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

2025-05-02

AI Summary

Data-driven constitutive modeling often yields non-physical, non-unique, or numerically unstable solutions due to the absence of physical constraints. To address this, this paper proposes a data-driven nonlocal constitutive learning framework that guarantees solution uniqueness. Its core contributions are: (1) incorporation of a monotonic gradient network constraint to theoretically enforce convexity of the energy density function, ensuring conditional uniqueness of solutions under small deformations; and (2) joint learning of the nonlocal kernel and nonlinear constitutive relation within a neural operator architecture, integrating monotonicity regularization, nonlocal integral modeling, and energy-consistency constraints. Experiments demonstrate that the model converges to ground-truth constitutive laws on synthetic data; achieves significantly lower displacement prediction errors under unseen loading conditions compared to standard neural networks; and successfully learns robust homogenized constitutive models directly from molecular dynamics data.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

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.

Problem

• Ensuring solution uniqueness in data-driven nonlinear material models
• Improving generalization across different boundary conditions and domains
• Learning nonlocal constitutive relations with physical interpretability

Innovation

• Monotone Peridynamic Neural Operator for nonlinear materials
• Ensures solution uniqueness via monotone gradient network
• Learns nonlocal kernel and nonlinear constitutive relation