Related papers: A Score-based Diffusion Model Approach for Adaptiv…
Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…
The study presents a general framework for discovering underlying Partial Differential Equations (PDEs) using measured spatiotemporal data. The method, called Sparse Spatiotemporal System Discovery ($\text{S}^3\text{d}$), decides which…
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,…
Diffusion models have emerged as the principal paradigm for generative modeling across various domains. During training, they learn the score function, which in turn is used to generate samples at inference. They raise a basic yet unsolved…
Deep-learning video super-resolution has progressed rapidly, but climate applications typically super-resolve (increase resolution) either space or time, and joint spatiotemporal models are often designed for a single pair of…
Score-based generative models have demonstrated significant practical success in data-generating tasks. The models establish a diffusion process that perturbs the ground truth data to Gaussian noise and then learn the reverse process to…
Diffusion processes are a class of stochastic differential equations (SDEs) providing a rich family of expressive models that arise naturally in dynamic modelling tasks. Probabilistic inference and learning under generative models with…
Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore researchers often seek simpler…
We propose self-diffusion, a novel framework for solving inverse problems without relying on pretrained generative models. Traditional diffusion-based approaches require training a model on a clean dataset to learn to reverse the forward…
We present a concise derivation for several influential score-based diffusion models that relies on only a few textbook results. Diffusion models have recently emerged as powerful tools for generating realistic, synthetic signals --…
Rapidly developing machine learning methods has stimulated research interest in computationally reconstructing differential equations (DEs) from observational data which may provide additional insight into underlying causative mechanisms.…
Diffusion models are powerful tools for sampling from high-dimensional distributions by progressively transforming pure noise into structured data through a denoising process. When equipped with a guidance mechanism, these models can also…
Predicting the behavior of complex systems is critical in many scientific and engineering domains, and hinges on the model's ability to capture their underlying dynamics. Existing methods encode the intrinsic dynamics of high-dimensional…
We introduce a novel framework for uncertainty quantification of solution operators associated with stochastic partial differential equations (SPDEs). Although SPDEs play a central role in modeling complex physical systems under…
This paper is devoted to a system of stochastic partial differential equations (SPDEs) that have a slow component driven by fractional Brownian motion (fBm) with the Hurst parameter $H >1/2$ and a fast component driven by fast-varying…
Accurate forecasting of spatiotemporal data remains challenging due to complex spatial dependencies and temporal dynamics. The inherent uncertainty and variability in such data often render deterministic models insufficient, prompting a…
Stemming from the stochastic Lotka-Volterra or predator-prey equations, this work aims to model the spatial inhomogeneity by using stochastic partial differential equations (SPDEs). Compared to the classical models, the SPDE model is more…
We formulate a new class of stochastic partial differential equations (SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which allow the high-order integral-partial differential operators into both drift and diffusion…
The reconstruction and inference of stochastic dynamical systems from data is a fundamental task in inverse problems and statistical learning. While surrogate modeling advances computational methods to approximate these dynamics, standard…
Multi-fidelity surrogate learning is important for physical simulation related applications in that it avoids running numerical solvers from scratch, which is known to be costly, and it uses multi-fidelity examples for training and greatly…