Neural operators solve inverse problems for constitutive model discovery
Computational Engineering, Finance, and Science
Summary
The authors developed two types of neural networks, called Physics-Augmented Neural Operators (PANO) and Constitutive Artificial Neural Operators (CANO), to quickly figure out the material properties from experimental data. Instead of slow trial-and-error methods, their networks learn to directly connect displacement and force measurements to a material’s strain-energy function, which tells us how it responds to stress. They use a mathematical tool called Laplacian eigenfunctions to handle data in a way that is flexible and resistant to noise. After training on simulated data, these networks can predict material behavior instantly, even when tested on new, noisy, or incomplete data from different sample sizes. This method aims to make material characterization more efficient by bypassing traditional optimization steps.
Authors
Moritz Flaschel, Burigede Liu, Ellen Kuhl
Abstract
Characterizing the mechanical response of materials traditionally requires solving optimization problems in which model parameters are calibrated or trained to minimize the discrepancy between model predictions and experimental data. This process can be computationally expensive and time-consuming. To overcome this limitation, we propose two neural operator architectures that directly map experimentally measured data to the constitutive functions governing the mechanical response of the material: Physics-Augmented Neural Operators (PANO) and Constitutive Artificial Neural Operators (CANO). The proposed neural operators approximate the mapping between the infinite-dimensional input space of full-field displacement measurements and net reaction forces, and the infinite-dimensional output space of hyperelastic strain-energy density functions. The displacement fields are encoded through Laplacian eigenfunctions to obtain discretization-independent and noise-robust predictions. Our framework constrains the output space to physically admissible material models that satisfy fundamental physical requirements by design. The neural operators are trained on simulated data tuples of displacement fields and reaction forces for a range of material models. Once trained, the neural operators enable near-instantaneous material characterization and require only a single forward pass to infer the strain-energy density function from a given experimental dataset. We test the predictive power of the neural operators for unseen data, noisy data, data with missing information, data from different spatial discretizations, and data from geometries of different sizes.