Learning Constitutive Relations using Symmetric Positive Definite Neural Networks

Abstract

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. We demonstrate the effectiveness of SPD-NN on hyperelastic, elasto-plastic, and multiscale fiber-reinforced plate problems from solid mechanics. The generality and robustness of the SPD-NN make it a promising tool for a wide range of constitutive modeling applications.

arXiv comment31 pages, 20 figures

References in corpus (7)

• On the difficulty of training Recurrent Neural Networks
• Solving high-dimensional partial differential equations using deep learning
• A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials
• Exploring the 3D architectures of deep material network in data-driven multiscale mechanics
• Learning Constitutive Relations from Indirect Observations Using Deep Neural Networks
• On the Computational Efficiency of Training Neural Networks
• Effect of Depth and Width on Local Minima in Deep Learning

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