Complex systems often involve higher-order interactions which require us to go beyond their description in terms of pairwise networks. Triadic interactions are a fundamental type of higher-order interaction that occurs when one node regulates the interaction between two other nodes. Triadic interactions are found in a large variety of biological systems, from neuron-glia interactions to gene-regulation and ecosystems. However, triadic interactions have so far been mostly neglected. In this article, we propose {the Triadic Perceptron Model (TPM)} that demonstrates that triadic interactions can modulate the mutual information between the dynamical state of two linked nodes. Leveraging this result, we formulate the Triadic Interaction Mining (TRIM) algorithm to extract triadic interactions from node metadata, and we apply this framework to gene expression data, finding new candidates for triadic interactions relevant for Acute Myeloid Leukemia. Our work reveals important aspects of higher-order triadic interactions that are often ignored, yet can transform our understanding of complex systems and be applied to a large variety of systems ranging from biology to climate.