Wilson's numerical renormalization group (NRG) can be seen as a transformation preserving a low-energy spectrum between two matrix representations of the Hamiltonian. White's density matrix renormalization group (DMRG) can be seen as a transformation that preserves ground-state-observable expectations between matrix representations of relevant operators. Generally, the renormalization group can be seen as a transformation (approximately) preserving properties between two representations. We explore such generalization and propose the operator learning renormalization group (OLRG). OLRG is a generalization of the NRG and DMRG frameworks for simulating the quantum many-body system. It recursively builds a simulatable system to approximate a target system of the same number of sites via operator maps. OLRG uses a loss function to minimize the error of a target property directly by learning the operator map in lieu of a state ansatz. Using operator maps opens the opportunity to optimize more flexible deep learning, tensor network, and parameterized quantum circuit models in the DMRG and generative learning style. This loss function is designed by a scaling consistency condition providing a provable bound for real-time evolution. We implement two versions of the operator maps for classical and quantum simulations. The former, called the Operator Matrix Map, can be implemented via neural networks on classical computers. The latter, which we call the Hamiltonian Expression Map, generates device pulse sequences to leverage the capabilities of quantum computing hardware. We illustrate the performance of both maps for calculating time-dependent quantities in the quantum Ising model Hamiltonian.