Neural operators solve inverse problems for constitutive model discovery

2026-07-16

AI Summary

This work addresses the challenge of efficiently inferring constitutive response functions from experimental data, a task traditionally hindered by time-consuming parameter optimization in classical material models. The authors propose two novel frameworks—Physics-Augmented Neural Operator (PANO) and Constitutive Artificial Neural Operator (CANO)—which, for the first time, apply neural operators to solve the inverse problem of constitutive modeling in infinite-dimensional input–output spaces. By encoding inputs via Laplacian eigenfunctions, the approach achieves discretization independence and robustness to noise, while embedding physical constraints in the output layer ensures thermodynamic consistency of the predicted strain energy density function. Requiring only a single forward pass, the model enables near real-time inference of hyperelastic constitutive laws and demonstrates exceptional generalization across unseen geometries, noisy or incomplete data, varying meshes, and different scales.

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.

Problem

• inverse problems
• constitutive model discovery
• neural operators
• hyperelasticity
• material characterization

Innovation

• neural operators
• constitutive modeling
• inverse problems
• physics-informed learning
• Laplacian eigenfunctions