The objective of this project was to develop new methods for rapidly computing accurate bounds on the solutions of nonlinear ordinary differential equations (ODEs) subject to bounded uncertainties. Toward this end, our key insight was that the conservatism of fast interval methods can be dramatically reduced through the use of model redundancy. Specifically, prior work showed that bounds produced by interval methods often enclose large regions of state-space that violate redundant relations implied by the dynamics, such as conservation laws. Furthermore, such relations (known as 'solution invariants') can be exploited to obtain much sharper bounds in many cases. Motivated by these observations, we pursued new bounding approaches for general nonlinear systems based on the deliberate introduction of redundant model equations to reduce conservatism. The work was organized around three major tasks. The most significant accomplishments in each task are summarized below: Task 1: Develop a fast and accurate state bounding algorithm that exploits pre-existing model redundancy. A new bounding theorem was proven that enables the use of nonlinear invariants within fast bounding methods based on differential inequalities (DI) for the first time. An efficient new bounding algorithm was also developed to implement this theory. In aggregate, these advances extend the redundancy-based DI bounding approach to systems satisfying a much more general class of invariants and have enabled efficient computation of very sharp bounds for several test cases. Task 2: Develop a theoretical framework for the introduction of redundancy into arbitrary dynamic models to effectively reduce conservatism. A framework was developed for introducing solution invariants into arbitrary systems by lifting them into a higher-dimensional state space. Critically, this enables the methods from Task 1 to be applied.