On-chip photonic neural networks (PNN) are emerging as an attractive solution for artificial neural networks due to their high computing density, low energy consumption, and compact size. Matrix-vector multiplication (MVM) plays a key role in on-chip PNN, and can achieve high-speed multiply-accumulate operation. Most current schemes implement MVM by adopting wavelength division multiplexing technology to accumulate the power of different wavelengths together. This requires multiple laser sources. Additionally, both positive and negative domain MVM are inevitable for realizing precise PNNs, but because of the innate limitations of light, effective solutions to perform negative value computing are still inadequate. Here, we propose and demonstrate a PNN accelerator based on mode division multiplexing technology to reduce the use of multi-wavelength lasers. We show that it can satisfactorily tackle real-number-field computing (including positive and negative domains) based on a novel, to our knowledge, transformation mapping approach. As a proof-of-concept, we demonstrate a fabricated accelerator for image convolution and letter pattern detection, achieving a computing density of 1.37TOPS/mm2 under the 22.38 Gbaud modulation rate.