To extend the IRK-PINN algorithm for handling divergent trajectories, particularly in systems governed by the Coulomb potential, several strategies can be employed. One approach is to incorporate regularization techniques that penalize the divergence of trajectories during the training process. This could involve adding a term to the loss function that quantifies the divergence of predicted trajectories from physically viable solutions, thereby guiding the neural network to focus on stable paths.
Another strategy is to implement a multi-fidelity training approach, where the model is trained on a combination of high-fidelity data (from accurate numerical simulations) and low-fidelity data (from less accurate models). This can help the IRK-PINN learn the underlying dynamics more robustly, especially in regions of phase space where trajectories are prone to divergence.
Additionally, adaptive sampling techniques can be utilized to focus training on regions of phase space that exhibit complex behavior, such as near singularities or points of high curvature in the potential landscape. By dynamically adjusting the training data based on the model's performance, the IRK-PINN can better capture the nuances of the system's dynamics.
Finally, exploring alternative neural network architectures, such as recurrent neural networks (RNNs) or attention mechanisms, may provide the flexibility needed to model the intricate relationships between phase-space variables, thereby improving the handling of divergent trajectories.