Classifying patents by their relevance to the UN Sustainable Development Goals (SDGs) is crucial for tracking how innovation addresses global challenges. However, the absence of a large, labeled dataset limits the use of supervised learning. Existing methods, such as keyword searches, transfer learning, and citation-based heuristics, lack scalability and generalizability. This paper frames patent-to-SDG classification as a weak supervision problem, using citations from patents to SDG-tagged scientific publications (NPL citations) as a noisy initial signal. To address its sparsity and noise, we develop a composite labeling function (LF) that uses large language models (LLMs) to extract structured concepts, namely functions, solutions, and applications, from patents and SDG papers based on a patent ontology. Cross-domain similarity scores are computed and combined using a rank-based retrieval approach. The LF is calibrated via a custom positive-only loss that aligns with known NPL-SDG links without penalizing discovery of new SDG associations. The result is a silver-standard, soft multi-label dataset mapping patents to SDGs, enabling the training of effective multi-label regression models. We validate our approach through two complementary strategies: (1) internal validation against held-out NPL-based labels, where our method outperforms several baselines including transformer-based models, and zero-shot LLM; and (2) external validation using network modularity in patent citation, co-inventor, and co-applicant graphs, where our labels reveal greater thematic, cognitive, and organizational coherence than traditional technological classifications. These results show that weak supervision and semantic alignment can enhance SDG classification at scale.