Piecewise Neural Network Model to Obtain Large Interval Solution to the Lorenz System

This study evaluates the accuracy and efficiency of Piecewise Neural Networks (PWNNs) for solving the Lorenz system over extended time intervals, comparing them to Physics-Informed Neural Networks (PINNs). Although PINNs have shown promise in modeling complex systems by incorporating physical laws into the learning process, they often struggle with error propagation over long durations. The PWNN approach addresses this limitation by dividing the time domain into smaller sub-intervals, training individual neural networks for each segment, and then seamlessly combining the solutions. The results indicate that PWNN outperforms PINN in various performance metrics, yielding significantly lower root mean squared error (RMSE) values. This improvement is particularly notable in later time intervals, where PINNs show higher cumulative error. In the final sub-intervals, farthest from the initial condition, PWNNs achieved up to an 80% reduction in RMSE for solution variables, with overall reductions of 70% for x(t), 60% for y(t), and 75% for z(t) across the entire domain compared to PINNs. The findings highlight the efficiency and accuracy of PWNNs over PINNs, providing precise solutions without additional training iterations or data, offering a targeted approach to address long-domain solutions of the Lorenz system.