Learning Constitutive Relations using Symmetric Positive Definite Neural Networks

Understand

We present the Cholesky-factored symmetric positive definite neural network (SPD-NN) for modeling constitutive relations in dynamical equations.

• Instead of directly predicting the stress, the SPD-NN trains a neural network to predict the Cholesky factor of a tangent stiffness matrix, based on which the stress is calculated in the incremental form.
• As a result of the special structure, SPD-NN weakly imposes convexity on the strain energy function, satisfies time consistency for path-dependent materials, and therefore improves numerical stability, especially when the SPD-NN is used in finite element simulations.
• Depending on the types of available data, we propose two training methods, namely direct training for strain and stress pairs and indirect training for loads and displacement pairs.