Related papers: Comments on "Bayesian Solution Uncertainty Quantif…
We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under…
In the paper we presnt a solution of problem 31 of chaptr 8 of the monograph of K. Denecke and S. L. Wismath: Hypridentities and clones, Gordon & Breach, 2000, by presnting two main theorems on hyperbasis for hypervarieties and…
High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…
Preliminary version of Chapter 2 in the book "Encyclopedia of Special functions: The Askey-Bateman Project, Vol. 2: Multivariate special functions", T. H. Koornwinder and J. V. Stokman (eds.), Cambridge University Press, 2021.
Using probabilistic methods, we establish a-priori estimates for two classes of quasilinear parabolic systems of partial differential equations (PDEs). We treat in particular the case of a nonlinearity which has quadratic growth in the…
We show that some hard to detect properties of quadratic ODEs (eg certain preserved integrals and measures) can be deduced more or less algorithmically from their Kahan discretization, using Darboux Polynomials (DPs). Somewhat similar…
We make comments on Gaidarzhy {\it et al.}'s [{\it Phys. Rev. Lett.} 94, 030402 (2005)] letter.
We compare the results of our two papers with the results of the paper Aratyn H., Gomes J.F., Zimerman A.H., Higher order Painlev\'e equations and their symmetries via reductions of a class of integrable models, J. Phys. A: Math. Theor., V.…
This paper considers the problem of uniqueness of the solutions to a class of Markovian backward stochastic differential equations (BSDEs) which are also connected to certain nonlinear partial differential equation (PDE) through a…
A variational quantum algorithm for numerically solving partial differential equations (PDEs) on a quantum computer was proposed by Lubasch et al. In this paper, we generalize the method introduced by Lubasch et al. to cover a broader class…
Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations…
Numerical models based on partial differential equations (PDE), or integro-differential equations, are ubiquitous in engineering and science, making it possible to understand or design systems for which physical experiments would be…
Deep neural networks (NNs) are powerful black box predictors that have recently achieved impressive performance on a wide spectrum of tasks. Quantifying predictive uncertainty in NNs is a challenging and yet unsolved problem. Bayesian NNs,…
Contribution to the discussion of the paper "Causal inference using invariant prediction: identification and confidence intervals" by Peters, B\"uhlmann and Meinshausen, to appear in the Journal of the Royal Statistical Society, Series B.
Mechanistic models with differential equations are a key component of scientific applications of machine learning. Inference in such models is usually computationally demanding, because it involves repeatedly solving the differential…
Bayesian networks provide a probabilistic semantics for qualitative assertions about likelihood. A qualitative reasoner based on an algebra over these assertions can derive further conclusions about the influence of actions. While the…
Partial Differential Equations (PDEs) describe several problems relevant to many fields of applied sciences, and their discrete counterparts typically involve the solution of sparse linear systems. In this context, we focus on the analysis…
Symbolic regression with polynomial neural networks and polynomial neural ordinary differential equations (ODEs) are two recent and powerful approaches for equation recovery of many science and engineering problems. However, these methods…
The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical…
A unified approach, for solving a wide class of single and many-body quantum problems, commonly encountered in literature is developed based on a recently proposed method for finding solutions of linear differential equations. Apart from…