Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networks

TL;DR

This work addresses the challenge of reliably homogenizing 2D periodic composites with discontinuous coefficients by embedding a dual formulation into the PINN framework to produce guaranteed upper and lower bounds on the homogenized conductivity $A^*$. It investigates both strong- and weak-form (VPINN) formulations, using a periodic neural-network architecture to enforce boundary conditions and, in the VPINN case, test-function bases (spectral and neural) to handle discontinuities. The key findings show that strong-form PINNs perform well only for smoothly varying materials and can fail catastrophically for sharp interfaces, while VPINNs offer robustness to discontinuities but require careful selection of test bases and may incur higher computational demands; in all cases, the dual bounds provide a reliable diagnostic and error bound for the homogenized parameters. The primal-dual framework thus enhances the reliability and applicability of PINN-based homogenization to micromechanics, with potential extensions to 3D problems and elasticity, where the divergence-free constraint becomes more complex.

Abstract

Physics-informed neural networks (PINNs) have shown promise in solving partial differential equations (PDEs) relevant to multiscale modeling, but they often fail when applied to materials with discontinuous coefficients, such as media with piecewise constant properties. This paper introduces a dual formulation for the PINN framework to improve the reliability of the homogenization of periodic thermo-conductive composites, for both strong and variational (weak) formulations. The dual approach facilitates the derivation of guaranteed upper and lower error bounds, enabling more robust detection of PINN failure. We compare standard PINNs applied to smoothed material approximations with variational PINNs (VPINNs) using both spectral and neural network-based test functions. Our results indicate that while strong-form PINNs may outperform VPINNs in controlled settings, they are sensitive to material discontinuities and may fail without clear diagnostics. In contrast, VPINNs accommodate piecewise constant material parameters directly but require careful selection of test functions to avoid instability. Dual formulation serves as a reliable indicator of convergence quality, and its integration into PINN frameworks enhances their applicability to homogenization problems in micromechanics.

Figures (26)

• Figure 1: Scheme of the primal PINN architecture introduced by JIANG2023115972. The network inputs the microscopic coordinates $[x_1, x_2]$ and outputs the fluctuating temperature $\widetilde{u}_{\rm NN}$. Identical architecture is used for the dual formulation, where the output is the stream function $\widetilde{w}_{\rm NN}$.
• Figure 2: Smooth approximations $\bm{A}_{s,\varepsilon}(\bm{x})$ of the material distribution depending on the parameter $\varepsilon$.
• Figure 3: Benchmark primal and dual solutions for $\bm{\xi}=[1,0]^{\rm{T}}$, obtained with FEM, using discretization into $128 \times 128$ unique DoFs.
• Figure 4: Comparison of the primal and dual estimates (-$\bullet$-) and guaranteed bounds (-$\star$-) of the PINN solutions depending on the number of network parameters and the transition parameter $\varepsilon$ of the material approximation.
• Figure 5: Training loss of the primal and dual solutions and the gap between the estimates, for the PINNs with the transition parameter $\varepsilon=1/30$.