A nonlocal physics-informed deep learning framework using the peridynamic differential operator

1. Introduction

• PINNs use feed-forward networks with space and time inputs, enforce governing equations through automatic differentiation, and support PDE solution and parameter identification.The introduction situates this framework among applications to computational science and engineering.Source: §1 · p. 2
• Existing PINNs can lose accuracy around sharp stress or strain gradients, including those associated with mixed displacement–traction boundaries.The proposed remedy is to add long-range interactions alongside local space and time inputs.Source: §1 · p. 2
• PDDO provides nonlocal interactions and replaces spatial derivatives in governing equations with nonlocal representations.Its functions are generated in discrete form during preprocessing, so the authors say they do not degrade deep-learning performance.Source: §1 · p. 2
• The paper applies local and nonlocal PINNs to rigid-punch indentation, an example chosen to illustrate sharp gradients from mixed boundary conditions.The paper proceeds from the established PINN framework to the PDDO extension and then the indentation example.Source: §1 · p. 2

2.1. Basics of the PINN framework

• A PINN represents the approximate solution with a feed-forward network and incorporates PDE residuals and training data in its loss.Inputs may be spatial and temporal coordinates; automatic differentiation evaluates the PDE derivatives, while training uses iterative optimization.Source: §2.1 · p. 3
• Training may stop when epochs no longer improve the loss, and the original random-location, one-epoch strategy may not suit fixed, limited sensor locations.The paper uses SciANN, a high-level Keras wrapper, for PINN implementation.Source: §2.1 · p. 4

2.2. Solid mechanics with elastoplastic deformation

• The solid-mechanics formulation adopts von Mises elastoplasticity with a yield surface and separates deviatoric strain into elastic and plastic components.Under plane strain, εzz remains zero; for elastoplasticity, σzz is not predefined.Source: §2.2 · p. 5

2.3. Local PINN for elastoplasticity

• For local elastoplastic PINNs, coordinates are inputs and displacement, strain, and stress components are outputs; the loss measures field-data errors, equilibrium, and constitutive relations.The same loss framework supports PDE solution and parameter identification, with material parameters trained as variables during identification.Source: §2.3 · pp. 5–6

• The nonlocal PINN replaces single-point input with x’s family Hx, whose interaction extent is set by horizon δx and weighted by relative position.Source: §3 · pp. 6–7
• PDDO represents fields and their derivatives through interactions over Hx, yielding integral expressions for arbitrary-order derivatives that can be evaluated by numerical quadrature.The paper reports accurate derivative evaluation in the domain interior and near its boundaries.Source: §3 · p. 7
• The network maps x and its family points to corresponding field values, from which nonlocal f(x) and its derivatives are constructed.Source: §3 · p. 8
• As the influence horizon δx approaches zero, the nonlocal architecture recovers the local PINN architecture.Source: §3 · p. 8
• AD-PDDO-PINN uses PDDO for the function and automatic differentiation for derivatives, whereas PDDO-PINN uses PDDO for both.Source: §3 · pp. 8–9
• PDDO-PINN permits ReLU and architectures unavailable to local PINN, and may improve per-epoch efficiency by avoiding extended automatic-differentiation graph computations.Source: §3 · p. 9

4.1. Problem description

• The plane-strain test indents a 1 m square elastoplastic body with a 0.2 m rigid punch and 1 mm prescribed displacement, combining displacement and traction boundary conditions.The setup specifies E = 70 GPa, ν = 0.3, σY 0 = 0.1 GPa, and H0 = 0.5 GPa.Source: §4 · p. 9, §4.1 · p. 9, Figure 4 · p. 10
• FEM reference fields show plastic deformation localized beneath the punch, with plastic-strain components mostly zero except near punch corners, where sharp gradients occur.The synthetic data are generated using COMSOL on a uniform 100 × 100 element mesh.Source: §4.1 · p. 10, Figure 5 · p. 10, Figure 6 · p. 11

4.2. Local PINN results

• For linear elasticity, local PINN captures high-gradient regions near punch corners but diffuses them; its λ = 40.2 GPa and µ = 26.2 GPa estimates have less than 3% error.The prediction differences and parameter estimates are shown in Figs. 7 and 8.Source: §4.2 · p. 11, Figure 7 · p. 12, Figure 8 · p. 12
• For elastoplastic deformation, local PINN has poor quantitative agreement and estimates Hp = 0.00 GPa, failing to identify the hardening parameter.The authors speculate that localized plastic deformation leaves few sampling points contributing to the loss function.Source: §4.2 · p. 12, Figure 9 · p. 13, Figure 10 · p. 13

4.3. Nonlocal PINN results

• The nonlocal PINN variants use long-range family points as input; the second-order Taylor-series expansion yields up to 49 family members, and AD-PDDO-PINN differentiates with respect to each member.The nonlocal networks approximate displacement and stress fields for the indentation problem; local and nonlocal network architectures are identical.Source: §4.3 · p. 13, §4.3.1 · p. 14

4.3.1. AD-PDDO-PINN

• AD-PDDO-PINN more accurately captures the elastoplastic solution, including high-gradient regions, and estimates λ = 40.4 GPa, µ = 26.9 GPa, σY 0 = 0.10 GPa, and Hp = 1.03 GPa.It identifies the hardening parameter accurately, unlike local PINN.Source: Figure 11 · p. 14, §4.3.1 · p. 14, Figure 12 · p. 15

4.3.2. PDDO-PINN

• PDDO-PINN outperforms local PINN but is less accurate than AD-PDDO-PINN; its estimates are λ = 40.3 GPa, µ = 26.9 GPa, σY 0 = 0.0999 GPa, and Hp = 1.25 GPa.It avoids automatic differentiation, making derivative evaluation through Eq. (4.5) faster per training epoch.Source: Figure 13 · p. 15, §4.3.2 · p. 15, Figure 14 · p. 16

5. Discussion and Conclusions

• The nonlocal approaches converge faster and reach a lower final L/L0 than local PINN in linear-elastic and nonlinear elastoplastic cases.Figure 15 plots normalized loss against epochs and shows learning rate on a second axis.Source: Figure 15 · p. 16, §5 · p. 16
• PDDO-PINN is introduced in automatic-differentiation and analytical-derivative versions, and recovers local PINN as the interaction range δx approaches zero.The PD functions can be incorporated into the network without degrading modern deep-learning performance.Source: §5 · p. 16
• Nonlocal PINN captures stress and strain concentrations with global functions and outperforms local PINN in solution accuracy and estimated model parameters.The indentation problem has mixed displacement–traction boundary conditions that produce localized deformation and sharp gradients.Source: §5 · p. 17
• The authors identify network size, PDDO approximation order, and training optimization as unresolved choices for the approach.They suggest nonlocal PINN may support simulation and discovery of PDEs whose solutions develop sharp gradients.Source: §5 · p. 17

Appendix A. PDDO Derivation

• Each point may have a differently positioned, sized, or shaped interaction family, and its weight function specifies interaction strength.Decreasing family size makes interactions more local; family size and shape affect the degree of nonlocality.Source: Appendix A · p. 17
• Orthogonality conditions applied to the Taylor-series expansion yield nonlocal representations of a function and its derivatives.The appendix identifies the expansion as second order in two dimensions.Source: Appendix A · pp. 17–18
• PD functions are built as linear combinations of polynomial basis functions, with their coefficients determined through an algebraic system.After solving a = A−1b, the PD functions can be constructed.Source: Appendix A · pp. 18–19
• The PDDO recovers local differentiation as the horizon size approaches zero.The derivation multiplies expansion terms by PD functions and integrates over the interaction family.Source: Appendix A · pp. 17, 19