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Artificial neural networks for solving ordinary and partial differential equations.
I E Lagaris1, A Likas, D I Fotiadis
1Department of Computer Science, University of Ioannina, GR 45110 Ioannina, Greece.
IEEE Transactions on Neural Networks
|February 8, 2008
Summary
This study introduces a novel artificial neural network (ANN) method for solving differential equations. The approach ensures initial/boundary conditions are met, offering efficient solutions for ordinary and partial differential equations.
Area of Science:
- Computational Mathematics
- Artificial Intelligence
- Numerical Analysis
Background:
- Solving differential equations is crucial in science and engineering.
- Existing numerical methods can be computationally intensive.
- Artificial neural networks offer a potential alternative for complex problems.
Purpose of the Study:
- To develop a novel artificial neural network (ANN) based method for solving initial and boundary value problems.
- To ensure that the proposed ANN method inherently satisfies initial and boundary conditions.
- To demonstrate the method's applicability across various types of differential equations.
Main Methods:
- A trial solution is constructed as a sum of two parts: one satisfying boundary conditions and another utilizing a feedforward neural network.
- The neural network is trained to satisfy the differential equation while preserving the boundary conditions.
- The method is applied to ordinary differential equations (ODEs) and partial differential equations (PDEs).
Main Results:
- The ANN method successfully solves various model differential equations.
- Comparisons with the Galerkin finite element method show comparable or improved results for PDEs.
- The approach is versatile, applicable to single ODEs, coupled ODE systems, and PDEs.
Conclusions:
- The proposed ANN method provides an effective way to solve differential equations with enforced boundary conditions.
- The method shows promise for significant speed gains with specialized hardware like neuroprocessors.
- This technique offers a powerful new tool for computational mathematics and scientific simulations.
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