Monotonicity constraints in neural networks: A survey and future directions

Monotonicity constraints are important in many machine learning applications where domain knowledge indicates that predictions should follow monotonic relationships with respect to certain input features. Incorporating monotonicity constraints into neural networks has attracted increasing attention. However, existing neural networks differ significantly in how they implement monotonicity and lack a systematic analysis of monotonicity design. This paper provides a comprehensive study of monotonicity constraints in neural networks. First, we formalize monotonicity and clarify its relationship to related concepts. Based on this, we propose a unified taxonomy that organizes existing methods into two categories: structure-based monotonicity approaches, which guarantee monotonicity through architectural design and parameter constraints, and non-structural monotonicity approaches, which induce, verify, or certify monotonicity without encoding it directly into the architecture. Second, we conduct a systematic empirical comparison of representative methods across 12 real-world tabular datasets using a unified evaluation protocol. The results show that carefully designed structure-based approaches can achieve predictive performance comparable to unconstrained neural networks while guaranteeing monotonic behavior, whereas naive constraint implementations often suffer from limited expressive capacity. Non-structural approaches provide greater flexibility and can adapt to existing architectures, but their effectiveness and guarantee strength depend strongly on the specific mechanism. Overall, this work offers a unified perspective on designing monotonicity constraints in neural networks. The source code and datasets are available at https://github.com/junpeng-sun/MonNN.