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 analysis.We demonstrate that conventional mac...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 analysis.We demonstrate that conventional machine learning models and algorithms,such as the random feature model,the two-layer neural network model and the residual neural network model,can all be recovered(in a scaled form)as particular discretizations of different continuous formulations.We also present examples of new models,such as the flow-based random feature model,and new algorithms,such as the smoothed particle method and spectral method,that arise naturally from this continuous formulation.We discuss how the issues of generalization error and implicit regularization can be studied under this framework.展开更多
Hybrid molecule/cluster statistical thermodynamics （HMCST） method is an efficient tool to simulate nano-scale systems under quasi-static loading at finite temperature. In this paper, a self-adaptive algorithm is dev...Hybrid molecule/cluster statistical thermodynamics （HMCST） method is an efficient tool to simulate nano-scale systems under quasi-static loading at finite temperature. In this paper, a self-adaptive algorithm is developed for this method. Explicit refinement criterion based on the gradient of slip shear deformation and a switching criterion based on generalized Einstein approximation is proposed respectively. Results show that this self-adaptive method can accurately find clusters to be refined or transferred to molecules, and efficiently refine or trans- fer the clusters. Furthermore, compared with fully atomistic simulation, the high computational efficiency of the self-adaptive method appears very attractive.展开更多
基金supported by a gift to Princeton University from iFlytek and the Office of Naval Research(ONR)Grant(Grant No.N00014-13-1-0338)。
文摘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 analysis.We demonstrate that conventional machine learning models and algorithms,such as the random feature model,the two-layer neural network model and the residual neural network model,can all be recovered(in a scaled form)as particular discretizations of different continuous formulations.We also present examples of new models,such as the flow-based random feature model,and new algorithms,such as the smoothed particle method and spectral method,that arise naturally from this continuous formulation.We discuss how the issues of generalization error and implicit regularization can be studied under this framework.
基金supported by the National Natural Science Foundation of China (Nos.10932011,10772181,10732090,10772012 and 11021262)the National Basic Research Program of China (No.2007CB814803)
文摘Hybrid molecule/cluster statistical thermodynamics （HMCST） method is an efficient tool to simulate nano-scale systems under quasi-static loading at finite temperature. In this paper, a self-adaptive algorithm is developed for this method. Explicit refinement criterion based on the gradient of slip shear deformation and a switching criterion based on generalized Einstein approximation is proposed respectively. Results show that this self-adaptive method can accurately find clusters to be refined or transferred to molecules, and efficiently refine or trans- fer the clusters. Furthermore, compared with fully atomistic simulation, the high computational efficiency of the self-adaptive method appears very attractive.
