Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equation

The article presents a two-stage multi-grade deep learning (TS-MGDL) method designed to address optimization challenges in solving nonlinear partial differential equations (PDEs), specifically the viscous Burgers' equation. The approach involves training shallow networks progressively by freezing previously learned grades and refining them in a second stage, which enhances interpretability and stability while proving a monotonic reduction in loss. Numerical experiments indicate that TS-MGDL achieves predictive error reductions of up to 60 times compared to single-grade learning (SGL), highlighting its potential for improving the accuracy of neural network solutions for complex PDEs.