相关论文: Entropy-dissipation Informed Neural Network for Mc…

This thesis develops numerical and theoretical approaches for understanding and analyzing singularity formation in Partial Differential Equations (PDEs). The singularity formation in the Navier-Stokes Equation (NSE) is famously challenging…

As the limit equations of mean-field particle systems perturbed by common environmental noise, the McKean-Vlasov stochastic differential equations with common noise have received a lot of attention. Moreover, past dependence is an…

We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs' limitations in…

We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erd\H{o}s-R\'enyi graph with parameter $p_n\in (0, 1]$,…

Several important learning tasks can be formulated as minimizing an entropy-regularized objective over an appropriate space of probability distributions. Mean-field Langevin dynamics (MFLD) facilitate computation in this general context,…

This paper concerns the nonparametric estimation problem of the distribution-state dependent drift vector field in an interacting $N$-particle system. Observing single-trajectory data for each particle, we derive the mean-field rate of…

The unsteady Micropolar Navier-Stokes Equations (MNSE) are a system of parabolic partial differential equations coupling linear velocity and pressure with angular velocity: material particles have both translational and rotational degrees…

In this work, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are…

The existence of random dynamical systems for McKean--Vlasov SDEs is established. This is approached by considering the joint dynamics of the corresponding nonlinear Fokker-Planck equation governing the law of the system and the underlying…

We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or…

The multipole-expansion (MPE) model is an implicit solvation model used to efficiently incorporate solvent effects in quantum chemistry. Even within the recent direct approach, the multipole basis used in MPE to express the dielectric…

The Fokker-Planck Equation (FPE) is a fundamental tool for the investigation of kinematic aspects of a wide range of systems. For systems governed by the non-additive entropy $S_q$, the Plastino-Plastino Equation (PPE) is the correct…

We establish a quantitfied overdamped limit for kinetic Vlasov-Fokker-Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is…

The paper is concerned with a McKean-Vlasov type SDE with drift in anisotropic Besov spaces with negative regularity and with degenerate diffusion matrix under the weak H{\"o}rmander condition. The main result is of existence and uniqueness…

Based on the assumption of the existence and uniqueness of the invariant measure for McKean-Vlasov stochastic differential equations (MV-SDEs), a self-interacting process that depends only on the current and historical information of the…

This paper is mainly concerned with the large deviation principle of the fractional McKean-Vlasov stochastic reaction-diffusion equation defined on R^n with polynomial drift of any degree. We first prove the well-posedness of the underlying…

Deep neural networks trained through end-to-end learning have achieved remarkable success across various domains in the past decade. However, the end-to-end learning strategy, originally designed to minimize predictive loss in a black-box…

We discuss the construction and approximation of solutions to a nonlinear McKean-Vlasov equation driven by a singular self-excitatory interaction of the mean-field type. Such an equation is intended to describe an infinite population of…

In this paper, we consider the density estimation problem associated with the stationary measure of ergodic It\^o diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take an…

The numerical approximation of solutions to the compressible Euler and Navier-Stokes equations is a crucial but challenging task with relevance in various fields of science and engineering. Recently, methods from deep learning have been…