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

TL;DR

This paper introduces a PDE energy driven iterative framework that efficiently and stably solves PDEs without traditional matrix discretizations or neural network training, demonstrating promising results across various equations.

Contribution

The work presents a novel physically constrained diffusion iteration method for PDEs that avoids classical discretizations and learning-based approaches, ensuring stability and accuracy.

Findings

Stable convergence to physical solutions from random initial fields.

Accurate resolution of sharp gradients and controlled MSE.

Competitive accuracy and stability compared to analytical solutions.

Abstract

Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require costly training and often suffer from limited generalization. In this work, we proposes a PDE energy driven framework that solves PDEs through physically constrained diffusion iterations, without relying on classical matrix based finite element assembly or data driven neural network training. The proposed method evolves arbitrary random initial fields through PDE energy driven implicit iterations combined with Gaussian smoothing, while strictly enforcing boundary conditions at each iteration. The proposed formulation is applied to representative one dimensional Poisson, Heat, and viscous Burgers equations, covering both steady state and transient problems. Numerical results demonstrate stable convergence to the unique physical solution from random initializations, with accurate resolution of sharp gradients and controlled Mean Squared Error (MSE) across a wide range of discretization parameters. Detailed comparisons with analytical solutions indicate that the framework achieves competitive accuracy and stability. Overall, the proposed framework provides a fast, flexible, and physically consistent alternative to traditional numerical solvers, offering a potential pathway for scalable PDE solutions in both research and engineering applications.