An improved neural network (NN) approach is presented for the simultaneous development of accurate potential-energy hypersurfaces and corresponding force fields that can be utilized to conduct ab initio molecular dynamics and Monte Carlo studies on gas-phase chemical reactions. The method is termed as combined function derivative approximation (CFDA). The novelty of the CFDA method lies in the fact that although the NN has only a single output neuron that represents potential energy, the network is trained in such a way that the derivatives of the NN output match the gradient of the potential-energy hypersurface. Accurate force fields can therefore be computed simply by differentiating the network. Both the computed energies and the gradients are then accurately interpolated using the NN. This approach is superior to having the gradients appear in the output layer of the NN because it greatly simplifies the required architecture of the network. The CFDA permits weighting of function fitting relative to gradient fitting. In every test that we have run on six different systems, CFDA training (without a validation set) has produced smaller out-of-sample testing error than early stopping (with a validation set) or Bayesian regularization (without a validation set). This indicates that CFDA training does a better job of preventing overfitting than the standard methods currently in use. The training data can be obtained using an empirical potential surface or any ab initio method. The accuracy and interpolation power of the method have been tested for the reaction dynamics of H+HBr using an analytical potential. The results show that the present NN training technique produces more accurate fits to both the potential-energy surface as well as the corresponding force fields than the previous methods. The fitting and interpolation accuracy is so high (rms error=1.2 cm(-1)) that trajectories computed on the NN potential exhibit point-by-point agreement with corresponding trajectories on the analytic surface.

A general method for the development of potential-energy hypersurfaces is presented. The method combines a many-body expansion to represent the potential-energy surface with two-layer neural networks (NN) for each M-body term in the summations. The total number of NNs required is significantly reduced by employing a moiety energy approximation. An algorithm is presented that efficiently adjusts all the coupled NN parameters to the database for the surface. Application of the method to four different systems of increasing complexity shows that the fitting accuracy of the method is good to excellent. For some cases, it exceeds that available by other methods currently in literature. The method is illustrated by fitting large databases of ab initio energies for Si(n) (n=3,4,...,7) clusters obtained from density functional theory calculations and for vinyl bromide (C(2)H(3)Br) and all products for dissociation into six open reaction channels (12 if the reverse reactions are counted as separate open channels) that include C-H and C-Br bond scissions, three-center HBr dissociation, and three-center H(2) dissociation. The vinyl bromide database comprises the ab initio energies of 71 969 configurations computed at MP4(SDQ) level with a 6-31G(d,p) basis set for the carbon and hydrogen atoms and Huzinaga's (4333/433/4) basis set augmented with split outer s and p orbitals (43321/4321/4) and a polarization f orbital with an exponent of 0.5 for the bromine atom. It is found that an expansion truncated after the three-body terms is sufficient to fit the Si(5) system with a mean absolute testing set error of 5.693x10(-4) eV. Expansions truncated after the four-body terms for Si(n) (n=3,4,5) and Si(n) (n=3,4,...,7) provide fits whose mean absolute testing set errors are 0.0056 and 0.0212 eV, respectively. For vinyl bromide, a many-body expansion truncated after the four-body terms provides fitting accuracy with mean absolute testing set errors that range between 0.0782 and 0.0808 eV. These errors correspond to mean percent errors that fall in the range 0.98%-1.01%. Our best result using the present method truncated after the four-body summation with 16 NNs yields a testing set error that is 20.3% higher than that obtained using a 15-dimensional (15-140-1) NN to fit the vinyl bromide database. This appears to be the price of the added simplicity of the many-body expansion procedure.

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A generalized method that permits the parameters of an arbitrary empirical potential to be efficiently and accurately fitted to a database is presented. The method permits the values of a subset of the potential parameters to be considered as general functions of the internal coordinates that define the instantaneous configuration of the system. The parameters in this subset are computed by a generalized neural network (NN) with one or more hidden layers and an input vector with at least 3n-6 elements, where n is the number of atoms in the system. The Levenberg-Marquardt algorithm is employed to efficiently affect the optimization of the weights and biases of the NN as well as all other potential parameters being treated as constants rather than as functions of the input coordinates. In order to effect this minimization, the usual Jacobian employed in NN operations is modified to include the Jacobian of the computed errors with respect to the parameters of the potential function. The total Jacobian employed in each epoch of minimization is the concatenation of two Jacobians, one containing derivatives of the errors with respect to the weights and biases of the network, and the other with respect to the constant parameters of the potential function. The method provides three principal advantages. First, it obviates the problem of selecting the form of the functional dependence of the parameters upon the system's coordinates by employing a NN. If this network contains a sufficient number of neurons, it will automatically find something close to the best functional form. This is the case since Hornik et al., [Neural Networks 2, 359 (1989)] have shown that two-layer NNs with sigmoid transfer functions in the first hidden layer and linear functions in the output layer are universal approximators for analytic functions. Second, the entire fitting procedure is automated so that excellent fits are obtained rapidly with little human effort. Third, the method provides a procedure to avoid local minima in the multidimensional parameter hyperspace. As an illustrative example, the general method has been applied to the specific case of fitting the ab initio energies of Si(5) clusters that are observed in a molecular dynamics (MD) simulation of the machining of a silicon workpiece. The energies of the Si(5) configurations obtained in the MD calculations are computed using the B3LYP procedure with a 6-31G(**) basis set. The final ab initio database, which comprises the density functional theory energies of 10 202 Si(5) clusters, is fitted to an empirical Tersoff potential containing nine adjustable parameters, two of which are allowed to be the functions of the Si(5) configuration. The fitting error averaged over all 10 202 points is 0.0148 eV (1.43 kJ mol(-1)). This result is comparable to the accuracy achieved by more general fitting methods that do not rely on an assumed functional form for the potential surface.

Previous methods proposed for obtaining analytic potential-energy surfaces (PES) from ab initio electronic structure calculations are not self-starting. They generally require that the sampling of configuration space important in the reaction dynamics of the process being investigated be initiated by using chemical intuition or a previously developed semiempirical potential-energy surface. When the system under investigation contains four or more atoms undergoing three- and four-center reactions in addition to bond scission processes, obtaining a sufficiently converged initial sampling can be very difficult due to the extremely large volume of configuration space that is important in the reaction dynamics. It is shown that by combining direct dynamics (DD) with previously reported molecular dynamics (MD), novelty sampling (NS), and neural network (NN) methods, an analytical surface suitable for MD computations for large systems may be obtained. Application of the method to the investigation of N-O bond scission and cis-trans isomerization reactions of HONO followed by comparison of the resulting neural network potential-energy surface to one obtained by using a semiempirical potential to initiate the sampling shows that the two potential surfaces are the same within the fitting accuracy of the surfaces. It is concluded that the combination of direct dynamics, molecular dynamics, novelty sampling, and neural network fitting provides a self-starting, robust, and accurate DD/MD/NS/NN method for the execution of first-principles, ab initio, molecular dynamics studies in systems containing four or more atoms which are undergoing simultaneous two-, three-, and four-center reactions.