Some recent progress in singular stochastic partial differential equations

Abstract

Stochastic PDEs are ubiquitous in mathematical modeling. Yet, many such equations are too singular to admit classical treatment. In this article we review some recent progress in defining, approximating and studying the properties of a few examples of such equations. We focus mainly on the dynamical Φ 4 equation, KPZ equation and Parabolic Anderson Model, as well as touch on a few other equations which arise mainly in physics.

Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation

“…Since the KPZ equation is strictly speaking a singular SPDE (see, e.g., [32,39,45]), it has to be regularized in some way. From a physical point of view, this can be done by either introducing a smallest length-scale (e.g., in form of a lattice-spacing δ [38]) or, in Fourierspace, by defining an upper cutoff wave number [35].…”

“…As we have already mentioned in Sect. 3, the solution h(x, t) to the KPZ equation from ( 1) is a very rough function for all times t. The spatial regularity of h(x, t) cannot be higher than that of the solution to the corresponding Edwards-Wilkinson equation, h (0) (x, t), i.e., the KPZ equation with a vanishing coupling constant, λ = 0 in (1) (see, e.g., [39,45,49]). For h (0) (x, t) it can be checked that for all t > 0…”

Gaussian fluctuations from the 2D KPZ equation

“…There is a lot of activities on the study of singular SPDEs over the past decade. We refer to the reviews [12,13,25] and the references therein. For the KPZ equation, progresses in d ≥ 3 can be found in [15,20], where results similar to Theorem 1.1 were proved.…”

A Partial Differential Equation-Based Image Restoration Method in Environmental Art Design

“…The nature of the solution on the "microelement" is homogeneous, and using the same approximator will give similar prediction accuracy. Therefore, the prediction accuracy of classical numerical methods is consistent [9] over a given definition domain. In fact, through numerical approximation theory, numerical methods can obtain consistent a priori error estimates.…”

“…PINN considers the use of deep neural networks to approximate the unknown solutions of partial differential equations. This paper also focuses on considering this deep neural networkbased approach for solving partial differential equations, i.e., image restoration-informed neural network approach [9]. In a real image restoration problem, we need the amount of image restoration at some moment, on some spatial point or some small region.…”