Related papers: Conditioning non-linear and infinite-dimensional d…
In this article we show that a finite dimensional stochastic differential equation driven by a L\'evy process can be formulated as a stochastic partial differential equation. We prove the existence and uniqueness of strong solutions of such…
The Carleman approach is well-known in the field of deterministic classical dynamics as a method to replace a finite number $d$ of non-linear differential equations by an infinite-dimensional linear system. Here this approach is applied to…
Building on recent advances in scientific machine learning and generative modeling for computational fluid dynamics, we propose a conditional score-based diffusion model designed for multi-scenarios fluid flow prediction. Our model…
The study of multidimensional stochastic processes involves complex computations in intricate functional spaces. In particular, the diffusion processes, which include the practically important Gauss-Markov processes, are ordinarily defined…
Stochastic differential equations (SDEs) provide a natural framework for modelling intrinsic stochasticity inherent in many continuous-time physical processes. When such processes are observed in multiple individuals or experimental units,…
We introduce a class of measure-valued processes, which -- in analogy to their finite dimensional counterparts -- will be called measure-valued polynomial diffusions. We show the so-called moment formula, i.e.~a representation of the…
Diffusion models have gained attention for their ability to represent complex distributions and incorporate uncertainty, making them ideal for robust predictions in the presence of noisy or incomplete data. In this study, we develop and…
We consider the problem of parameter estimation for a class of continuous-time state space models. In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a…
Generating graph-structured data requires learning the underlying distribution of graphs. Yet, this is a challenging problem, and the previous graph generative methods either fail to capture the permutation-invariance property of graphs or…
Score-based generative modelling (SGM) has proven to be a very effective method for modelling densities on finite-dimensional spaces. In this work we propose to extend this methodology to learn generative models over functional spaces. To…
We extend the work of Tanase-Nicola and Kurchan on the structure of diffusion processes and the associated supersymmetry algebra by examining the responses of a simple statistical system to external disturbances of various kinds. We…
Although diffusion models have successfully extended to function-valued data, stochastic interpolants -- which offer a flexible way to bridge arbitrary distributions -- remain limited to finite-dimensional settings. This work bridges this…
Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest…
We investigate nonequilibrium steady-state dynamics in both continuous- and discrete-state stochastic processes. Our analysis focuses on planar diffusion dynamics and their coarse-grained approximations by discrete-state Markov chains.…
Temporal data such as time series can be viewed as discretized measurements of the underlying function. To build a generative model for such data we have to model the stochastic process that governs it. We propose a solution by defining the…
Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make…
The diffusion bridge, which is a diffusion process conditioned on hitting a specific state within a finite period, has found broad applications in various scientific and engineering fields. However, simulating diffusion bridges for modeling…
We propose an unbiased Monte-Carlo estimator for $\mathbb{E}[g(X_{t_1}, \cdots, X_{t_n})]$, where $X$ is a diffusion process defined by a multi-dimensional stochastic differential equation (SDE). The main idea is to start instead from a…
Score-based diffusion models have emerged as powerful techniques for generating samples from high-dimensional data distributions. These models involve a two-phase process: first, injecting noise to transform the data distribution into a…
We present a novel simulation-free framework for training continuous-time diffusion processes over very general objective functions. Existing methods typically involve either prescribing the optimal diffusion process -- which only works for…