Score based models solve physics constrained image reconstruction problems
Backward SDEs-based Diffusion for Physics-Constrained Generation
Machine Learning
Summary
Some computer programs can generate images by learning patterns from many examples, but making sure these generated images fit physical rules or measurements is tricky. The authors propose a new way to adjust these image-generating models so that their outputs always respect the required physical constraints, like matching a real measurement exactly. They do this by solving a special kind of math problem (called a backward stochastic differential equation) that links the measurement rules to the image-generating process. They tested their method on simple examples and in a practical medical imaging problem, showing better results and consistent accuracy.
What this means in practice
- •For medical imaging teams: Improve sparse-view CT image quality while strictly enforcing measurement correctness using pretrained score models without retraining.
- •For engineering simulation groups: Generate physically consistent simulations or reconstructions constrained by domain-specific measurements using pretrained diffusion priors with guaranteed feasibility.
Authors
Zihao Wang
Abstract
Pretrained score-based diffusion models provide strong unconditional priors, yet enforcing measurement or physics consistency in inverse problems is often handled by heuristic guidance, intermittent projections, or task-specific conditional training, with limited guarantees of feasibility at the end of inference. We propose terminal-conditioned inversion for score-based SDE priors. Given a frozen Score-SDE prior and a task-defined terminal feasibility specification, we construct an associated backward stochastic differential equation whose adapted solution defines a principled inverse map from the terminal requirement to a prior state at a chosen noise level. Under standard regularity conditions, we establish existence and uniqueness of the adapted solution and obtain terminal consistency by construction. We further develop a practical neural BSDE solver that composes arbitrary pretrained diffusion priors with domain constraints without modifying the score-defined coefficients, producing an anchored prior state that enables neighborhood sampling for uncertainty characterization. Experiments on toy datasets validate stable terminal-conditioned inversion and distributionally consistent neighborhood sampling. As a real-world case study, we apply the framework to sparse-view CT reconstruction and achieve improved reconstruction quality over representative training-free baselines while satisfying strict measurement feasibility under the prescribed terminal specification. Project is available in: \href{https://laplacelab.github.io/BSDEDiffusion/}{https://laplace.center/icmlbsdeI/}