The Diffusion Manifold Method for probabilistic inverse problems

This paper proposes the Diffusion Manifold Method for solving probabilistic inverse problems without requiring prior distributions. The approach enables the identification of probability structures in random sources directly from response measurements of stochastic dynamical systems. By leveraging a known system surrogate model, the method transfers the probability density of random sources to an embedded trivial Riemannian submanifold in the response space, establishing a manifold version of the principle of preservation of probability. We develop a convex optimization framework based on random source space partitioning that directly identifies the joint probability density function (PDF) of dependent random sources through least-squares formulation. Numerical verification demonstrates that the proposed method effectively captures fine geometric features of the joint PDF while maintaining robustness against noisy measurement data, and is capable of solving ill-posed inverse problems.