Potential research questions

Questions related to hyperbolic PDEs

• Do PINNs tend to yield strong solutions when there exist both strong and weak solutions?
• How can we encourage PINNs to yield entropy-satisfying weak solutions?
• Are recurrent NNs or something similar useful for incorporating causality?

Questions related to structure preservation

• What are effective ways to enforce conservation or dissipation of a scalar quantity (e.g. energy or entropy)?
• Does enforcing this kind of structure result in better or worse solutions of the problem?

Literature

• Shock capturing PINNs for hyperbolic problems by under-resolution near shocks

• Biasing PINNs towards entropy and TVD solutions

• Attention-based PINNs for hyperbolic problems

• Learning operators from/to general function spaces(DeepONets)

• Understanding and mitigating gradient flow pathologies in PINNs

• NH-PINN: Neural homogenization-based physics-informed neural network for multiscale problems

Using conserved quantities in training

• Modified physics-informed neural network method based on the conservation law constraint and its prediction of optical solitons

• A Modified Physics Informed Neural Networks for Solving the Partial Differential Equation with Conservation Laws