dorsal/arxiv
View SchemaArtificial Neural Networks for Solving Ordinary and Partial Differential Equations
| Authors | I. E. Lagaris, A. Likas, D. I. Fotiadis |
|---|---|
| Categories | |
| ArXiv ID | physics/9705023 |
| URL | https://arxiv.org/abs/physics/9705023 |
| DOI | 10.1109/72.712178 |
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 boundary (or initial) conditions and contains no adjustable parameters. The second part is constructed so as not to affect the boundary conditions. This part involves a feedforward neural network, containing adjustable parameters (the weights). Hence by construction the boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ODE's, to systems of coupled ODE's and also to PDE's. In this article we illustrate the method by solving a variety of model problems and present comparisons with finite elements for several cases of partial differential equations.
{
"annotation_id": "3bac4d9f-3b12-4289-88e9-1d6dd3e3d355",
"date_created": "2026-03-02T18:01:20.662000Z",
"date_modified": "2026-03-02T18:01:20.662000Z",
"file_hash": "33302e5ff7e73a3ed2401894014fdd3745ba8fbbab0776bab9bb52de3ebe1454",
"private": false,
"record": {
"abstract": "We present a method to solve initial and boundary value problems using\nartificial neural networks. A trial solution of the differential equation is\nwritten as a sum of two parts. The first part satisfies the boundary (or\ninitial) conditions and contains no adjustable parameters. The second part is\nconstructed so as not to affect the boundary conditions. This part involves a\nfeedforward neural network, containing adjustable parameters (the weights).\nHence by construction the boundary conditions are satisfied and the network is\ntrained to satisfy the differential equation. The applicability of this\napproach ranges from single ODE\u0027s, to systems of coupled ODE\u0027s and also to\nPDE\u0027s. In this article we illustrate the method by solving a variety of model\nproblems and present comparisons with finite elements for several cases of\npartial differential equations.",
"arxiv_id": "physics/9705023",
"authors": [
"I. E. Lagaris",
"A. Likas",
"D. I. Fotiadis"
],
"categories": [
"physics.comp-ph",
"comp-gas",
"nlin.CG",
"quant-ph"
],
"doi": "10.1109/72.712178",
"title": "Artificial Neural Networks for Solving Ordinary and Partial Differential Equations",
"url": "https://arxiv.org/abs/physics/9705023"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "d3b4708e-45c4-467f-b377-eaee809311e4",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}