Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions
This addresses the issue of non-physical solutions in material modeling for researchers and engineers, though it is incremental as it builds on existing neural operator methods.
The paper tackles the problem of ensuring well-posedness in data-driven constitutive models for nonlinear materials by introducing the monotone peridynamic neural operator (MPNO), which guarantees solution uniqueness and shows superior generalization with smaller displacement errors in downstream tasks.
Data-driven methods have emerged as powerful tools for modeling the responses of complex nonlinear materials directly from experimental measurements. Among these methods, the data-driven constitutive models present advantages in physical interpretability and generalizability across different boundary conditions/domain settings. However, the well-posedness of these learned models is generally not guaranteed a priori, which makes the models prone to non-physical solutions in downstream simulation tasks. In this study, we introduce monotone peridynamic neural operator (MPNO), a novel data-driven nonlocal constitutive model learning approach based on neural operators. 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 gradient induces convexity of the learnt energy density function, thereby guaranteeing solution uniqueness of MPNO in small deformation regimes. To validate our approach, we evaluate MPNO's performance on both synthetic and real-world datasets. On synthetic datasets with manufactured kernel and constitutive relation, we show that the learnt model converges to the ground-truth as the measurement grid size decreases both theoretically and numerically. Additionally, our MPNO exhibits superior generalization capabilities than the conventional neural networks: it yields smaller displacement solution errors in down-stream tasks with new and unseen loadings. Finally, we showcase the practical utility of our approach through applications in learning a homogenized model from molecular dynamics data, highlighting its expressivity and robustness in real-world scenarios.