Convex physics-informed kernel for machine learning in the ultimate capacity assessment of beam sections

We propose a machine-learning framework for the ultimate limit state capacity check of reinforced concrete (RC) sections subject to axial force and biaxial bending. Analytical capacity domains are generated through a fiber-free integration approach and employed as reference data to train Support Vector Machine (SVM) classifiers using polynomial and radial kernels. It is shown that the pure numerical calibration of SVMs fails to capture the convex and compact nature typical of the RC domains and produces surrogate limit state functions exhibiting non-convexity and, sometimes, non-limited regions. Convexity and boundedness are therefore treated as essential mechanical admissibility requirements, rather than as optional numerical regularizations. In order to avoid such issues and to preserve the physical significance of the domain, a convexification and coercivization procedure for polynomial kernels, aiming at enforcing analytical convexity and closure of the decision surfaces, is developed. It consists of building convexified kernels by projecting the associated quadratic form onto the cone of positive semidefinite matrices and reconstructing their quartic term as a convex combination of elementary atoms. The resulting SVMs yield compact, bounded, and mechanically interpretable decision boundaries, offering a physics-informed alternative to standard data-driven surrogates. It is shown that unconstrained kernels may exhibit higher apparent accuracy while generating mechanically inadmissible decision surfaces, thus highlighting the limits of accuracy-based assessment alone in safety-critical applications. The proposed approach is general and extendable to other materials and kernel degrees, thus bridging mechanics and statistical learning toward the development of mechanics-informed classifiers for structural capacity assessment. The proposed framework is specifically conceived as a sectional-level post-processing tool, enabling repeated ultimate limit-state verifications at negligible computational cost once global stress resultants are known.