TSONN: Time-stepping-oriented neural network for solving partial differential equations

Understand

Deep neural networks (DNNs), especially physics-informed neural networks (PINNs), have recently become a new popular method for solving forward and inverse problems governed by partial differential equations (PDEs).

• However, these methods still face challenges in achieving stable training and obtaining correct results in many problems, since minimizing PDE residuals with PDE-based soft constraint make the problem ill-conditioned.
• Different from all existing methods that directly minimize PDE residuals, this work integrates time-stepping method with deep learning, and transforms the original ill-conditioned optimization problem into a series of well-conditioned sub-problems over given pseudo time intervals.
• The convergence of model training is significantly improved by following the trajectory of the pseudo time-stepping process, yielding a robust optimization-based PDE solver.