A Mathematical Certification for Positivity Conditions in Neural Networks With Applications to Partial Monotonicity and Trustworthy AI

A. Polo-Molina, D. Alfaya, J. Portela

IEEE Transactions on Neural Networks and Learning Systems Vol. 37, nº. 2, pp. 981 - 996

Resumen:

Artificial neural networks (ANNs) have become a powerful tool for modeling complex relationships in large-scale datasets. However, their closed box nature poses trustworthiness challenges. In certain situations, ensuring trust in predictions might require following specific partial monotonicity constraints. However, certifying if an already-trained ANN is partially monotonic is challenging. Therefore, ANNs are often disregarded in some critical applications, such as credit scoring, where partial monotonicity is required. To address this challenge, this article presents a novel algorithm (LipVor) that certifies if a closed box model, such as an ANN, is positive based on a finite number of evaluations. Consequently, since partial monotonicity can be expressed as a positivity condition on partial derivatives, LipVor can certify whether an ANN is partially monotonic. To do so, for every positively evaluated point, the Lipschitzianity of the closed box model is used to construct a specific neighborhood, where the function remains positive. Next, based on the Voronoi diagram of the evaluated points, a sufficient condition is stated to certify if the function is positive in the domain. Unlike prior methods, our approach certifies partial monotonicity without constrained architectures or piecewise linear activations. Therefore, LipVor could open up the possibility of using unconstrained ANN in some critical fields. Moreover, some other properties of an ANN, such as convexity, can be posed as positivity conditions, and therefore, LipVor could also be applied.

Resumen divulgativo:

El paper presenta el algoritmo LipVor, que revoluciona la confianza en la inteligencia artificial. Permite certificar matemáticamente si una red neuronal cumple requisitos de fiabilidad, como la monotonía. Este avance abre la puerta al uso seguro de IA en sectores críticos como el financiero.

Palabras Clave: Artificial neural networks (ANNs), mathematical certification, partial monotonicity, trustworthy AI

Índice de impacto JCR-JIF y cuartil WoS: 9,700 - Q1 (2025)

Referencia DOI: https://doi.org/10.1109/TNNLS.2025.3614431

Publicado en papel: Febrero 2026.

Publicado on-line: Octubre 2025.

Cita:
A. Polo-Molina, D. Alfaya, J. Portela, "A Mathematical Certification for Positivity Conditions in Neural Networks With Applications to Partial Monotonicity and Trustworthy AI", IEEE Transactions on Neural Networks and Learning Systems, Vol. 37, nº. 2, pp. 981 - 996, Febrero 2026. [Online: Octubre 2025] doi: 10.1109/TNNLS.2025.3614431