DeepVFM: Neural Networks discovery of virtual fields for stable inverse constitutive identification of hyperelastic materials

Many scientific measurement problems reduce to recovering unknown parameters of a known governing equation from observed field data. In solid mechanics, this means identifying constitutive material parameters (i.e. stiffness) from full-field displacement measurements. The Virtual Fields Method (VFM) (M. Grediac 2006) is a technique that can leverage full-field data and the principle of virtual work to find these parameters. However, it can be challenging to identify the test functions that comply with the boundary conditions of the problem and satisify the VFM, and the accuracy of the VFM depends on these test functions. In recent years, physics-informed neural networks (Raissi, 2019) and neural operators (Lu, 2021) have gained substantial attention as forward and inverse solvers of partial differential equations. These models are trained using physics-informed loss functions that enforce governing equations, boundary conditions, and constitutive constraints during optimization. Here, we leverage the VFM's principle of virtual work and combine it with physics-informed training to learn optimal virtual fields directly from full-field displacement data. We present DeepVFM a physics-informed deep learning framework that learns optimal virtual fields directly from displacement data, allowing for direct calculation of the material parameters. We leverage neural networks to estimate these optimal test functions, parametrized as polynomials. We design a custom branch-trunk architecture that takes displacement field data as input and outputs the polynomial coefficients defining the virtual field. We use the principle of virtual work, evaluated as the residual between external and internal virtual work, as the training loss. We enforce kinematic admissibility with an envelope function applied to the raw network output, zeroing the virtual field wherever the true boundary conditions require it, rather than penalizing violations in the loss. We tested DeepVFM on synthetic data generated in a finite element simulation software, FEBio, using a Neo-Hookean constitutive model, across uniaxial and biaxial loading configurations on a unit cube. To assess robustness and generalization, we conduct sensitivity studies (ablations) across the polynomial basis order, the number of virtual fields, network width, applied load magnitude, and multiple material models. These experiments characterize DeepVFM's accuracy under ideal conditions and its reliability under the variability expected in practice. We also provide a benchmark against user-defined virtual field sets to compare our method against the classic VFM. DeepVFM achieves accurate identification of both Young's modulus and Poisson's ratio, with relative error below 5% upon convergence for most tested materials, though accurate recovery requires a minimum displacement-to-noise ratio, below which identification degrades. In conclusion, we present DeepVFM, a physics-informed deep learning framework that can identify constitutive parameters by automating virtual field selection and calculate the material parameters. Future work will focus on extending DeepVFM to more complex geometries, heterogeneous materials, as well as to more complex constitutive models, including anisotropic, viscoelastic, and elasto-plastic formulations.