Yi Bing

Yi Bing, Liu Jia, Fu Jinyang et al.

Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training strategies, which tightly couple learning with specific governing equations and limit generalization across problem settings. In this work, we propose a diffusion model with physics-guided inference for solving PDEs, in which the diffusion model is trained using standard data-driven procedures, while physical laws are incorporated exclusively during the reverse inference stage. The reverse diffusion dynamics is guided by a PDE residual energy function, combined with Gaussian smoothing and explicit boundary enforcement, yielding a physically consistent stochastic iteration that is independent of the training process. From a numerical standpoint, the proposed framework can be interpreted as a diffusion-inspired implicit solver that converges to the PDE solution even when initialized from random noise and perturbed by stochastic fluctuations. The method is validated on classical PDE equation such as Poisson, Diffusion, and Burgers equations with varying coefficients. Numerical results demonstrate robust convergence, high accuracy, and strong generalization without retraining, highlighting the proposed framework as a unified alternative to classical numerical solvers and physics-informed neural networks.

Yi Bing, Zheng Ran, Fu Jinyang et al.

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.