Diffusion models excel at creating visually-convincing images, but they often struggle to meet subtle constraints inherent in the training data. Such constraints could be physics-based (e.g., satisfying a PDE), geometric (e.g., respecting symmetry), or semantic (e.g., including a particular number of objects). When the training data all satisfy a certain constraint, enforcing this constraint on a diffusion model makes it more reliable for generating valid synthetic data and solving constrained inverse problems. However, existing methods for constrained diffusion models are restricted in the constraints they can handle. For instance, recent work proposed to learn mirror diffusion models (MDMs), but analytical mirror maps only exist for convex constraints and can be challenging to derive. We propose *neural approximate mirror maps* (NAMMs) for general, possibly non-convex constraints. Our approach only requires a differentiable distance function from the constraint set. We learn an approximate mirror map that transforms data into an unconstrained space and a corresponding approximate inverse that maps data back to the constraint set. A generative model, such as an MDM, can then be trained in the learned mirror space and its samples restored to the constraint set by the inverse map. We validate our approach on a variety of constraints, showing that compared to an unconstrained diffusion model, a NAMM-based MDM substantially improves constraint satisfaction. We also demonstrate how existing diffusion-based inverse-problem solvers can be easily applied in the learned mirror space to solve constrained inverse problems.

Plug-and-play diffusion priors (PnPDP) have emerged as a promising research direction for solving inverse problems. However, current studies primarily focus on natural image restoration, leaving the performance of these algorithms in scientific inverse problems largely unexplored. To address this gap, we introduce \textsc{InverseBench}, a framework that evaluates diffusion models across five distinct scientific inverse problems. These problems present unique structural challenges that differ from existing benchmarks, arising from critical scientific applications such as optical tomography, medical imaging, black hole imaging, seismology, and fluid dynamics. With \textsc{InverseBench}, we benchmark 14 inverse problem algorithms that use plug-and-play diffusion priors against strong, domain-specific baselines, offering valuable new insights into the strengths and weaknesses of existing algorithms. To facilitate further research and development, we open-source the codebase, along with datasets and pre-trained models, at [https://devzhk.github.io/InverseBench/](https://devzhk.github.io/InverseBench/).

Photoacoustic tomography (PAT) is a rapidly-evolving medical imaging modality that combines optical absorption contrast with ultrasound imaging depth. One challenge in PAT is image reconstruction with inadequate acoustic signals due to limited sensor coverage or due to the density of the transducer array. Such cases call for solving an ill-posed inverse reconstruction problem. In this work, we use score-based diffusion models to solve the inverse problem of reconstructing an image from limited PAT measurements. The proposed approach allows us to incorporate an expressive prior learned by a diffusion model on simulated vessel structures while still being robust to varying transducer sparsity conditions.

Priors are essential for reconstructing images from noisy and/or incomplete measurements. The choice of the prior determines both the quality and uncertainty of recovered images. We propose turning score-based diffusion models into principled image priors ("score-based priors") for analyzing a posterior of images given measurements. Previously, probabilistic priors were limited to handcrafted regularizers and simple distributions. In this work, we empirically validate the theoretically-proven probability function of a score-based diffusion model. We show how to sample from resulting posteriors by using this probability function for variational inference. Our results, including experiments on denoising, deblurring, and interferometric imaging, suggest that score-based priors enable principled inference with a sophisticated, data-driven image prior.

An object's interior material properties, while invisible to the human eye, determine motion observed on its surface. We propose an approach that estimates heterogeneous material properties of an object from a monocular video of its surface vibrations. Specifically, we show how to estimate Young's modulus and density throughout a 3D object with known geometry. Knowledge of how these values change across the object is useful for simulating its motion and characterizing any defects. Traditional nondestructive testing approaches, which often require expensive instruments, generally estimate only homogenized material properties or simply identify the presence of defects. In contrast, our approach leverages monocular video to (1) identify image-space modes from an object's sub-pixel motion, and (2) directly infer spatially-varying Young's modulus and density values from the observed modes. We demonstrate our approach on both simulated and real videos.

We introduce a perceptually motivated approach to bandwidth expansion for speech. Our method pairs a new 3-way split variant of the FFTNet neural vocoder structure with a perceptual loss function, combining objectives from both the time and frequency domains. Mean opinion score tests show that it outperforms baseline methods from both domains, even for extreme bandwidth expansion.