Super-resolution for evolutionary probability density of stochastic dynamic responses using physics-guided diffusion model

Abstract

The evolutionary probability density (EPD) is of great value in explaining and controlling the dynamic behavior of stochastic systems. However, for real complex systems, solving the generalized probability density evolution equation (GDEE) is a formidable task. In recent years, the development of physical-informed neural networks (PINNs) has provided new insights for solving the GDEE. Nevertheless, PINNs require random variables to follow a deterministic probability distribution; otherwise, the model must be retrained if the distribution changes. Diffusion models, known for their ability to effectively capture complex data distributions and their strong generalization capabilities, have been widely applied to image super-resolution tasks. Inspired by this, a physics-guided diffusion model is proposed in this work, where the GDEE is embedded in the proposed model as the prior physical knowledge. This model can reconstruct high-fidelity EPD of stochastic dynamic responses from low-fidelity EPD input, which is obtained using fewer data points through any distribution fitting method, thereby reducing computational resource consumption. Specifically, when the probability distribution of the random parameters of the system changes, the model parameters do not need to be retrained. Finally, three analytical models with exact solutions and a real-world engineering case are selected to validate the accuracy and effectiveness of the proposed method.