We introduce Pulse, a framework for processing continuous queries over models of continuous-time data, which can compactly and accurately represent many real-world activities and processes. Pulse implements several query operators, including filters, aggregates and joins, that work by solving simultaneous equation systems, which in many cases is significantly cheaper than processing a stream of tuples. As such, Pulse translates regular queries to work on continuous-time inputs, to reduce computational overhead and latency while meeting user-specified error bounds on query results. For error bound checking, Pulse uses an approximate query inversion technique that ensures the solver executes infrequently and only in the presence of errors, or no previously known results. We first discuss the high-level design of Pulse, which we fully implemented in a stream processing system. We then characterise Pulse's behavior through experiments with real data, including financial data from the New York Stock Exchange, and spatial data from the Automatic Identification System for tracking naval vessels. Our results verify that Pulse is practical and demonstrates significant performance gains for a variety of workload and query types.