A Randomized PDE Energy driven Iterative Framework for Efficient and Stable PDE Solutions

This work introduces a PDE energy-driven iterative framework for solving partial differential equations efficiently and stably, without relying on traditional matrix-based discretizations or costly data-driven neural network training. It evolves random initial fields through physically constrained diffusion iterations and Gaussian smoothing, strictly enforcing boundary conditions, and demonstrates stable convergence on Poisson, Heat, and viscous Burgers equations.