Despite their elegant formulation and lightweight memory cost, neural ordinary differential equations (NODEs) suffer from known representational limitations. In particular, the single flow learned by NODEs cannot express all homeomorphisms from a given data space to itself, and their static weight parametrization restricts the type of functions they can learn compared to discrete architectures with layer-dependent weights. Here, we describe a new module called neurally-controlled ODE (N-CODE) designed to improve the expressivity of NODEs. The parameters of N-CODE modules are dynamic variables governed by a trainable map from initial or current activation state, resulting in forms of open-loop and closed-loop control, respectively. A single module is sufficient for learning a distribution on non-autonomous flows that adaptively drive neural representations. We provide theoretical and empirical evidence that N-CODE circumvents limitations of previous models and show how increased model expressivity manifests in several domains. In supervised learning, we demonstrate that our framework achieves better performance than NODEs as measured by both training speed and testing accuracy. In unsupervised learning, we apply this control perspective to an image Autoencoder endowed with a latent transformation flow, greatly improving representational power over a vanilla model and leading to state-of-the-art image reconstruction on CIFAR-10.

In many reinforcement learning problems, parameters of the model may vary with its phase while the agent attempts to learn through its interaction with the environment. For example, an autonomous car's reward on selecting a path may depend on traffic conditions at the time of the day or the transition dynamics of a drone may depend on the current wind direction. Many such processes exhibit a cyclic phase-structure and could be represented with a control policy parameterized over a circular or cyclic phase space. Attempting to model such phase variations with a standard data-driven approach (e.g. deep networks) without explicitly modeling the phase of the model can be challenging. Ambiguities may arise as the optimal action for a given state can vary depending on the phase. To better model cyclic environments, we propose phase-parameterized policies and value function approximators that explicitly enforce a cyclic structure to the policy or value space. We apply our phase-parameterized reinforcement learning approach to both feed-forward and recurrent deep networks in the context of trajectory optimization and locomotion problems. Our experiments show that our proposed approach has superior modeling performance than traditional function approximators in cyclic environments.