Sampling the distribution of bosons that have undergone a random unitary evolution is strongly believed to be a computationally hard problem. Key to outperforming classical simulations of this task is to increase both the number of input photons and the size of the network. We propose driven boson sampling, in which photons are input within the network itself, as a means to approach this goal. When using heralded single-photon sources based on parametric down-conversion, this approach offers an $sim e$-fold enhancement in the input state generation rate over scattershot boson sampling, reaching the scaling limit for such sources. More significantly, this approach offers a dramatic increase in the signal-to-noise ratio with respect to higher-order photon generation from such probabilistic sources, which removes the need for photon number resolution during the heralding process as the size of the system increases.
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