Functions bandlimited in frequency are free of the curse of dimensionality

Mulero-Martínez, Juan Ignacio

doi:10.1016/j.neucom.2006.05.010

Cited by 11 publications

(8 citation statements)

References 23 publications

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“…When the network is truncated-by exploiting the fast decay of Gaussians-it can be proved that the mean square error is of order O 1/γ (N) n , where γ is a proper function greater than 1. As a result, functions that are band-limited in frequency are free from the curse of dimensionality [14].…”

Section: Dimensionality Reductionmentioning

confidence: 99%

Robust GRBF Static Neurocontroller With Switch Logic for Control of Robot Manipulators

Mulero-Martínez

2012

IEEE Trans. Neural Netw. Learning Syst.

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A new Gaussian radial basis function static neurocontroller is presented for stable adaptive tracking control. This is a two-stage controller acting in a supervisory fashion by means of a switch logic and allowing arbitration between a neural network (NN) and a robust proportional-derivative controller. The structure is intended to reduce the effects of the curse of dimensionality in multidimensional systems by fully exploiting the mechanical properties of the robot manipulator. A new factorization of the Coriolis/centripetal matrix is used, leading to an NN model that is much smaller than the dynamic ones. By resorting to the extended multivariate Shannon theorem and the computation of the effective bandwidth of the revolute robot manipulators, the network parameters are tuned. Stability and convergence properties are analyzed. This provides the assurance of reliability and effectiveness to make such controller viable. A robot manipulator with two degrees of freedom is employed to study the adaptive features of the neural control algorithm. Finally, the effectiveness of the proposed method is compared to the nonadaptive case.

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Section: Dimensionality Reductionmentioning

confidence: 99%

Robust GRBF Static Neurocontroller With Switch Logic for Control of Robot Manipulators

Mulero-Martínez

2012

IEEE Trans. Neural Netw. Learning Syst.

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“…for functions whose Fourier spectrum decays rapidly beyond a certain frequency). 95 Note that f in Eq. 1 can be replaced with a vector f with components fk, in that case the coefficients will also depend on k, cnk, while the basis can remain the same.…”

Section: 𝜎𝜎(∑ 𝑐𝑐 𝑛𝑛 𝜎𝜎( 𝑁𝑁 𝑛𝑛=0mentioning

confidence: 99%

“…When the argument of σ is expressed as b n | w n – x | 2 , one obtains a so-called radial basis function (RBF) NN. RBF NNs are also universal approximators , and also in principle allow avoiding exponential scaling, specifically, for band-limited functions (i.e., for functions whose Fourier spectrum decays rapidly beyond a certain frequency) …”

Section: Neural Network For Potential Energy Surfacesmentioning

confidence: 99%

“…RBF NNs are also universal approximators 94,95 and also in principle allow avoiding exponential scaling, specifically, for band-limited functions (i.e., for functions whose Fourier spectrum decays rapidly beyond a certain frequency). 96 Note that f in eq 1 can be replaced with a vector f with components f k ; in that case, the coefficients will also depend on k, c nk , while the basis can remain the same. This would be a NN with multiple outputs that could be for example energies and forces or potential values of different states.…”

Section: Introduction To Neural Network Used In Pes Fittingmentioning

confidence: 99%

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Neural Network Potential Energy Surfaces for Small Molecules and Reactions

2020

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We review progress in neural network (NN) based methods for the construction of interatomic potentials from discrete samples (such as ab initio energies) for applications in classical and quantum dynamics including reaction dynamics and computational spectroscopy. The main focus is on methods for building molecular potential energy surfaces (PES) in internal coordinates that explicitly include all many-body contributions, even though some of the methods we review limit the degree of coupling, due either to a desire to limit computational cost or to limited data. Explicit and direct treatment of all many-body contributions is only practical for sufficiently small molecules, which are therefore our primary focus. This includes small molecules on surfaces. We consider direct, single NN PES fitting as well as more complex methods which impose structure (such as a multi-body representation) on the PES function, either through the architecture of one NN or by using multiple NNs. We show how NNs are effective in building representations with low-dimensional functions, including dimensionality reduction. We consider NN based approaches to build PESs in the sums-of-product form important for quantum dynamics, ways to treat symmetry, and issues related to sampling data distributions and the relation between PES errors and errors in observables. We highlight combinations of NNs with other ideas such as permutationally invariant polynomials or sums of environment-dependent atomic contributions which have recently emerged as powerful tools for building highly accurate PESs for relatively large molecular and reactive systems.

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“…It was suggested that RBFNNs escape the curse of dimensionality [66]. The energy was a fitting parameter that was adjusted to be equal to that computed from the wavefunction.…”

Section: Machine Learning For Solution Of the Vibrational Schrödingermentioning

confidence: 99%

Machine learning for the solution of the Schrödinger equation

Manzhos

2020

Mach. Learn.: Sci. Technol.

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Machine learning (ML) methods have recently been increasingly widely used in quantum chemistry. While ML methods are now accepted as high accuracy approaches to construct interatomic potentials for applications, the use of ML to solve the Schrödinger equation, either vibrational or electronic, while not new, is only now making significant headway towards applications. We survey recent uses of ML techniques to solve the Schrödinger equation, including the vibrational Schrödinger equation, the electronic Schrödinger equation and the related problems of constructing functionals for density functional theory (DFT) as well as potentials which enter semi-empirical approximations to DFT. We highlight similarities and differences and specific difficulties that ML faces in these applications and possibilities for cross-fertilization of ideas.© 2020 The Author(s). Published by IOP Publishing Ltd Mach. Learn.: Sci. Technol. 1 (2020) 013002 S Manzhos approximator properties, with modifications specific to the problem of solving the SE or building functionals, including adapted architectures of ML methods and data selection schemes. It is therefore important to recognize similarities and differences in the requirements that can be addressed via ML among different applications.In this Perspective, we survey recent uses of ML techniques to solve the Schrödinger equation, including the vibrational Schrödinger equation and the electronic problem: the electronic Schrödinger equation and the related problems of constructing exchange-correlation and kinetic energy functionals for Kohn-Sham (KS-) and orbital-free (OF-) density functional theory (DFT) as well as use of ML for semi-empirical approximations used in density functional based approaches such as DFTB (density functional tight binding) and dispersion-corrected DFT (DFT-D). We only consider the use of ML for the solution of the vibrational and electronic SE or the KS equation or the Hohenberg-Kohn (HK) equation and not methods that aim to avoid such solutions (such as those directly mapping the molecular structure to the spectrum or energy or properties without solving for the density or wavefunction [29][30][31][32][33][34][35][36]); we also do not consider uses of ML for other types of quantum mechanical modelling or other types of differential or integral equations [37][38][39][40][41][42].We do not aim to present a comprehensive review but rather a survey allowing similarities and differences of ML uses in all these applications, as well as promising directions for future research, to transpire. This is not a review of ML methods; the key machine learning techniques that found use in quantum chemistry are well reviewed elsewhere [3-5, 43, 44], their description will not be repeated there; the reader is advised to consult the literature for their introduction, such as [45,46] for neural networks, [47,48] for Gaussian process regression (GPR), [49] for kernel ridge regression (KRR; note the similarity in the form of the function representation between GPR and KRR), [50] for...

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Functions bandlimited in frequency are free of the curse of dimensionality

Cited by 11 publications

References 23 publications

Robust GRBF Static Neurocontroller With Switch Logic for Control of Robot Manipulators

Robust GRBF Static Neurocontroller With Switch Logic for Control of Robot Manipulators

Neural Network Potential Energy Surfaces for Small Molecules and Reactions

Machine learning for the solution of the Schrödinger equation

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