  On-chip optical neural networks (ONNs) have recently emerged as an attractive
hardware accelerator for deep learning applications, characterized by high
computing density, low latency, and compact size. As these networks rely
heavily on massive matrix multiplication, photonic computing cores for matrix
computation become crucial components for on-chip ONNs, which harness the
degree of freedoms (DOFs) in photonics including space, wavelength and mode
dimensions. However, previous photonic computing devices have not fully
utilized the orthogonality and the conversion characteristic of the waveguide
modes, which as we show here, allows for the simultaneous parallel computing of
several independent matrix-vector multiplications within the same device. In
this work, we propose an inverse-designed photonic computing core for parallel
matrix-vector multiplication. The matrices are implemented through a mode
conversion process, where the input fundamental modes are simultaneously
converted into several orthogonal output modes. Specifically, we target the
complex-valued conversion matrices between input and output modes and inversely
design the dielectric distribution within the device to achieve parallel
matrix-vector multiplication. As a demonstration, the proposed photonic
computing core supports simultaneous parallel computing of two independent
matrix-vector multiplications, with an ultra-compact footprint and high
computing precision (relative error < 8%) at 1550 nm wavelength. The
inverse-designed photonic computing devices hold great potential for
high-performance on-chip ONNs with low energy consumption and high computing
density.
