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

Researchers have proposed new methods to solve partial differential equations (PDEs) using deep learning techniques, achieving significant improvements in accuracy and computational efficiency.

Two new methods have been introduced to tackle the challenges of solving PDEs. The first, called Two-Stage Multi-Grade Deep Learning (TS-MGDL), uses a two-stage process where shallow networks are trained progressively and then refined ^[1]. TS-MGDL has been applied to the viscous Burgers' equation, a nonlinear PDE with steep gradients and shock-like solutions, and has shown a significant reduction in predictive errors by up to a factor of 60 compared to single-grade learning (SGL). Meanwhile, a second method, Physics-Informed Broad Learning System (PIBLS), reformulates PDE solving as a direct least-squares optimization, achieving one to three orders of magnitude faster computation than conventional Physics-Informed Neural Networks (PINNs) ^[2]. PIBLS also achieves higher solution accuracy than PINNs and provides a computationally efficient paradigm for scientific machine learning, making it suitable for real-time simulation and design optimization tasks.