PDE Energy Driven Framework Solves Equations Without Training

A new framework solves partial differential equations (PDEs) through physically constrained diffusion iterations, eliminating the need for matrix-based discretization or neural network training. The method evolves random initial fields using PDE energy driven implicit iterations with Gaussian smoothing, strictly enforcing boundary conditions at each step. It was tested on one-dimensional Poisson, Heat, and viscous Burgers equations for steady-state and transient problems. Numerical results confirm stable convergence. The approach addresses efficiency and generalization issues in traditional solvers and learning-based methods.