Structure preserving neural networks based on ODEs

Neural networks have gained much interest because of their effectiveness in many applications. However, their mathematical properties are generally not well understood. In the presence of some underlying geometric structure in the data or in the function to approximate, it is often desirable to consider this in the design of the neural network. In this work, we start with a non-autonomous ODE and build neural networks using a suitable, structure-preserving, numerical time-discretisation. The structure of the neural network is then inferred from the properties of the ODE vector field. To support the flexibility of the approach, we go through the derivation of volume-preserving, mass-preserving and Lipschitz constrained neural networks. Finally, a mass-preserving network is applied to the problem of approximating the dynamics of a conservative dynamical system. On the other hand, a Lipschitz constrained network is demonstrated to provide improved adversarial robustness to a CIFAR-10 classifier.