Wang, Y.-R.; Chen, G.-W. Predicting Multiple Numerical Solutions to the Duffing Equation Using Machine Learning. Appl. Sci.2023, 13, 10359.
Wang, Y.-R.; Chen, G.-W. Predicting Multiple Numerical Solutions to the Duffing Equation Using Machine Learning. Appl. Sci. 2023, 13, 10359.
Wang, Y.-R.; Chen, G.-W. Predicting Multiple Numerical Solutions to the Duffing Equation Using Machine Learning. Appl. Sci.2023, 13, 10359.
Wang, Y.-R.; Chen, G.-W. Predicting Multiple Numerical Solutions to the Duffing Equation Using Machine Learning. Appl. Sci. 2023, 13, 10359.
Abstract
This study uses machine learning to predict the convergence results of the Duffing equation with and without damping. The Duffing equation represents a nonlinear second-order differential equation with interesting behavior in undamped free vibration and forced vibration with damping. Convergence alternates randomly between 1 and -1 in undamped free vibration, depending on initial conditions. For forced vibration with damping, multiple factors influence vibration patterns. We utilize the fourth-order Runge-Kutta method to collect convergence results for both conditions. Machine learning techniques, specifically the long short term memory (LSTM) and LSTM-Neural Network (LSTM-NN) method, are employed to predict these convergence values. The LSTM-NN model is a hybrid approach that combines the LSTM method with the addition of hidden layers of neurons. Both the LSTM and LSTM-NN models are thoroughly explored and analyzed in this research. The research process involves three stages: data preprocessing, training, and verification. The results show that the LSTM-NN model becomes more adept at predicting binary datasets, boasting an impressive accuracy of up to 98%. However, when it comes to predicting multiple solutions, the traditional LSTM method outperforms the LSTM-NN approach.
Keywords
Duffing equation; deep learning; neural networks; recurrent neural networks; long short term memory.
Subject
Engineering, Mechanical Engineering
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
