Growing diversity and complexity of on-chip photonic applications requires rapid design of components with stateof-the-art operation metrics. Here, we demonstrate a highly flexible and efficient method for designing several classes of compact and low-loss integrated optical devices. By leveraging a datadriven approach, we represent devices in the form of cascaded eigenmode scattering matrices, through a data-driven eigenmode expansion method. We perform electromagnetic computations using parallel data processing techniques, demonstrating simulation of individual device responses in tens of milliseconds with physical accuracies matching 3D-FDTD. We then couple these simulations with nonlinear optimization algorithms to design silicon-based waveguide tapers, power splitters, and waveguide crossings with state-of-the-art performance and near-lossless operation. These three sets of devices highlight the broad computational efficiency of the design methodology shown, and the applicability of the demonstrated data-driven eigenmode expansion approach to a wide set of photonic design problems.