Physics-informed DeepONet for superelastic strain decomposition: application to Ti–Zr–Nb–Sn alloys

Superelastic β -Ti alloys rely on stress-induced martensitic transformation, yielding history-dependent and microstructure-sensitive responses. Constitutive analysis of metastable β – α’’ systems still often rely on phenomenological models whose internal variables are difficult to calibrate and interpret from macroscopic data alone. This study develops a physics-informed operator-learning framework that treats the martensitic fraction and plastic strain as explicit variables. A Lagoudas-type thermomechanical relation is retained at the macroscopic level, while a Deep Operator Network (DeepONet) is trained to infer their evolution from strain–time histories using a physics-informed loss. Phase-dependent elastic modulus and the maximum transformation strain are guided by in situ X-ray diffraction and cyclic tensile tests on Ti- x Zr-8Nb-2Sn ( x = 40, 45, 50 at.%) alloys. An initial architecture study shows that a physics-informed DeepONet achieves lower global and turning-point stress errors than a fully connected neural network of comparable capacity, but also reveals that a single internal variable tends to absorb both recoverable and irreversible deformation. The constitutive relation is therefore refined to include plastic strain and combined with a constrained double-head DeepONet, which preserves stress-prediction accuracy while producing a smoother, monotonic martensitic fraction and an irrecoverable strain consistent with transformation-induced plasticity. The constrained model reproduces cyclic stress–strain curves and accurately captures turning points, and the inferred strain decomposition shows close agreement with independent stepped tensile-reheating tests. These results indicate that physics-informed operator networks can provide a data-efficient and interpretable route to constitutive modeling of β – α’’ superelastic alloys, compatible with the thermodynamic formulations and lattice-resolved measurements.