Related papers: Gradient Estimates for Nonlinear Diffusion Semigro…

We construct deterministic particle solutions for linear and fast diffusion equations using a nonlocal approximation. We exploit the $2$-Wasserstein gradient flow structure of the equations in order to obtain the nonlocal approximating PDEs…

This paper investigates the well-posedness and small-noise asymptotics of a class of stochastic partial differential equations defined on a bounded domain of $\mathbb{R}^d$, where the diffusion coefficient depends nonlinearly and…

The gradient discretisation method (GDM) is a generic framework for designing and analysing numerical schemes for diffusion models. In this paper, we study the GDM for the porous medium equation, including fast diffusion and slow diffusion…

Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…

The output distribution of a neural network (NN) over the entire input space captures the complete input-output mapping relationship, offering insights toward a more comprehensive NN understanding. Exhaustive enumeration or traditional…

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…

A novel multi-level method for partial differential equations with uncertain parameters is proposed. The principle behind the method is that the error between grid levels in multi-level methods has a spatial structure that is by good…

In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for…

This paper addresses the nonparametric estimation of the drift function over a compact domain for a time-homogeneous diffusion process, based on high-frequency discrete observations from $N$ independent trajectories. We propose a neural…

Motivated by recent work on approximation of diffusion equations by deterministic interacting particle systems, we develop a nonlocal approximation for a range of linear and nonlinear diffusion equations and prove convergence of the method…

The cutoff method, which cuts off the values of a function less than a given number, is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations. A convergence analysis is given for a broad…

In the field of inverse estimation for systems modeled by partial differential equations (PDEs), challenges arise when estimating high- (or even infinite-) dimensional parameters. Typically, the ill-posed nature of such problems…

Partial differential equations (PDEs) provide a mathematical foundation for simulating and understanding intricate behaviors in both physical sciences and engineering. With the growing capabilities of deep learning, data$-$driven approaches…

We develop a class of non-Gaussian translation processes that extend classical stochastic differential equations (SDEs) by prescribing arbitrary absolutely continuous marginal distributions. Our approach uses a copula-based transformation…

We present an explicit method for simulating stochastic differential equations (SDEs) that have variable diffusion coefficients and satisfy the detailed balance condition with respect to a known equilibrium density. In Tupper and Yang…

This paper proposes a regularized pairwise difference approach for estimating the linear component coefficient in a partially linear model, with consistency and exact rates of convergence obtained in high dimensions under mild scaling…

In this paper we focus on the global-in-time existence and the pointwise estimates of solutions to the initial value problem for the semilinear dissipative wave equation in multi-dimensions. By using the method of Green function combined…

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…

Standard causal inference characterizes treatment effect through averages, but the counterfactual distributions could be different in not only the central tendency but also spread and shape. To provide a comprehensive evaluation of…