Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems

By Tianping Chen, Hong Chen. 1995

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Citations in the corpus

MIONet: Learning multiple-input operators via tensor product (2022, background): “One main difference between DeepONet and FNO is that the theory of FNO requires that the input function v and the output function u should be defined on the same domain [16, 17], but the theory of DeepONet does not have this restriction [3, 27, 6, 19, 32].”
Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators (2019, methodology): “Next, we state the following theorem due to Chen & Chen [5], see appendix for more details.”
Learning the solution operator of parametric partial differential equations with physics-informed DeepONets (2021, unknown): “Motivated by the universal approximation theorem for operators ( 35 , 36 ), the architecture features two neural networks coined as the “branch” and “trunk” networks, respectively; the automatic differentiation of which enables us to learn the solution operator of arbitrary PDEs.”
Analysis of Financial Time Series (2005, unknown): “They can approximate any continuous function uniformly on compact sets by increasing the number of nodes in the hidden layer; see Hornik, Stinchcombe, and White (1989), Hornik (1993), and Chen and Chen (1995). This property of neural networks is the universal approximation property of the multilayer”
A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data (2021, methodology): “On this point, it is worth noted that DeepONet was based from the onset on the theorem of Chen & Chen [2], whereas the formulation of FNO was not theoretically justified originally, and the recent theoretical work covers only invariant kernels.”
Stochastic Configuration Networks: Fundamentals and Algorithms (2017, background): “OVER the past decades, neural networks have been successfully applied for data modeling due to its universal approximation power for nonlinear maps [1]–[4], and learning capability from a collection of training samples [5], [6].”
Choose a Transformer: Fourier or Galerkin (2021, methodology): “The pioneering model, DeepONet [60], attributes architecturally to a universal approximation theorem for operators [18].”
Neural Operator: Learning Maps Between Function Spaces (2021, methodology): “Following Chen and Chen (1995), DeepONet(s) define the functional Gj as maps between finite dimensional spaces.”
Hierarchical Models in the Brain (2008, background): “However, unlike back propagation of errors or universal approximation in neural networks [22], DEM is not simply a nonlinear function approximation device.”
On universal approximation and error bounds for Fourier Neural Operators (2021, methodology): “Comparison of FNOs with DeepOnets In this section, we will compare Ψ-FNOs with another operator learning framework, namely DeepOnets of Chen and Chen (1995); Lu et al.”
A physics-informed variational DeepONet for predicting the crack path in brittle materials (2021, unknown): “The idea of DeepONet is motivated by the universal approximation theorem for operators [48], which states that a neural network with a single hidden layer can approximate accurately any linear/non-linear continuous function or operator.”
Fourier Neural Operator with Learned Deformations for PDEs on General Geometries (2022, methodology): “• DeepONet: the deep operator network [10] is a line of works designed with respect to the linear approximation of operator as shown in [28].”