We experimentally solve the problem of maximizing capacity under a total supply power constraint in a massively parallel submarine cable context, i.e., for a spatially uncoupled system in which fiber Kerr nonlinearity is not a dominant limitation. By using multi-layer neural networks trained with extensive measurement data acquired from a 12-span 744-km optical fiber link as an accurate digital twin of the true optical system, we experimentally maximize fiber capacity with respect to the transmit signal's spectral power distribution based on a gradient-descent algorithm. By observing convergence to approximately the same maximum capacity and power distribution for almost arbitrary initial conditions, we conjecture that the capacity surface is a concave function of the transmit signal power distribution. We then demonstrate that eliminating gain flattening filters (GFFs) from the optical amplifiers results in substantial capacity gains per Watt of electrical supply power compared to a conventional system that contains GFFs.