Title: On the Forward Invariance of Neural ODEs. Abstract: To ensure robust and trustworthy decision-making, it is highly desirable to enforce constraints over a neural network's parameters and its inputs automatically by back-propagating output specifications. This way, we can guarantee that the network makes reliable decisions under perturbations. Here, we propose a new method for achieving a class of specification guarantees for neural Ordinary Differentiable Equations (ODEs) by using invariance set propagation. An invariance of a neural ODE is defined as an output specification, such as to satisfy mathematical formulae, physical laws, and system safety. We use control barrier functions to specify the invariance of a neural ODE on the output layer and propagate it back to the input layer. Through the invariance backpropagation, we map output specifications onto constraints on the neural ODE parameters or its input. The satisfaction of the corresponding constraints implies the satisfaction of output specifications. This allows us to achieve output specification guarantees by changing the input or parameters while maximally preserving the model performance. We demonstrate the invariance propagation on a comprehensive series of representation learning tasks, including spiral curve regression, autoregressive modeling of joint physical dynamics, convexity portrait of a function, and safe neural control of collision avoidance for autonomous vehicles.