Systematically investigating interactions among molecules of the same type across different contexts is crucial for unraveling disease mechanisms and developing potential therapeutic strategies. The "A-A-B" triplet paradigm provides a principled approach to model such context-specific interactions, and leveraging third-order tensor to capture such type ternary relationships is an efficient strategy. However, effectively modeling both multilinear and nonlinear characteristics to accurately identify such triplets using tensor-based methods remains a challenge. In this paper, we propose a novel Convolutional Neural Tensor Completion (ConvNTC) framework that collaboratively learns the multilinear and nonlinear representations to model triplet-based network interactions. ConvNTC consists of a multilinear module and a nonlinear module. The former is a tensor decomposition approach that integrates multiple constraints to learn the tensor factor embeddings. The latter contains three components: an embedding generator to produce position-specific index embeddings for each tensor entry in addition to the factor embeddings, a convolutional encoder to perform nonlinear feature mapping while preserving the tensor's rank-one property, and a Kolmogorov-Arnold Network (KAN) based predictor to effectively capture high-dimensional relationships aligned with the intrinsic structure of real-world data. We evaluate ConvNTC on two types triplet datasets of the "A-A-B" type: miRNA-miRNA-disease and drug-drug-cell. Comprehensive experiments against 11 state-of-the-art methods demonstrate the superiority of ConvNTC in terms of triplet prediction. ConvNTC reveals promising prognostic values of the miRNA-miRNA interactions on breast cancer and detects synergistic drug combinations in cancer cell lines.