ODE-DPS: ODE-based Diffusion Posterior Sampling for Inverse Problems in Partial Differential Equation

TL;DR

ODE-DPS introduces a Bayesian, score-based diffusion framework for PDE inverse problems, casting posterior inference as conditional generation via reverse-time dynamics and deriving an efficient ODE-based posterior sampler. By training a time-conditioned score model on partial prior data and approximating the likelihood, the method constructs an adaptive reverse process that yields samples from the posterior without requiring paired data for each new problem. An adaptive residual norm further improves boundary accuracy, and a U-Net score network enables scalable, task-agnostic inversion. Empirical results on inverse heat and wave problems show substantial gains in inversion accuracy and robustness over traditional regularization methods, indicating strong potential for practical PDE inverse applications. The approach reduces the need for retraining across problem settings, offering improved efficiency and interpretability in Bayesian PDE inversion.

Abstract

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.

Figures (10)

• Figure 1: The inversion results of the problem (\ref{['heq1']}). (a) denotes the ground truth of source $f$. (b), (c) respectively provide the inversion results by Landweber iteration regularization and Tikhonov regularization method. (d), (e) respectively denotes the difference between true $f$ and the inversion source by Landweber iteration regularization and Tikhonov regularization method. The relative $l_2$ errors of the true and inversion source obtained by Landweber iteration and Tikhonov regularization are 34.4% and 37.9% respectively. Here, we set the regularization parameter $\alpha=0.005$ for Tikhonov regularization.
• Figure 2: The inversion results of the problem (\ref{['heq1']}). (a) denotes the ground truth of source $f$. (b) denotes the inversion results by Diffusion Posterior Sampling algorithm \ref{['al11']}. (c) denotes the difference between true $f$ and the inversion source. The relative $l_2$ error of the true and inversion source is 18.4%. In algorithm \ref{['al11']}, we set $N=1000$.
• Figure 3: The Simplified U-Net model structure.
• Figure 4: The inversion results of source $f_1$. (a): the true $f_1$ and measurement data $u^{\delta}(x,y,T;f_1)$. (b)(c)(d)(e): the inversion $f_1$ and errors using different regularization methods. (f): the errors between the true $f_1$ and the inversion $f_{1,rec}$. (g): the errors between the measurement data $u^{\delta}(x,y,T;f_1)$ and $u(x,y,T;f_{1,rec})$. Here, we set the noise level $\varepsilon=0.05$ in (\ref{['5.1']}), the stopping parameter $\tau=1.01$ in discrepancy principle, the regularization parameter $\alpha=0.005$ in Tikhonov regularization.
• Figure 5: The inversion results of source $f_2$. (a): the true $f_2$ and measurement data $u^{\delta}(x,y,T;f_2)$. (b)(c)(d)(e): the inversion $f_2$ and errors using different regularization methods. (f): the errors between the true $f_2$ and the inversion $f_{2,rec}$. (g): the errors between the measurement data $u^{\delta}(x,y,T;f_2)$ and $u(x,y,T;f_{2,rec})$. Here, we set the noise level $\varepsilon=0.05$ in (\ref{['5.1']}), the stopping parameter $\tau=1.01$ in discrepancy principle, the regularization parameter $\alpha=0.005$ in Tikhonov regularization.

Theorems & Definitions (1)

• Remark 1