Research proposes PDE energy-driven framework for solving partial differential equations through physically constrained diffusion iterations without matrix-based discretization or neural network training

A framework solves PDEs (partial differential equations — mathematical descriptions of how physical systems change) by evolving random initial fields through energy-driven implicit iterations combined with Gaussian smoothing, while enforcing boundary conditions at each iteration.

The method was applied to one-dimensional Poisson, Heat, and viscous Burgers equations covering both steady-state and transient problems, and demonstrated stable convergence to unique physical solutions from random initializations with controlled Mean Squared Error (MSE) across discretization parameters.

The framework offers a fast, flexible, and physically consistent alternative to traditional numerical solvers, providing a potential pathway for scalable PDE solutions in research and engineering applications.