3.4k citations · 4.6k across the 14 of their papers we have counts for
14 papers
Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients
Xiaojie Wang, Yuying Zhao, Zhongqiang Zhang
We present an error analysis of weak convergence of one-step numerical schemes for stochastic differential equations (SDEs) with super-linearly growing coefficients. Following Mils…
A Modified Walk-on-sphere Method for High Dimensional Fractional Poisson Equation
Caiyu Jiao, Changpin Li, Hexiang Wang +1
We develop walk-on-sphere for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk-on-sphere method is based on probabilistic represen tati…
Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients
Xiaojie Wang, Yuying Zhao, Zhongqiang Zhang
We present an error analysis of weak convergence of one-step numerical schemes for stochastic differential equations (SDEs) with super-linearly growing coefficients. Following Mils…
A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data
Lu Lu, Xuhui Meng, Shengze Cai +4
Neural operators can learn nonlinear mappings between function spaces and offer a new simulation paradigm for real-time prediction of complex dynamics for realistic diverse applica…
Convergence rate of DeepONets for learning operators arising from advection-diffusion equations
Beichuan Deng, Yeonjong Shin, Lu Lu +2
We present convergence analysis of operator learning in [Chen and Chen 1995] and [Lu et al. 2020], where continuous operators are approximated by a sum of products of branch and tr…
Error estimates of residual minimization using neural networks for linear PDEs
Yeonjong Shin, Zhongqiang Zhang, George Em Karniadakis
We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm re…
hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
Ehsan Kharazmi, Zhongqiang Zhang, George Em Karniadakis
We formulate a general framework for hp-variational physics-informed neural networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep neural networks and hp-ref…
DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
Lu Lu, Pengzhan Jin, George Em Karniadakis
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a sin…
Regularity and spectral methods for two-sided fractional diffusion equations with a low-order term
Zhaopeng Hao, Guang Lin, Zhongqiang Zhang
We study regularity and numerical methods for two-sided fractional diffusion equations with a lower-order term. We show that the regularity of the solution in weighted Sobolev spac…
Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions
Fanhai Zeng, Zhongqiang Zhang, George Em Karniadakis
Starting with the asymptotic expansion of the error equation of the shifted Grünwald--Letnikov formula, we derive a new modified weighted shifted Grünwald--Letnikov (WSGL) formula…
Implicit-Explicit difference schemes for nonlinear fractional differential equations with non-smooth solutions
Wanrong Cao, Fanhai Zeng, Zhongqiang Zhang +1
We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order . From the known structure of th…
Wiener chaos vs stochastic collocation methods for linear advection-diffusion equations with multiplicative white noise
Zhongqiang Zhang, Michael V. Tretyakov, Boris Rozovskii +1
We compare Wiener chaos and stochastic collocation methods for linear advection-reaction-diffusion equations with multiplicative white noise. Both methods are constructed based on…
Order-preserving strong schemes for SDEs with locally Lipschitz coefficients
Zhongqiang Zhang, Heping Ma
We introduce a class of explicit balanced schemes for stochastic differential equations with coefficients of superlinearly growth satisfying a global monotone condition. The first…
A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications
M. V. Tretyakov, Z. Zhang
A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDE) which coefficients are allowed to grow polynomially at infinity a…