Related papers: RED-DiffEq: Regularization by denoising diffusion …
Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at…
Diffusion models have become a successful approach for solving various image inverse problems by providing a powerful diffusion prior. Many studies tried to combine the measurement into diffusion by score function replacement, matrix…
Diffusion models have recently emerged as powerful generative priors for solving inverse problems. However, training diffusion models in the pixel space are both data-intensive and computationally demanding, which restricts their…
We further develop a new framework, called PDE Acceleration, by applying it to calculus of variations problems defined for general functions on $\mathbb{R}^n$, obtaining efficient numerical algorithms to solve the resulting class of…
Diffusion-based models have demonstrated impressive accuracy and generalization in solving partial differential equations (PDEs). However, they still face significant limitations, such as high sampling costs and insufficient physical…
Regularization by denoising (RED) is a powerful framework for solving imaging inverse problems. Most RED algorithms are iterative batch procedures, which limits their applicability to very large datasets. In this paper, we address this…
This paper aims to develop and analyze a numerical scheme for solving the backward problem of semilinear subdiffusion equations. We establish the existence, uniqueness, and conditional stability of the solution to the inverse problem by…
REgularization by Denoising (RED) is an attractive framework for solving inverse problems by incorporating state-of-the-art denoising algorithms as the priors. A drawback of this approach is the high computational complexity of denoisers,…
Long-horizon spatiotemporal prediction remains a challenging problem due to cumulative errors, noise amplification, and the lack of physical consistency in existing models. While diffusion models provide a probabilistic framework for…
Deep neural networks have recently achieved significant advancements in remote sensing superresolu-tion (SR). However, most existing methods are limited to low magnification rates (e.g., 2 or 4) due to the escalating ill-posedness at higher…
Convection-diffusion equations arise in a variety of applications such as particle transport, electromagnetics, and magnetohydrodynamics. Simulation of the convection-dominated regime for these problems, even with high-fidelity techniques,…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
In this article, we develop and present a novel regularization scheme for ill-posed inverse problems governed by nonlinear time-dependent partial differential equations (PDEs). In our recent work, we introduced a bi-level regularization…
We develop a domain-decomposition model reduction method for linear steady-state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equations with random diffusivities, and the…
The inverse diffusion curve problem focuses on automatic creation of diffusion curve images that resemble user provided color fields. This problem is challenging since the 1D curves have a nonlinear and global impact on resulting color…
Building on the well-known total-variation (TV), this paper develops a general regularization technique based on nonlinear isotropic diffusion (NID) for inverse problems with piecewise smooth solutions. The novelty of our approach is to be…
This paper introduces PDEformer, a neural solver for partial differential equations (PDEs) capable of simultaneously addressing various types of PDEs. We propose to represent the PDE in the form of a computational graph, facilitating the…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…
Full waveform inversion (FWI) has the potential to provide high-resolution subsurface model estimations. However, due to limitations in observation, e.g., regional noise, limited shots or receivers, and band-limited data, it is hard to…