Modern microelectronic processors have migrated towards parallel computing architectures with many-core processors. However, such expansion comes with diminishing returns exacted by the high cost of data movement between individual processors. The use of optical interconnects 1,2 has burgeoned as a promising technology that can address the limits of this data transfer. While recent pushes to enhance optical communication have focused on developing wavelength-division multiplexing technology, this approach will eventually saturate the usable bandwidth, and new dimensions of data transfer will be paramount to fulfill the ever-growing need for speed 3-6 . Here we demonstrate an integrated intra-and inter-chip multi-dimensional communication scheme enabled by photonic inverse design. Using broad-band inverse-designed mode-division multiplexers, we combine wavelength-and mode-multiplexing of data at a rate exceeding terabit-per-second. Crucially, as we take advantage of an orthogonal optical basis, our approach is inherently scalable to a multiplicative enhancement over the current state of the art.