Distributed approximating functional approach to fitting and predicting potential surfaces. 1. Atom-atom potentials

Frishman, Anatoli M.; Hoffman, David K.; Rakauskas, R. J.; Kouri, Donald J.

doi:10.1016/s0009-2614(96)00150-9

Cited by 25 publications

(3 citation statements)

References 8 publications

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“…Fitting procedures that are not physically motivated have the virtue of simplicity. The most popular procedures of this type are spline, − interpolating moving least squares (IMLS), , reproducing kernel Hilbert space (RKHS), , modified Sheppard interpolation (MSI), − distributed approximating functionals (DAF), − and neural network (NN) algorithms. − These approaches are systematically improvable, work well even if coupling is large, and they are easy to use; e.g., most parameter values are completely determined by the numerical algorithm and do not need to be estimated on the basis of experimental results or intuition. However, in all of these methods, there are parameters that must be chosen by the user.…”

Section: Introductionmentioning

confidence: 99%

“…The popular Taylor-expansion based MSI method requires potential derivatives and is used almost exclusively with second-order expansions, as higher derivatives are difficult to obtain from ab initio calculations. The RKHS method is easiest to use with tensor product grids ,, which make it difficult to fit surfaces for systems with more than three atoms. ,,− With a trajectory-based point selection scheme, the MSI approach can fit dynamically relevant parts of PESs of four-atom molecules ,− and some five- and six-atom systems. − For the cited four-atom systems, the potential fitting errors of these methods are of the order of 10 1 −10 2 cm -1 . Similar errors are obtained with physically motivated fitting functions. ,, Recently, permutation invariant polynomials and interatomic distances have been used to fit several potentials.…”

Section: Introductionmentioning

confidence: 99%

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A Nested Molecule-Independent Neural Network Approach for High-Quality Potential Fits

et al. 2005

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It is shown that neural networks (NNs) are efficient and effective tools for fitting potential energy surfaces. For H2O, a simple NN approach works very well. To fit surfaces for HOOH and H2CO, we develop a nested neural network technique in which we first fit an approximate NN potential and then use another NN to fit the difference of the true potential and the approximate potential. The root-mean-square error (RMSE) of the H2O surface is 1 cm(-1). For the 6-D HOOH and H2CO surfaces, the nested approach does almost as well attaining a RMSE of 2 cm(-1). The quality of the NN surfaces is verified by calculating vibrational spectra. For all three molecules, most of the low-lying levels are within 1 cm(-1) of the exact results. On the basis of these results, we propose that the nested NN approach be considered a method of choice for both simple potentials, for which it is relatively easy to guess a good fitting function, and complicated (e.g., double well) potentials for which it is much harder to deduce an appropriate fitting function. The number of fitting parameters is only moderately larger for the 6-D than for the 3-D potentials, and for all three molecules, decreasing the desired RMSE increases only slightly the number of required fitting parameters (nodes). NN methods, and in particular the nested approach we propose, should be good universal potential fitting tools.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Nested Molecule-Independent Neural Network Approach for High-Quality Potential Fits

et al. 2005

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“…However, standard interpolation methods (such as splines or orthogonal polynomial methods) are difficult to apply to systems with more than a few degrees of freedom. In recent years, a number of techniques have been developed to address this problem, and substantial progress has been made. − Nevertheless, more work is needed to develop an accurate, efficient method that can be routinely applied to reacting systems with many degrees of freedom.…”

Section: Introductionmentioning

confidence: 99%

Neural Network Models of Potential Energy Surfaces: Prototypical Examples

Witkoskie

Doren

2004

J. Chem. Theory Comput.

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Neural networks can be used generate potential energy hypersurfaces by fitting to a data set of energies at discrete geometries, as might be obtained from ab initio calculations. Prior work has shown that this method can generate accurate fits in complex systems of several dimensions. The present paper explores fundamental properties of neural network potential representations in some simple prototypes of one, two, and three dimensions. Optimal fits to the data are achieved by adjusting the network parameters using an extended Kalman filtering algorithm, which is described in detail. The examples provide insight into the relationships between the form of the function being fit, the amount of data needed for an adequate fit, and the optimal network configuration and number of neurons needed. The quality of the network interpolation is substantially improved if gradients as well as the energy are available for fitting. The fitting algorithm is effective in providing an accurate interpolation of the underlying potential function even when random noise is added to the data used in the fit.

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Uniform Approximation of Functions with Discrete Approximation Functionals

Chandler

Gibson

1999

Journal of Approximation Theory

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Hoffman, Kouri, and collaborators have calculated nonrelativistic quantum scattering amplitudes by numerically evaluating Feynman path integrals. They observed that the errors introduced by their numerical scheme were uniform in coordinate space, implying that their scheme accurately reproduces both the shape and the phase of functions. Furthermore, they observed that the size and the uniform nature of the errors were preserved when the functions were allowed to evolve in time under the action of the kinetic energy operator. In this paper it is established that these observed properties of the errors are not numerical artifacts but follow from analytical properties of a general class of approximations that include those of Hoffman, Kouri, and collaborators as a special case. Academic Press

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Distributed approximating functional approach to fitting and predicting potential surfaces. 1. Atom-atom potentials

Cited by 25 publications

References 8 publications

A Nested Molecule-Independent Neural Network Approach for High-Quality Potential Fits

A Nested Molecule-Independent Neural Network Approach for High-Quality Potential Fits

Neural Network Models of Potential Energy Surfaces: Prototypical Examples

Uniform Approximation of Functions with Discrete Approximation Functionals

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