IEEE Transactions on Neural Networks / 1998

Artificial neural networks for solving ordinary and partial differential equations

I.E. Lagaris, A. Likas, D.I. Fotiadis

Popular and Landmark PapersProbabilistic ML

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE's), to systems of coupled ODE's and also to partial differential equations (PDE's). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galekrkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.

2,170 citations0 influential

Full paper

Read the original paper

Open PDF Source page

Learning resources

arXiv PDFPDF arXiv abstract pagearXiv Google Scholar referencesGoogle Scholar Paper pageOpenAlex Papers with Code searchPapers with Code YouTube explanationsYouTube

Reading state

Discuss in ChatGPT

Uses your own ChatGPT account. The paper context is copied into a tutor prompt before ChatGPT opens.

Preview prompt

You are my AI/ML research paper instructor. I want to deeply understand the paper below.

First, teach it in layers:
1. One-paragraph intuition.
2. Problem statement and why it mattered.
3. Key method, architecture, or algorithm.
4. Important equations or mechanisms, explained intuitively.
5. Experiments and evidence.
6. Limitations, assumptions, and failure modes.
7. How this paper influenced later AI/ML/Deep Learning/GenAI work.
8. A 30-minute study plan with checkpoints.
9. Quiz me with 5 questions and wait for my answers.

When something is not available in the attached context, say what is missing and infer carefully.

### Paper attached as context
Title: Artificial neural networks for solving ordinary and partial differential equations
Authors: I.E. Lagaris, A. Likas, D.I. Fotiadis
Year: 1998
Venue: IEEE Transactions on Neural Networks
Categories: Popular and Landmark Papers, Probabilistic ML
Citations: 2,170
Paper URL: https://arxiv.org/abs/physics/9705023
Open PDF: https://arxiv.org/pdf/physics/9705023

Abstract:
We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE's), to systems of coupled ODE's and also to partial differential equations (PDE's). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galekrkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.

Learning resources:
- PDF: arXiv PDF (https://arxiv.org/pdf/physics/9705023)
- arXiv: arXiv abstract page (https://arxiv.org/abs/physics/9705023)
- Google Scholar: Google Scholar references (https://scholar.google.com/scholar?q=Artificial%20neural%20networks%20for%20solving%20ordinary%20and%20partial%20differential%20equations)
- OpenAlex: Paper page (https://doi.org/10.1109/72.712178)
- Papers with Code: Papers with Code search (https://paperswithcode.com/search?q=Artificial%20neural%20networks%20for%20solving%20ordinary%20and%20partial%20differential%20equations)
- YouTube: YouTube explanations (https://www.youtube.com/results?search_query=Artificial%20neural%20networks%20for%20solving%20ordinary%20and%20partial%20differential%20equations+paper+explained)