Neuro-Spectral Architectures for Causal Physics-Informed Networks

A new framework for solving nonlinear partial differential equations (PDEs) has been proposed, integrating perturbation theory with one-shot transfer learning in Physics-Informed Neural Networks (PINNs). This method decomposes nonlinear PDEs into linear subproblems, which are solved using a Multi-Head PINN. The approach allows for quick adaptation to new PDE instances with varying conditions, achieving errors around 1e-3 and adaptation times under 0.2 seconds, demonstrating comparable accuracy to classical solvers.

This study introduces a data augmentation strategy designed to enhance the training of neural networks, thereby improving prediction accuracy. By generating synthetic data using a compartmental model and incorporating uncertainty, the model is calibrated with available data and integrated with deep learning techniques. The findings indicate that neural networks trained on these augmented datasets demonstrate significantly better predictive performance, particularly with Physics-Informed Neural Networks (PINNs) and Nonlinear Autoregressive (NAR) models.