High-fidelity simulation of turbulent combustion requires solving a species transport equation on a very high-resolution grid. The computational cost of solving this problem scales with the number of species. The computational cost can be prohibitive for many practical fuels with O(1000) species. There are several data-driven dimension reduction techniques that can extract correlated structures between the species during an offline stage. However, these techniques cannot extrapolate to unseen conditions. To this end, we present a reduced-order modeling framework, in which the correlated structures are extracted directly from the species transport equation –– bypassing the need to generate data. These structures are exploited by building on-the-fly reduced-order models (ROM). The correlated structures are represented by a set of time-dependent orthonormal bases and their evolution is prescribed by the physics of the problem. We present several demonstration cases including reduced-order modeling of reactive species transport equation in turbulent combustion as well as sensitivity analysis and uncertainty quantification in fluid dynamics problems.