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In this article, the solution of a statistical inverse problem $M = AU+\mathcal{E}$ by the Bayesian approach is studied where $U$ is a function on the unit circle $\mathbb{T}$, i.e., a periodic signal. The mapping $A$ is a smoothing linear…
Bayesian solution of an inverse problem for indirect measurement $M = AU + {\mathcal{E}}$ is considered, where $U$ is a function on a domain of $R^d$. Here $A$ is a smoothing linear operator and $ {\mathcal{E}}$ is Gaussian white noise. The…
Score-based diffusion models generate new samples by learning the score function associated with a diffusion process. While the effectiveness of these models can be theoretically explained using differential equations related to the…
Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models,…
We introduce Functional Diffusion Processes (FDPs), which generalize score-based diffusion models to infinite-dimensional function spaces. FDPs require a new mathematical framework to describe the forward and backward dynamics, and several…
This study presents a conditional flow matching framework for solving physics-constrained Bayesian inverse problems. In this setting, samples from the joint distribution of inferred variables and measurements are assumed available, while…
Inverse problems are prevalent across various disciplines in science and engineering. In the field of computer vision, tasks such as inpainting, deblurring, and super-resolution are commonly formulated as inverse problems. Recently,…
Generative diffusion models learn probability densities over diverse image datasets by estimating the score with a neural network trained to remove noise. Despite their remarkable success in generating high-quality images, the internal…
Temporal prediction is inherently uncertain, but representing the ambiguity in natural image sequences is a challenging high-dimensional probabilistic inference problem. For natural scenes, the curse of dimensionality renders explicit…
The statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect observation is studied in the Bayesian framework. In practice, one often considers only…
Stochastic differential equations (SDEs) provide a flexible framework for modeling temporal dynamics in partially observed systems. A central task is to calibrate such models from data, which requires inferring latent trajectories and…
Recently, score-based diffusion models have shown satisfactory performance in MRI reconstruction. Most of these methods require a large amount of fully sampled MRI data as a training set, which, sometimes, is difficult to acquire in…
We present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems. We adopt the Bayesian inference formulation: given observational data and their…
We propose a conditional stochastic interpolation (CSI) method for learning conditional distributions. CSI is based on estimating probability flow equations or stochastic differential equations that transport a reference distribution to the…
Causal modeling has long been an attractive topic for many researchers and in recent decades there has seen a surge in theoretical development and discovery algorithms. Generally discovery algorithms can be divided into two approaches:…
This is an expository article on the score-based diffusion models, with a particular focus on the formulation via stochastic differential equations (SDE). After a gentle introduction, we discuss the two pillars in the diffusion modeling --…
We present the Score-based Autoencoder for Multiscale Inference (SAMI), a method for unsupervised representation learning that combines the theoretical frameworks of diffusion models and VAEs. By unifying their respective evidence lower…
We propose a novel approach for learning causal response representations. Our method aims to extract directions in which a multidimensional outcome is most directly caused by a treatment variable. By bridging conditional independence…
This work explores the theoretical and practical foundations of denoising diffusion probabilistic models (DDPMs) and score-based generative models, which leverage stochastic processes and Brownian motion to model complex data distributions.…
We present an ``equation-free'' multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast…