Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses

TL;DR

The paper introduces the Peridynamic Neural Operator (PNO), a data-driven nonlocal constitutive model that learns ordinary state-based peridynamic bond laws from full-field displacement and loading data. By factoring the nonlocal constitutive law into neural networks for the kernel $\omega^{NN}$ and the scalar state $\sigma^{NN}$, the PNO yields a physically consistent forward model that preserves momentum balance and objectivity while remaining resolution- and discretization-independent. Across synthetic (MD graphene, anisotropic hyperelastic FEM) and real (DIC-based tissue) datasets, PNO achieves higher accuracy than traditional constitutive models, demonstrates robustness to noise, and generalizes to different geometries and loadings. The work provides a scalable, physics-informed pathway to learn complex material behavior directly from spatial measurements, with potential extensions to heterogeneity and fracture.

Abstract

Neural operators, which can act as implicit solution operators of hidden governing equations, have recently become popular tools for learning the responses of complex real-world physical systems. Nevertheless, most neural operator applications have thus far been data-driven and neglect the intrinsic preservation of fundamental physical laws in data. In this work, we introduce a novel integral neural operator architecture called the Peridynamic Neural Operator (PNO) that learns a nonlocal constitutive law from data. This neural operator provides a forward model in the form of state-based peridynamics, with objectivity and momentum balance laws automatically guaranteed. As applications, we demonstrate the expressivity and efficacy of our model in learning complex material behaviors from both synthetic and experimental data sets. We show that, owing to its ability to capture complex responses, our learned neural operator achieves improved accuracy and efficiency compared to baseline models that use predefined constitutive laws. Moreover, by preserving the essential physical laws within the neural network architecture, the PNO is robust in treating noisy data. The method shows generalizability to different domain configurations, external loadings, and discretizations.

Figures (14)

• Figure 1: Demonstration of typical training, validation, and testing samples from MD data sets in Example I: Graphene. Colors show the $u_1$ displacement component.
• Figure 2: Demonstration of data generation for Example I: Graphene. Left: hexagongal atomic lattice in MD. Center: full MD mesh. Right: 21x21 nodes coarse-grained nodes. Colors within the small circle show coarse-grained bond forces on the node at the center of the circle.
• Figure 3: The body force and the corresponding displacement field components for a training and a validation sample of the coarse-grained graphene data set.
• Figure 4: PNO prediction results of Example I: Graphene. Comparison of the displacement field (given body force and boundary conditions) and the prediction of body force (given displacement field) versus the corresponding ground truth for a test sample.
• Figure 5: PNO prediction results of Example I: Graphene. Left: $L^2$-errors of the proposed PNO model, in comparison with the optimal LPS model you2022data. Right: The learned kernel function $\omega^{NN}(|{\boldsymbol\xi}|/\delta)$ of the PNO model.