Analysing omics data requires computational methods to effectively handle its complexity and to derive meaningful hypotheses about molecular mechanisms. While data-driven statistical and machine learning methods can identify patterns from omics data across multiple samples, they typically require a large number of samples and they often lack interpretability and alignment with existing biological knowledge. In contrast, knowledge-based network methods integrate molecular data with prior knowledge to provide results that are biologically interpretable, but they lack both a unified mathematical framework, leading to ad-hoc solutions specific to particular data types or prior knowledge, limiting their generalisability, and a common modelling interface for programmatic manipulation, restricting method extensions. Furthermore, existing methods generally cannot perform joint network inference across multiple samples or conditions, which restricts their capacity to capture shared mechanisms, making these methods more sensitive to noise and prone to overfitting. To address these limitations, we introduce CORNETO (Constrained Optimisation for the Recovery of NETworks from Omics), a unified framework for knowledge-driven network inference. CORNETO redefines the joint inference task as a constrained optimisation problem with a penalty that induces structured sparsity, allowing for simultaneous network inference across multiple samples. The framework is highly flexible and supports a wide variety of prior knowledge networks—undirected, directed and signed graphs, as well as hypergraphs—enabling the generalisation and improvement of many network inference methods, despite their seemingly different assumptions. We demonstrate its utility by presenting novel extensions of methods for signalling, metabolism and protein-protein interactions. We show how these new methods improve the performance of traditional techniques on a diverse set of biological tasks using simulated and real data. CORNETO is available as an open-source Python package (github.com/saezlab/corneto), facilitating researchers in extending, reusing, and harmonising methods for network inference.