Despite linear models being introduced in the latest Versatile Video Coding (VVC) standard to exploit the correlation among luma and chroma channels for removing redundancy, these models cannot take into account the nonlinearity of components, resulting in degraded intra prediction precision. In this paper, a neural network-based method is proposed for cross-channel chroma intra prediction to enhance the coding efficiency. Specifically, the neighboring reference and co-located samples are separately input into the proposed network to exploit spatial and cross-channel correlations fully. Furthermore, in order to acquire a more compact representation of residual signals, a transform-based loss is employed to enhance the effectiveness of the compression. The proposed method is integrated into VVC, competing with the intrinsic chroma prediction regarding rate-distortion optimization to enhance coding performance further. The extensive experimental results demonstrate the superiority of the proposed method over the VVC test model (VTM) 7.0, achieving average bitrate savings of 2.68% and 2.11% for U and V chroma components, respectively.