Context-aware diffusion models for solving partial differential equations

Solving partial differential equations (PDEs) enables computers to simulate and understand continuously evolving physical processes in nature, forming a critical foundation that bridges physical laws and intelligent computation. Recently, deep learning has emerged as a research hotspot for solving PDEs due to its powerful nonlinear representation capacity and efficient approximation capabilities. However, it still faces challenges such as poor generalization and unsatisfactory solution accuracy. To this end, this paper proposes a context-aware diffusion model-driven framework for solving differential equations. First, we establish an implicit denoising diffusion mechanism tailored for solving differential equations. Second, we integrate contextual information, which includes demonstration data (i.e., known condition-solution pairs) and the current problem conditions, into the denoising network through a cross-attention mechanism. This integration provides precise guidance for solution generation throughout the recursive denoising process. Experimental results are reported across three categories of differential equation problems: linear, nonlinear, and variational problems. Our method demonstrates competitive or superior solution accuracy compared with representative neural operator baselines, including FNO, DeepONet, ICON, OFormer, UNO, and GNO.