In this section, we want to give a brief introduction to neural networks (NNs). It is written for readers who are not familiar with neural networks but are curious about how they can be applied to practical problems in chemical reaction dynamics. The field of neural networks covers a very broad area. It is not possible to discuss all types of neural networks. Instead, we will concentrate on the most common neural network architecture, namely, the multilayer perceptron (MLP). We will describe the basics of this architecture, discuss its capabilities, and show how it has been used on several different chemical reaction dynamics problems (for introductions to other types of networks, the reader is referred to References 105-107). For the purposes of this document, we will look at neural networks as function approximators. As shown in Figure 3-1, we have some unknown function that we wish to approximate. We want to adjust the parameters of the network so that it will produce the same response as the unknown function, if the same input is applied to both systems. For our applications, the unknown function may correspond to the relationship between the atomic structure variables and the resulting potential energy and forces. The multilayer perceptron neural network is built up of simple components. We will begin with a single-input neuron, which we will then extend to multiple inputs. We will next stack these neurons together to produce layers. Finally, we will cascade the layers together to form the network. A single-input neuron is shown in Figure 3-2. The scalar input p is multiplied by the scalar weight w to form wp, one of the terms that is sent to the summer. The other input, 1, is multiplied by a bias b and then passed to the summer. The summer output n, often referred to as the net input, goes into a transfer function f, which produces the scalar neuron output a.