Approximation of Functionals by Neural Network Without Curse of Dimensionality
Copyright policy )Approximation of Functionals by Neural Network Without Curse of Dimensionality
In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is $O(1/\sqrt{m})$ where $m$ is the size of networks, which overcomes the curse of dimensionality. The key idea of the approximation is to define a Barron spectral space of functionals.
- Hong Kong University of Science and Technology China (People's Republic of)
- Hong Kong Polytechnic University China (People's Republic of)
- Hong Kong University of Science and Technology (香港科技大學) China (People's Republic of)
- Department of Mathematics
FOS: Computer and information sciences, Artificial intelligence, Computer Science - Machine Learning, Partial differential equations of mathematical physics and other areas of application, infinite-dimensional spaces, Numerical Analysis (math.NA), neural networks, Fourier series, Machine Learning (cs.LG), Barron spectral space, Approximations and expansions, Optimization and Control (math.OC), functionals, FOS: Mathematics, Mathematics - Numerical Analysis, Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems, Mathematics - Optimization and Control
FOS: Computer and information sciences, Artificial intelligence, Computer Science - Machine Learning, Partial differential equations of mathematical physics and other areas of application, infinite-dimensional spaces, Numerical Analysis (math.NA), neural networks, Fourier series, Machine Learning (cs.LG), Barron spectral space, Approximations and expansions, Optimization and Control (math.OC), functionals, FOS: Mathematics, Mathematics - Numerical Analysis, Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems, Mathematics - Optimization and Control
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