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Physics Informed Neural Networks for Learning the Horizon Size in Bond-Based Peridynamic Models

Fabio V. Difonzo, Luciano Lopez, Sabrina F. Pellegrino

TL;DR

This work tackles the inverse problem of identifying the horizon size $\delta$ in bond-based peridynamics by employing Physics-Informed Neural Networks (PINNs) to learn $\delta$ across 1D and 2D domains for multiple kernel shapes including Gauss-type, V-shaped, distributed, and tent kernels. It provides a theoretical analysis showing one-sided convergence of SGD to a global minimum under wide-network assumptions and a $\mu$-PL$^*$ condition, with $\delta^*$ interpreted as an unstable gradient-flow equilibrium. Numerical experiments validate the theory, revealing that convergence to the true horizon depends on initialization and dimension, and demonstrate PINNs' viability for solving nonlocal inverse problems with horizon-parameter learning. The findings offer practical guidance on network width and initialization for horizon inference and contribute to the broader applicability of PINNs in nonlocal, kernel-based models.

Abstract

This paper broaches the peridynamic inverse problem of determining the horizon size of the kernel function in a one-dimensional model of a linear microelastic material. We explore different kernel functions, including V-shaped, distributed, and tent kernels. The paper presents numerical experiments using PINNs to learn the horizon parameter for problems in one and two spatial dimensions. The results demonstrate the effectiveness of PINNs in solving the peridynamic inverse problem, even in the presence of challenging kernel functions. We observe and prove a one-sided convergence behavior of the Stochastic Gradient Descent method towards a global minimum of the loss function, suggesting that the true value of the horizon parameter is an unstable equilibrium point for the PINN's gradient flow dynamics.

Physics Informed Neural Networks for Learning the Horizon Size in Bond-Based Peridynamic Models

TL;DR

This work tackles the inverse problem of identifying the horizon size in bond-based peridynamics by employing Physics-Informed Neural Networks (PINNs) to learn across 1D and 2D domains for multiple kernel shapes including Gauss-type, V-shaped, distributed, and tent kernels. It provides a theoretical analysis showing one-sided convergence of SGD to a global minimum under wide-network assumptions and a -PL condition, with interpreted as an unstable gradient-flow equilibrium. Numerical experiments validate the theory, revealing that convergence to the true horizon depends on initialization and dimension, and demonstrate PINNs' viability for solving nonlocal inverse problems with horizon-parameter learning. The findings offer practical guidance on network width and initialization for horizon inference and contribute to the broader applicability of PINNs in nonlocal, kernel-based models.

Abstract

This paper broaches the peridynamic inverse problem of determining the horizon size of the kernel function in a one-dimensional model of a linear microelastic material. We explore different kernel functions, including V-shaped, distributed, and tent kernels. The paper presents numerical experiments using PINNs to learn the horizon parameter for problems in one and two spatial dimensions. The results demonstrate the effectiveness of PINNs in solving the peridynamic inverse problem, even in the presence of challenging kernel functions. We observe and prove a one-sided convergence behavior of the Stochastic Gradient Descent method towards a global minimum of the loss function, suggesting that the true value of the horizon parameter is an unstable equilibrium point for the PINN's gradient flow dynamics.
Paper Structure (5 sections, 9 theorems, 78 equations, 20 figures)

This paper contains 5 sections, 9 theorems, 78 equations, 20 figures.

Table of Contents

  1. Introduction to the peridynamic inverse problem
  2. Overview on PINNs
  3. One-sided convergence of the horizon learning process
  4. Numerical Experiments
  5. Conclusions

Key Result

Theorem 1.1

Let the initial data in eq:IC be given in $X$ and assume $C\in L^1(\mathbb{R})$. Then the initial-value problem associated with eq:periPDEsupport is locally well-posed with solution in $\mathcal{C}^2(X;[0,T])$, for any $T>0$.

Figures (20)

  • Figure 1: Qualitative behaviors of kernel functions defined in \ref{['eq:VKernel']} with $\lambda=1,\delta=10$, \ref{['eq:UKernel']} with $\lambda=7,\delta=1$ and \ref{['eq:tentKernel']} with $\delta=8$, respectively.
  • Figure 2: PINN structure used in this work, with $L$ layers, $N_l$ neurons per layer, $l=0,\ldots,L$.
  • Figure 3: Parameter learning, loss and gradient evolution for Example \ref{['ex:data2']} starting at $\delta=10.1$; the last graph is in logarithmic scale. The true value for the parameter is $\delta^*=10$. The loss function is the mean squared empirical risk $\mathcal{L}$ in \ref{['eq:PINN_lossFunction']} with constant learning rate.
  • Figure 4: Parameter learning, loss and gradient evolution for Example \ref{['ex:data2']} starting at $\delta=9.9$. The true value for the parameter is $\delta^*=10$. The loss function is the mean squared empirical risk $\mathcal{L}$ in \ref{['eq:PINN_lossFunction']} with constant learning rate.
  • Figure 5: Parameter learning evolution, loss and gradient for Example \ref{['ex:data2']} starting at $\delta=11$. The true value for the parameter is $\delta^*=10$. The loss function is the euclidean norm empirical risk $\mathcal{L}_2$ in \ref{['eq:PINN_lossFunction_norm']} with constant learning rate.
  • ...and 15 more figures

Theorems & Definitions (30)

  • Theorem 1.1: see Emmrich_Puhst_2015
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4: Local $\mu$-Polyak-Łojasiewicz condition LiuEtAl2022
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 20 more