Ke Chen

Scientific Machine Learning for PDEs

PDE operators are investigated and numerically approximated via neural networks.

Tackle the Curse of Dimensionality

Deep Operator Learning Lessens the Curse of Dimensionality for PDEs.
(with Chunmei Wang and Haizhao Yang), TMLR, arXiv:2301.12227 (2023).

Operator learning for PDE Inverse Problems

Pseudo-differential Integral Autoencoder Network for Inverse PDE Operators
(with Jasen Lai and Chunmei Wang), arXiv:2310.09856 (2023).

Let Data Talk: Data-regularized Operator Learning Theory for Inverse Problems
(with Chunmei Wang and Haizhao Yang), arXiv:2310.09854 (2023).

Inverse Problems in Optical Tomography

Optical tomography is a noninvasive method that reconstructs images from boundary measurements of light intensities transmitted and scattered through an object.

Stability Deterioration in Asymptotic Regimes

Stability of Stationary Inverse Transport Equation in Diffusion Scaling
(with Qin Li and Li Wang), Inverse Problems 34.2 (2018): 025004.
Stability of Inverse Transport Equation in Diffusion Scaling and Fokker-Planck Limit
(with Qin Li and Li Wang), SIAM Journal on Applied Mathematics 78.5 (2018): 2626-2647.

Online Computation of Optical Tomography

Online Learning in Optical Tomography: A Stochastic Approach
(with Qin Li and Jian-Guo Liu), Inverse Problems 34.7 (2018): 075010.

Random sketching methods for Inverse Problems

Structured random sketching for PDE inverse problems
(with Kit Newton, Stephan Wright, and Qin Li ), SIAM Journal on Matrix Analysis and Applications 41.4 (2020): 1742-1770.
Tensor-Structured Sketching for Constrained Least Squares
(with Ruhui Jin), SIAM Journal on Matrix Analysis and Applications 42.4 (2021): 1703-1731

Fast Direct Solver for PDE operators

We investigate the hierarchically low rank structure of the PDE operator and build its inverse in a fast and stable way.

Parabolic Equations

Fast and high-order approximation of parabolic equations using hierarchical direct solvers and implicit Runge-Kutta methods
(with Daniel Appeloö, Tracy Babb and Per-Gunnar Martinsson), arXiv: 2306.02526

PDE Compression via Random Sampling

Solving Multi-scale PDEs with a randomized solver that automatically captures the essential dimension.

General Framework for Multi-scale PDEs

Random Sampling and Efficient Algorithms for Multiscale PDEs
(with Qin Li, Jianfeng Lu and Stephen Wright), SIAM Journal on Scientific Computing 42.5 (2020): A2974-A3005.
Low-rank approximation for multiscale PDEs
(with Shi Chen, Qin Li, Jianfeng Lu and Stephen Wright), Notices of the American Mathematical Society 69.6 (2021).

Compression for Elliptic Equations

Randomized Sampling for Basis Functions Construction in Generalized Finite Element Methods
(with Qin Li, Jianfeng Lu and Stephen Wright), Multiscale Modeling & Simulation 18.2 (2020): 1153-1177.
Schwarz iteration method for elliptic equation with rough media based on random sampling,
(with Qin Li and Stephen Wright), Proceedings of International Consortium of Chinese Mathematics 2019.

Compression for Radiative Transfer Equation

A low-rank Schwarz method for radiative transfer equation with heterogeneous scattering coefficient
(with Qin Li, Jianfeng Lu and Stephen Wright), Multiscale Modeling & Simulation 19.2 (2021): 775-801

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