DiffNO: Neural Operator Learning Using Physically Structured Constrained Diffusion Model

doi:10.1007/978-3-031-99857-7_15

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DiffNO: Neural Operator Learning Using Physically Structured Constrained Diffusion Model

2026-01-01

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Zhichen Feng *

Xin Zhang

DOI：10.1007/978-3-031-99857-7_15

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Abstract

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We propose DiffNO, a novel framework that synergizes diffusion models with kernel-integrated neural operators to solve nonlinear partial differential equations (PDEs). Compared to deterministic architectures, diffusion models have a better chance of learning the complex mappings in the evolution of nonlinear PDEs due to the introduction of stochastic variables. However, it is challenging for diffusion models to directly learn complex nonlinear mappings on their own. To address these limitations, DiffNO constructs a kernel-integrated diffusion operator, incorporating the structure of Green's functions as prior knowledge into the diffusion drift term, thereby establishing a physically constrained stochastic evolution process. On a dataset containing various types of PDEs, especially including two highly nonlinear PDEs, the 2D Cahn-Hilliard and reaction-diffusion systems, our method significantly outperforms current state-of-the-art methods.

Keywords:

Neural operators

Diffusion Model

Green's Function

Data Analysis of Scientific Computing