Symmetry-enforcing neural networks with applications to constitutive
  modeling
Kévin Garanger Julie Kraus Julian J. Rimoli
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.
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.