No-Prop Learning based Physics-Informed Neural Networks for Nonlinear Materials

Constitutive models are essential for describing the complex constitutive response of elastoplastic materials, often requiring a numerical integration scheme for solution. Neural networks have emerged as potential substitutes for approximating the entire constitutive law or the expensive numerical solution step. We propose a hybrid constitutive model that replaces the expensive return mapping step with efficiently learned physics-informed neural networks. An efficient No-Propagation (No-Prop) based learning employing a least squares solution that targets just the output layer of the network is presented for modeling rate-independent plasticity with nonlinear isotropic hardening. The physics-informed neural network learns from the plastic residuals defined at unlabeled collocation points. The flexibility of the training scheme is showcased to adapt to different architectures and initializations of the fixed parameters in the network's hidden layers. Random initializations on single-hidden-layer networks are explored for improved performance through local neural networks defined on partitioned spaces. Use of pre-trained initializations for deep networks with multiple hidden layers is presented as an application to rapidly learn on different material properties through the reuse of the hidden representations. Constitutive model employing PINNs trained by the No-Prop scheme outperforms traditional gradient-based learning with up to two orders of magnitude faster training and largely improved accuracy of predictions. Additionally, we show that fast and accurate estimates of the tangent modulus can be obtained from simple closed-form expressions of the gradients of shallow networks, vastly reducing the computational cost of automatic differentiation in deep architectures.