Biological neural networks operate in the presence of task disruptions as they guide organisms toward goals. A familiar stream of stimulus-response causations can be disrupted by subtask streams imposed by the environment. For example, taking a familiar path to a foraging area might be disrupted by the presence of a predator, necessitating a "detour" to the area. The detour can be a known alternative path that must be dynamically composed with the original path to accomplish the overall task. In this project, overarching base paths are disrupted by independently learned path modules in the form of insertion, substitution, and deletion modifications to the base paths such that the resulting modified paths are novel to the network. The network's performance is then tested on these paths that have been learned in piecemeal fashion. In sum, the network must compose a new task on the fly. Several network architectures are tested: Time delay neural network (TDNN), Long short-term memory (LSTM), Temporal convolutional network (TCN), and Morphognosis, a hierarchical neural network. LSTM and Morphognosis perform significantly better for this task.

We prove two universal approximation theorems for a range of dropout neural networks. These are feed-forward neural networks in which each edge is given a random $\{0,1\}$-valued filter, that have two modes of operation: in the first each edge output is multiplied by its random filter, resulting in a random output, while in the second each edge output is multiplied by the expectation of its filter, leading to a deterministic output. It is common to use the random mode during training and the deterministic mode during testing and prediction. Both theorems are of the following form: Given a function to approximate and a threshold $\varepsilon>0$, there exists a dropout network that is $\varepsilon$-close in probability and in $L^q$. The first theorem applies to dropout networks in the random mode. It assumes little on the activation function, applies to a wide class of networks, and can even be applied to approximation schemes other than neural networks. The core is an algebraic property that shows that deterministic networks can be exactly matched in expectation by random networks. The second theorem makes stronger assumptions and gives a stronger result. Given a function to approximate, it provides existence of a network that approximates in both modes simultaneously. Proof components are a recursive replacement of edges by independent copies, and a special first-layer replacement that couples the resulting larger network to the input. The functions to be approximated are assumed to be elements of general normed spaces, and the approximations are measured in the corresponding norms. The networks are constructed explicitly. Because of the different methods of proof, the two results give independent insight into the approximation properties of random dropout networks. With this, we establish that dropout neural networks broadly satisfy a universal-approximation property.