Reproduce “TSONN: Time-stepping-oriented neural network for solving partial differential equations” on OpenResearch

About this paper

Wenbo Cao, Weiwei Zhang

TSONN (Time-stepping-Oriented Neural Network) introduces a robust framework for solving partial differential equations by integrating classical pseudo time-stepping methods into the deep learning optimization process. This approach enhances training stability and yields accurate solutions for complex problems like high Reynolds number flows and time-dependent equations, where standard Physics-informed Neural Networks often struggle.

Key insights

• Integrating time-stepping methods provides a more stable optimization landscape, allowing the neural network to converge reliably even for problems where direct minimization of PDE residuals (as in PINNs) fails.
• Implicit time-stepping schemes can be implemented within a neural network optimization framework as straightforwardly as explicit schemes, bypassing the need for local linearization or solving large algebraic systems common in traditional numerical methods.
• Explicit TSONN can surprisingly converge with pseudo time steps orders of magnitude larger than traditional numerical methods, particularly with a small number of inner iterations, suggesting an intrinsic filtering or artificial dissipation effect from the neural network optimization process.

Summary by alphaXiv.