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

TL;DR

This paper introduces a monotone peridynamic neural operator (MPNO) that guarantees solution uniqueness in nonlinear material modeling, improving physical interpretability, generalization, and robustness in data-driven constitutive models.

Contribution

The paper proposes a novel neural operator architecture that enforces solution uniqueness via monotone gradient networks, ensuring convex energy density functions in nonlinear material modeling.

Findings

MPNO converges to ground-truth models as measurement grid size decreases.

MPNO outperforms conventional neural networks in generalization to unseen loadings.

Demonstrated effectiveness in real-world molecular dynamics data applications.

Abstract

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.