Symmetry-enforcing neural networks with applications to constitutive modeling

TL;DR

This work tackles the challenge of enforcing material symmetries in data-driven constitutive modeling by introducing tensor-feature equivariant neural networks (TFENN), which enforce symmetry exactly at the neuron level for mappings between symmetric second-order tensors. TFENN employ symmetry-aware weights, biases, and activations (including eigenvalue-based tensor activations) to achieve $\mathcal{G}$-equivariance, and extend to recurrent architectures for history-dependent behavior. Demonstrated on isotropic neo-Hookean, tensegrity lattice metamaterials, and elasto-plastic microstructures, TFENN show superior data efficiency and accuracy compared with standard networks, with symmetry preserved to numerical precision and potential for symmetry-basis discovery through learnable rotations. The approach integrates with constitutive modeling and finite element analysis, offering a principled, scalable route to physics-consistent, tensor-valued learning across elastic and inelastic regimes and beyond.

Abstract

The use of machine learning techniques to homogenize the effective behavior of arbitrary microstructures has been shown to be not only efficient but also accurate. In a recent work, we demonstrated how to combine state-of-the-art micromechanical modeling and advanced machine learning techniques to homogenize complex microstructures exhibiting non-linear and history dependent behaviors (Logarzo et al., 2021). The resulting homogenized model, termed smart constitutive law (SCL), enables the adoption of microstructurally informed constitutive laws into finite element solvers at a fraction of the computational cost required by traditional concurrent multiscale approaches. In this work, the capabilities of SCLs are expanded via the introduction of a novel methodology that enforces material symmetries at the neuron level, applicable across various neural network architectures. This approach utilizes tensor-based features in neural networks, facilitating the concise and accurate representation of symmetry-preserving operations, and is general enough to be extend to problems beyond constitutive modeling. Details on the construction of these tensor-based neural networks and their application in learning constitutive laws are presented for both elastic and inelastic materials. The superiority of this approach over traditional neural networks is demonstrated in scenarios with limited data and strong symmetries, through comprehensive testing on various materials, including isotropic neo-Hookean materials and tensegrity lattice metamaterials. This work is concluded by a discussion on the potential of this methodology to discover symmetry bases in materials and by an outline of future research directions.

Figures (6)

• Figure 1: Representation of a TFENN with two hidden layers of eight two-dimensional tensor features each. Each connection between two features is associated to a weight tensor and each feature is associated with a bias tensor. The weights of the connections leading to a given feature, the bias tensor of this feature, and the activation at the corresponding layer fully characterize a neuron operation.
• Figure 2: Validation loss progress of neural networks trained on a neo-Hookean dataset with 20000 training samples. The minimum validation loss up to the current epoch for tensor feature-based and standard networks is shown by solid and dashed lines, respectively. The current validation loss is represented by the dimmed lines.
• Figure 3: Final validation loss of neural networks trained on full and reduced neo-Hookean datasets. The validation loss of TFENNs and standard networks is shown by solid and hatched bars, respectively. The ratio of the achieved validation loss to the one achieved by the tensor feature-based network with the least number of parameters, highlighted with a bar with a wider edge, is shown by the numbers above the bars.
• Figure 4: Unit cells used in the tensegrity lattice and elasto-plastic microstructure problems.
• Figure 5: Final validation loss of neural networks trained on full and reduced tensegrity cell datasets with different enforced symmetries. Plotting conventions as in \ref{['fig:neohookean_results']}.