Related papers: Score Shocks: The Burgers Equation Structure of Di…
Score-based generative models (SGMs) are powerful tools to sample from complex data distributions. Their underlying idea is to (i) run a forward process for time $T_1$ by adding noise to the data, (ii) estimate its score function, and (iii)…
We study the large-scale behaviour of a class of driven diffusive systems modelled by a Stochastic Partial Differential Equation, the Stochastic Burgers Equation (SBE) with general nonlinearity, at the critical dimension and in infinite…
Diffusion or score-based models recently showed high performance in image generation. They rely on a forward and a backward stochastic differential equations (SDE). The sampling of a data distribution is achieved by numerically solving the…
This paper compares theory and experiment for the kinetics of time-dependent sedimentation. We discuss non-interacting suspensions and colloids which may exhibit behavior similar to the one-dimensional motion of compressible gas. The…
This article focuses on a nonlinear Neumann boundary feedback control formulation for the viscous Burgers' equation and develops a class of finite difference schemes to achieve global stabilization. The proposed procedure, known as the…
This article addresses some asymptotic and numerical issues related to the solution of Burgers' equation, $-\epsilon u_{xx} + u_t + u u_x = 0$ on $(-1,1)$, subject to the boundary conditions $u(-1) = 1 + \delta$, $u(1) = -1$, and its…
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…
Dynamic response of the one-dimensional viscoelastic rod of finite length, that has one end fixed and the other subject to prescribed either displacement or stress, is analyzed by the analytical means of Laplace transform, yielding the…
Diffusion-based trajectory optimization has emerged as a powerful planning paradigm, but existing methods require either learned score networks trained on large datasets or analytical dynamics models for score computation. We introduce…
Adversarial Imitation Learning is traditionally framed as a two-player zero-sum game between a learner and an adversarially chosen cost function, and can therefore be thought of as the sequential generalization of a Generative Adversarial…
We provide full theoretical guarantees for the convergence behaviour of diffusion-based generative models under the assumption of strongly log-concave data distributions while our approximating class of functions used for score estimation…
Diffusion-based generative models have demonstrated a capacity for perceptually impressive synthesis, but can they also be great likelihood-based models? We answer this in the affirmative, and introduce a family of diffusion-based…
Score-based generative models (SGMs) synthesize new data samples from Gaussian white noise by running a time-reversed Stochastic Differential Equation (SDE) whose drift coefficient depends on some probabilistic score. The discretization of…
In this work we consider the problem of constructing initial conditions for a flow model such that the resulting flow evolution leads to a self-similar energy cascade consistent with Kolmogorov's statistical theory of turbulence. As a first…
Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic…
The Stochastic Burgers equation was introduced in [H. van Beijeren, R. Kutner and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985] as a continuous approximation of the fluctuations of the asymmetric simple exclusion process.…
The volume of a Wiener sausage constructed from a diffusion process with periodic, mean-zero, divergence-free velocity field, in dimension 3 or more, is shown to have a non-random and positive asymptotic rate of growth. This is used to…
Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…
A geometrically nonlinear theory for field dislocation thermomechanics based entirely on measurable state variables is proposed. Instead of starting from an ordering-dependent multiplicative decomposition of the total deformation gradient…
With significant advancements in diffusion models, addressing the potential risks of dataset bias becomes increasingly important. Since generated outputs directly suffer from dataset bias, mitigating latent bias becomes a key factor in…