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The Random Feature Method for Time-dependent Problems
Jingrun Chen, Weinan E, Yixin Luo
We present a framework for solving time-dependent partial differential equations (PDEs) in the spirit of the random feature method. The numerical solution is constructed using a sp…
Bridging Traditional and Machine Learning-based Algorithms for Solving PDEs: The Random Feature Method
Jingrun Chen, Xurong Chi, Weinan E +1
One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A long list of numerical methods have been propo…
MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs
Lulu Zhang, Tao Luo, Yaoyu Zhang +3
In this paper, we propose a a machine learning approach via model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator r…
Machine Learning from a Continuous Viewpoint
Weinan E, Chao Ma, Lei Wu
We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analys…
Pole-based approximation of Fermi-Dirac function
Lin Lin, Jianfeng Lu, Lexing Ying +1
Two approaches for the efficient rational approximation of the Fermi-Dirac function are discussed: one uses the contour integral representation and conformal mapping and the other…
Energy landscapes and rare events
Weinan E, Weiqing Ren, Eric Vanden-Eijnden
Many problems in physics, material sciences, chemistry and biology can be abstractly formulated as a system that navigates over a complex energy landscape of high or infinite dimen…