ODE-DPS: ODE-based Diffusion Posterior Sampling for Inverse Problems in Partial Differential Equation
This addresses the efficiency and applicability limitations of deep learning-based inversion methods for PDE inverse problems, though it appears incremental as an adaptation of diffusion models to this specific domain.
The paper tackles the problem of solving inverse problems in partial differential equations (PDEs) without requiring paired data or retraining for different conditions, introducing an unsupervised method based on score-based generative diffusion models that achieves efficient and robust inversion results across various PDE experiments.
In recent years we have witnessed a growth in mathematics for deep learning, which has been used to solve inverse problems of partial differential equations (PDEs). However, most deep learning-based inversion methods either require paired data or necessitate retraining neural networks for modifications in the conditions of the inverse problem, significantly reducing the efficiency of inversion and limiting its applicability. To overcome this challenge, in this paper, leveraging the score-based generative diffusion model, we introduce a novel unsupervised inversion methodology tailored for solving inverse problems arising from PDEs. Our approach operates within the Bayesian inversion framework, treating the task of solving the posterior distribution as a conditional generation process achieved through solving a reverse-time stochastic differential equation. Furthermore, to enhance the accuracy of inversion results, we propose an ODE-based Diffusion Posterior Sampling inversion algorithm. The algorithm stems from the marginal probability density functions of two distinct forward generation processes that satisfy the same Fokker-Planck equation. Through a series of experiments involving various PDEs, we showcase the efficiency and robustness of our proposed method.