High-rate systems are defined as dynamic systems experiencing high-rate and high-amplitude events. Examples include hypersonic vehicles and active impact mitigation strategies. The advanced operation of these mechanisms can only be achieved through control and feedback systems capable of operating in the sub-millisecond range, thus necessitating tight performance constraints. Additionally, high-rate systems are highly nonlinear and nonstationary, for which traditional real-time inference methods are incapable of providing credible predictions. Topological data analysis is gaining popularity for classifying complex time series. Its integration with architected machine learning algorithms shows promise in advancing the predictive capabilities for high-rate systems. However, topological data analysis is computationally expensive and cannot be applied in the sub-millisecond range. This project will investigate real-time topological data analysis capabilities by developing and integrating advances in mathematical, software, and hardware foundations. Successful completion of this project will yield theoretical foundations enabling the integration of topological data analysis with machine learning for modeling and forecasting time series, constituting a major leap from the pure algebraic topological approach. It is envisioned that the developed foundations, along with software and hardware artifacts, will find applications in supercomputing, high-speed data storage, connected vehicles, financial fraud mitigation, cyber-security, deep-fake detection, active blast shielding, and hypersonic vehicles. This project will broaden participation in computing by training multiple undergraduate and graduate students through a well-structured research and education plan that leverages existing programs and partnerships at the three partnering universities, including an undergraduate historically black college.