Solving the initial value problem of ordinary differential equations by Lie group based neural network method.

Area of Science:

• Applied Mathematics
• Computational Science
• Artificial Intelligence

Background:

• Solving initial value problems (IVPs) for ordinary differential equations (ODEs) is crucial in many scientific fields.
• Traditional numerical methods and existing artificial neural network (ANN) approaches have limitations in terms of accuracy and computational efficiency.
• Integrating Lie group theory with ANNs offers a potential avenue for improved ODE solving.

Purpose of the Study:

• To propose an alternative ANN approach for solving ODE IVPs by integrating feedforward neural networks (FNNs) with Lie group theory.
• To reduce the number of trainable parameters in the ANN model.
• To enhance the speed and accuracy of learning the true solution of ODEs.

Main Methods:

• The proposed method splits the trial solution of ODEs into two parts, utilizing Lie group expressions.
• The first part is solved using Lie group theory and established numerical/symbolic methods, requiring no network parameters.
• The second part employs an FNN with adjustable parameters, trained via error backpropagation to minimize a loss function.

Main Results:

• The combined FNN and Lie group approach significantly reduces the number of trainable parameters compared to existing methods.
• The method demonstrates faster and more accurate learning of the ODE solutions.
• Numerical application to physical oscillation problems shows promising results, supported by graphical representations.

Conclusions:

• The integration of Lie group theory with FNNs provides an effective and efficient method for solving ODE IVPs.
• This approach offers a significant improvement over existing ANN-based methods for differential equations.
• The reduced parameter count and enhanced accuracy suggest broad applicability in scientific and engineering domains.