Recent update in Physics-informed Neural Networks for Solving Evolutionary hyperbolic Partial Differential Equations

In the realm of solving hyperbolic partial differential equations (PDEs) through physics-informed Neural Networks (PINNs), my knowledge extends up to the most recent update. To my knowledge, the approach to addressing hyperbolic PDEs with PINNs involves the integration of governing equations, initial conditions, and boundary conditions into the loss function. This loss function plays a pivotal role in guiding the training process of the neural network, enabling it to approximate solutions to the PDE.

Notably, it's essential to consider various facets of neural network error, including optimization error, approximation error, and estimation error. The approximation error denotes the difference between the exact functional map and the neural network's mapping function on a given network architecture. Estimation error, on the other hand, emerges when the network is trained on a finite dataset to establish a mapping for the target domain. The generalization error encompasses both approximation and estimation errors, defining the accuracy of the neural network's predicted solution based on the provided dataset.

In the current landscape, considerable research efforts are dedicated to optimizing the coefficients associated with error terms. Additionally, there has been a development known as Conservative PINN (CPINN). However, for someone seeking to delve into this domain, it's understandable to feel confused about where to start. Seeking guidance from the community or experts in the field could be immensely beneficial in navigating the complexities and determining an effective starting point.