Evaluating single multiplicative neuron models in physics-informed neural networks for differential equations.

Area of Science:

• Computational Mathematics
• Artificial Intelligence
• Neural Network Architectures

Background:

• Artificial neural networks (ANNs) are powerful tools for estimation and classification.
• Physics-informed neural networks (PINNs) effectively solve differential equations by integrating boundary conditions into the loss function.
• Determining optimal ANN architecture (neurons, layers) remains a challenge.

Purpose of the Study:

• To investigate the application of the Single Multiplicative Neuron Model (SMNM) within the PINNs framework.
• To address the convergence issues encountered with the conventional SMNM in PINNs.
• To introduce and evaluate a modified 'mimic single multiplicative neuron model' (mimic-SMNM) for enhanced computational efficiency and convergence.

Main Methods:

• Implementation of the SMNM within the PINNs framework for a specific differential equation.
• Development of the mimic-SMNM architecture to retain conceptual advantages while ensuring convergence.
• Comparative analysis of real PINNs, the conventional SMNM, and the mimic-SMNM.

Main Results:

• The conventional SMNM failed to converge when applied to the differential equation.
• The real PINNs successfully solved the equation.
• The mimic-SMNM demonstrated architectural simplicity, computational feasibility, and convergence.
• The mimic-SMNM achieved a five-fold increase in computational speed compared to real PINNs after 30,000 epochs.

Conclusions:

• The conventional SMNM is not suitable for the PINNs framework due to convergence issues.
• The proposed mimic-SMNM provides an efficient and computationally feasible alternative for solving differential equations within the PINNs context.
• The mimic-SMNM offers significant speed advantages over standard PINNs for specific applications.