Inverse problems describe the process of estimating the causal factors from a
set of measurements or data. Mapping of often incomplete or degraded data to
parameters is ill-posed, thus data-driven iterative solutions are required, for
example when reconstructing clean images from poor signals. Diffusion models
have shown promise as potent generative tools for solving inverse problems due
to their superior reconstruction quality and their compatibility with iterative
solvers. However, most existing approaches are limited to linear inverse
problems represented as Stochastic Differential Equations (SDEs). This
simplification falls short of addressing the challenging nature of real-world
problems, leading to amplified cumulative errors and biases. We provide an
explanation for this gap through the lens of measure-preserving dynamics of
Random Dynamical Systems (RDS) with which we analyse Temporal Distribution
Discrepancy and thus introduce a theoretical framework based on RDS for SDE
diffusion models. We uncover several strategies that inherently enhance the
stability and generalizability of diffusion models for inverse problems and
introduce a novel score-based diffusion framework, the \textbf{D}ynamics-aware
S\textbf{D}E \textbf{D}iffusion \textbf{G}enerative \textbf{M}odel (D$^3$GM).
The \textit{Measure-preserving property} can return the degraded measurement to
the original state despite complex degradation with the RDS concept of
\textit{stability}. Our extensive experimental results corroborate the
effectiveness of D$^3$GM across multiple benchmarks including a prominent
application for inverse problems, magnetic resonance imaging. Code and data
will be publicly available.

逆问题与扩散模型、保度量动力学、随机动力系统和动力学感知的扩散生成模型 (D3GM) 有关，并揭示了以上几个策略的稳定性和广泛适用性，以及 D3GM 在多个基准测试中的有效性。