Decoupled Diffusion Solver for Inverse Problems on Function Spaces

Thomas Y.L. Lin, Jiachen Yao, Lufang Chiang, Julius Berner, Anima Anandkumar

Keywords: Diffusion Models, Posterior Sampling, Inverse Problems, Data Assimilation, Operator Learning, PDE, AI for Science

TL;DR: DDIS avoids guidance attenuation failure of joint modeling by decoupling diffusion prior learning from physics modeling, achieving SotA performance with superior data efficiency.

Abstract:

We propose a data-efficient, physics-aware generative framework in function space for inverse PDE problems.
Existing plug-and-play diffusion posterior samplers represent physics implicitly through joint coefficient-solution modeling, requiring substantial paired supervision.
In contrast, our Decoupled Diffusion Inverse Solver (DDIS) employs a decoupled design: an unconditional diffusion learns the coefficient prior, while a neural operator explicitly models the forward PDE for guidance.
This decoupling enables superior data efficiency and effective physics-informed learning, while naturally supporting Decoupled Annealing Posterior Sampling (DAPS) to avoid over-smoothing in Diffusion Posterior Sampling (DPS).
Theoretically, we prove that DDIS avoids the guidance attenuation failure of joint models when training data is scarce.
Empirically, DDIS achieves state-of-the-art performance under sparse observation, improving $l_2$ error by 11\% and spectral error by 54\% on average; when data is limited to 1\%, DDIS maintains accuracy with 40\% advantage in $l_2$ error compared to joint models.

Submission Number: 103