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

2026-04-30 · Source: cs.LG updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Engineering & Applied Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

This work introduces a novel "PDE energy driven framework" for efficiently and stably solving partial differential equations, addressing the limitations of traditional "matrix based discretizations" and "learning based methods" that require costly training. The proposed iterative method evolves arbitrary random initial fields through "physically constrained diffusion iterations" combined with "Gaussian smoothing", strictly enforcing boundary conditions without relying on "finite element assembly" or "neural network training". Applied to one-dimensional Poisson, Heat, and viscous Burgers equations, the framework demonstrates stable convergence, accurate resolution of sharp gradients, and controlled Mean Squared Error (MSE). Numerical results confirm competitive accuracy and stability compared to analytical solutions. This framework offers a fast, flexible, and physically consistent alternative for scalable PDE solutions in scientific and engineering applications.

Key takeaway

A novel PDE energy-driven iterative framework efficiently solves PDEs without relying on matrix-based discretization or neural network training. It achieves stable convergence and accurate resolution of sharp gradients by evolving random initial fields through physically constrained diffusion iterations with Gaussian smoothing. This offers a fast, flexible, and physically consistent alternative for researchers and engineers seeking scalable PDE solutions in scientific and engineering applications.

Topics

• PDE Solutions
• Iterative Framework
• PDE Energy
• Gaussian Smoothing
• Boundary Conditions

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.LG updates on arXiv.org.