Improving Information Propagation of Physics-Informed Neural Networks with General Causality Principle

Physics-informed neural networks (PINNs) with significant fitting ability have been regarded as a potential method for solving PDEs that may prevail the numeric method. While it has been recently found that PINNs may fail to learn the knowledge of PDEs when system multiscale, nonlinearity and inhomogeneity grow. One failure mode is that the information of physical laws is wrongly propagated with loss pathology. Respecting the temporal evolution of system, training with time causality is one effective way to mitigate such pathology. However, the spatial propagation across the solution domain is not considered. To understand, improve and manipulate the information propagation along both time and space axis, this paper proposed the formulation of general causality, integrated the space causality into time causality. Numerical experiments of two typical PDEs problems shows that the general causality improves the prediction performance of PINNs. The impact of causality resolution is further analyzed, indicating that the general causality has potential to solve more complex system with multiscale.