Natural projected flow: a PDE solver using neural networks

Solving PDEs with neural networks has attracted a lot of attentions in recent years especially with the introduction of Physics-Informed Neural Networks (PINNs). These methods typically utilize neural networks as approximate solutions and adjust their parameters to satisfy the PDE (approximately). Our method, called Natural Projected Flow, deviates from this approach by utilizing a semi-discrete formulation. This involves seeking a solution where the parameters of the neural network (representing the spatial variable) evolve over time. The crucial challenge lies in identifying the corresponding evolution equation for these parameters. Natural Projected Flow addresses this challenge by employing a L^2 projection of the flow of the PDE onto the manifold of neural networks. The effectiveness of our proposed numerical solver is demonstrated through applications to various classical PDEs, including diffusion and porous-media equations.