Hoe would this work?  I  imagine input neurons would be initial conditions and output neurons would be solution values at time T.  Somehow the ode needs to be part of the input also.  Do you have a links to the paper?

Hi Ronobir,

Very interesting question! You may find some papers by following the link below:

I hope it helps (if it did, please upvote!)

Best regards!

http://arxiv.org/pdf/physics/9705023.pdf

http://eprints.usq.edu.au/2781/1/Mai-Duy_Tran-Cong_Author's_version.pdf

http://ac.els-cdn.com/0895717794900957/1-s2.0-0895717794900957-main.pdf?_tid=3a21d1d8-de43-11e5-b388-00000aacb360&acdnat=1456681782_5cbe4b01df7fd60d9c3ce0f21a3041d1

http://ce.sharif.ir/courses/84-85/2/ce667/resources/root/Seminar_no_9/00167018.pdf

@Erick_Arguello,

I have already searched such papers. My question is comparatively less papers has been published in this area. So is it possible to phd in  this area? I mean has neural network perform good in solving differential equation?

The papers which I read , they all considered some types of function approximation by choosing a function with some parameters which is contrary to fdm and fem. So will this types of work have application in reality? 

@Erick_Arguello,

All papers your mentionted are published on 90' years, when started the "boom" of applications of  the recently invented  ANN technique. I think that for solving of ODE or PDE there are more powerful and successful numerical method than ANN, such FEM or FVM method.     

I realize this is a very old question, but since this is one of the google results on this topic, I'd like to also point out that there is some literature (also from the 90s, but resuming in 2018 or 2019) on learning not the solution to a given PDE or ODE, but learning the PDE or ODE itself. See work from 89 through 97 by Kevrekidis, Rico-Martinez, Farber, and others.

For example, see [1], which compares several methods for ODEs, but see in particular section III3 and Figure 9.

For PDEs, see [2], in particular Figure 3, which you may recognize as a PDE version of the four-state Runge-Kutta Figure 9 of [1].

As Diana Maria Thomas said, the approach of this family of papers is indeed to establish the learning problem as predicting the short-time flow of the ODE, and finding the ODE that does that. But the ODE is not part of the input--it is the thing which is learned. The fact that the composite neural network can predict the state at time T is not the point; the useful artifact of the training is the sub-network which evaluates the ODE or PDE.

[1] Rico-Martínez, Ramiro, Katharina Krischer, Ioannis G Kevrekidis, M. C. Kube, and J. L. Hudson. “Discrete- vs Continuous-Time Nonlinear Signal Processing of Cu Electrodissolution Data.” Chemical Engineering Communications 118, no. 1 (1992): 25–48. https://doi.org/10.1080/00986449208936084.

[2] González-García, R., Ramiro Rico-Martínez, and Ioannis G Kevrekidis. “Identification of Distributed Parameter Systems: A Neural Net Based Approach.” Computers & Chemical Engineering 22, no. 98 (March 1998): S965–68. https://doi.org/10.1016/S0098-1354(98)00191-4.

dear Tom S. Bertalan ,

Currently I am also working on the same topic, but I couldn't find the begin of problem, I gone through many books, neither of book is quite clear, even authors are not responding when i post a question, do you have any implementation of solution of differential/ Integral equation for which is solved in artificial neural networks.

recently I have gone through a article , in which to find solution of differential equation author replaced hidden layer as functional link layer and solved the problem. moreover they mentioned as they solved the solution in unsupervised learning manner. which is a big wonder for me. If you have any related information about this can you please kindly shared to me. hope so you may respond to this.