Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions.

Nonlocal continuum mechanics models, including peridynamics, have emerged as powerful tools for describing the mechanical responses of complex nonlinear materials. In typical applications of peridynamics, the functional form of the material model is prescribed in advance, based on the analyst's preferences and insight, creating the need for time-consuming calibration and validation for the particular material at hand. Although data-driven methods were proposed to streamline the modeling process, the well-posedness of these learned peridynamic models is generally not guaranteed, which creates the possibility of non-physical solutions in downstream simulation tasks. In this study, we address this challenge of developing an accurate data-driven model with known uniqueness properties. To do this, we introduce the monotone peridynamic neural operator (MPNO), a novel approach for learning a data-driven nonlocal constitutive model with guaranteed well-posedness for certain classes of problems. Our approach learns a nonlocal kernel together with a nonlinear constitutive relation, while ensuring solution uniqueness through a monotone gradient network. This architectural constraint on the gradient induces the convexity of the learnt energy density function. This guarantees the uniqueness of solutions in the small deformation regime. To validate our approach, we evaluate MPNO's performance on both synthetic and real-world datasets. On synthetic datasets generated with a manufactured kernel and constitutive relation, we show, both theoretically and numerically, that the learnt model converges to the ground-truth as the measurement grid size decreases. Additionally, our MPNO exhibits superior generalization capabilities in comparison with conventional neural networks. It yields smaller displacement solution errors in down-stream tasks with unseen loadings that are outside of the distribution of training samples. Finally, we showcase the practical utility of our approach through applications in learning a homogenized model from molecular dynamics data, highlighting the model's expressivity and physical interpretability in real-world scenarios.