al.  PyDEns: a Python Framework for Solving Differential Equations with Neural Networks , 2019, ArXiv.