VW-PINNs: A volume weighting method for PDE residuals in physics-informed neural networks

Understand

Physics-informed neural networks (PINNs) have shown remarkable prospects in the solving the forward and inverse problems involving partial differential equations (PDEs).

• The method embeds PDEs into the neural network by calculating PDE loss at a series of collocation points, providing advantages such as meshfree and more convenient adaptive sampling.
• However, when solving PDEs using nonuniform collocation points, PINNs still face challenge regarding inefficient convergence of PDE residuals or even failure.
• In this work, we first analyze the ill-conditioning of the PDE loss in PINNs under nonuniform collocation points.