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

arXiv: https://arxiv.org/abs/2509.07579 · HTML: https://mlsanity.com/paper/2509.07579/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 44 references that are in the map.
- First posted on arXiv 9 Sep 2025.

## Builds on (most cited first, 10 of 10)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape](https://mlsanity.com/paper/1907.02911/) (2019)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [Variational Physics Informed Neural Networks: the role of quadratures and test functions](https://mlsanity.com/paper/2109.02035/) (2021)

## Built on by (most cited first, 2 of 2)

- [Physics-informed Polynomial Chaos Expansion with Enhanced Constrained Optimization Solver and D-optimal Sampling](https://mlsanity.com/paper/2512.10873/) (2025)
- [Nonlinear elliptic homogenization with the parametric Deep Ritz method](https://mlsanity.com/paper/2609.05778/) (2026)

## Abstract (arXiv)

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.