Coupling among waveguides plays an important role in photonic integration, while it usually suffers from large wavelength dispersion and structural sensitivity that brings difficulties in broadband and robust photonic chip devices. Here, we report a new strategy of dispersion engineering of coupled waveguides by artificial gauge field (AGF) with curved trajectories, which gives rise to a dispersionless broadband coupler function in high-density silicon waveguides (waveguide pitch <λ/2). It is found that the artificial gauge field can generate an inverse dispersion to compensate for the dispersion of conventional waveguide coupling. As such, the coupling between the waveguides can be stable in a broad bandwidth. Based on this design, we demonstrate compact directional and 3dB couplers that show broadband dispersionless coupling of light with wavelength from 1400 to 1650 nm, which also exhibit robustness to considerably large structural variations ~150 nm (75% structural deviation). Furthermore, using the AGF coupler as the building block, we significantly demonstrate a three-level-cascaded waveguide network to route the broadband light to the desired ports, showing a tremendous advantage over the conventional counterparts. Our work exploits the artificial gauge field to integrated photonics and demonstrates the possibility of massive, broadband, robust and dense photonic integrations.