The classical reaction dynamics of a four-body, bimolecular reaction on a neural network (NN) potential-energy surface (PES) fitted to a database obtained solely from ab initio MP2/6-311G(d,p) calculations are reported, The present work represents the first reported application of ab initio NN methods to a four-body, bimolecular, gas-phase reaction where bond extensions reach 8.1 angstrom for the BeH + H-2 -> BeH2 + H reaction. A modified, iterative novelty sampling method is used to select data points based on classical trajectories computed on temporary NN surfaces. After seven iterations, the sampling process is found to converge after selecting 9604 configurations. Incorporation of symmetry increases this to 19 208 BeH3 configurations. The analytic PES for the system is obtained from the ensemble average of a five-member (6-60-1) NN committee. The mean absolute error (MAE) for the committee is 0.0046 eV (0.44 kJ mol(-1)). The total energy range of the BeH3 database is 147.0 kJ mol(-1). Therefore, this MAE represents a percent energy error of 0.30%. Since it is the gradient of the PES that constitutes the most important quantity in molecular dynamics simulations, the paper also reports mean absolute error for the gradient. This result is 0.026 eV angstrom(-1) (2.51 kJ mol(-1) angstrom(-1)). Since the gradient magnitudes span a range of 15.32 eV angstrom(-1) over the configuration space tested, this mean absolute gradient error represents a percent error of 0.17%. The mean percent absolute relative gradient error is 4.67%. The classically computed reaction cross sections generally increase with total energy. They vary from 0.007 to 0.030 angstrom(2) when H-2 is at ground. state, and from 0.05 to 0.10 angstrom(2) when H-2 is in the first excited state. Trajectory integration is very fast using the five-member NN PES. The average trajectory integration time is 1.07 s on a CPU with a clock speed of 2.4 GHz. Zero angular momentum collisions are also investigated and compared with previously reported quantum dynamics on the same system. The quantum reaction probabilities exhibit pronounced resonance effects that are absent in the classical calculations. The magnitudes of quantum and classical results are in fair accord with the classical results being about 30-40% higher due to the lack of quantum restrictions on the zero-point vibrational energy.