A mathematical guide to operator learning
Operator learning aims to discover properties of an underlying dynamical system or partial
differential equation (PDE) from data. Here, we present a step-by-step guide to operator …
differential equation (PDE) from data. Here, we present a step-by-step guide to operator …
Learning Green's functions associated with time-dependent partial differential equations
Neural operators are a popular technique in scientific machine learning to learn a
mathematical model of the behavior of unknown physical systems from data. Neural …
mathematical model of the behavior of unknown physical systems from data. Neural …
Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
GREEN’S MATRICES OF SECOND ORDER ELLIPTIC SYSTEMS WITH MEASURABLE
COEFFICIENTS IN TWO DIMENSIONAL DOMAINS 1. Introduction In t Page 1 TRANSACTIONS …
COEFFICIENTS IN TWO DIMENSIONAL DOMAINS 1. Introduction In t Page 1 TRANSACTIONS …
Uniform regularity estimates in parabolic homogenization
J Geng, Z Shen - Indiana University Mathematics Journal, 2015 - JSTOR
We consider a family of second-order parabolic systems in divergence form with rapidly
oscillating and time-dependent periodic coefficients, arising in the theory of homogenization …
oscillating and time-dependent periodic coefficients, arising in the theory of homogenization …
Gaussian bounds and asymptotic expansions of green function in parabolic homogenization
J Geng - Calculus of Variations and Partial Differential …, 2023 - Springer
For a family of second-order parabolic systems with rapidly oscillating and time-dependent
periodic coefficients, we investigate the asymptotic behavior of Green function. The …
periodic coefficients, we investigate the asymptotic behavior of Green function. The …
Perturbation theory for the parabolic regularity problem
M Ulmer - arxiv preprint arxiv:2408.12529, 2024 - arxiv.org
We show small and large Carleson perturbation results for the parabolic regularity boundary
value problem with boundary data in $\dot {L} _ {1, 1/2}^ p $. The operator we consider is …
value problem with boundary data in $\dot {L} _ {1, 1/2}^ p $. The operator we consider is …
Estimates for fundamental solutions of parabolic equations in non-divergence form
We construct the fundamental solution of second order parabolic equations in non-
divergence form under the assumption that the coefficients are of Dini mean oscillation in the …
divergence form under the assumption that the coefficients are of Dini mean oscillation in the …
Asymptotic expansions of fundamental solutions in parabolic homogenization
J Geng, Z Shen - Analysis & PDE, 2020 - msp.org
Asymptotic expansions of fundamental solutions in parabolic homogenization Page 1
ANALYSIS & PDE msp Volume 13 No. 1 2020 JUN GENG AND ZHONGWEI SHEN …
ANALYSIS & PDE msp Volume 13 No. 1 2020 JUN GENG AND ZHONGWEI SHEN …
Two-sided estimates on Dirichlet heat kernels for time-dependent parabolic operators with singular drifts in C1, α-domains
S Cho, P Kim, H Park - Journal of Differential Equations, 2012 - Elsevier
In this paper, we establish sharp two-sided estimates for the Dirichlet heat kernels of a large
class of time-dependent parabolic operators with singular drifts in C1, α-domain in Rd …
class of time-dependent parabolic operators with singular drifts in C1, α-domain in Rd …
Global estimates for Green's matrix of second order parabolic systems with application to elliptic systems in two dimensional domains
We establish global Gaussian estimates for the Green's matrix of divergence form, second
order parabolic systems in a cylindrical domain under the assumption that weak solutions of …
order parabolic systems in a cylindrical domain under the assumption that weak solutions of …