Machine learning methods are progressively investigated for a large amount of applications. Recently, the solution of partial differential equations (PDE) describing problems in elastostatics came into focus. The current work investigates two neural network-based classes of methods for their solution, namely the neural finite element method (FEM) and neural operator methods. The analysis of these approaches is carried out by means of numerical experiments with linear and nonlinear material behavior where the conventional FEM serves as a benchmark. The formulation of neural FEM allows for elegant integration of finite deformation hyperelasticity at medium training effort. Here, training data are replaced by the evaluation of the equilibrium PDE at sample points. In contrast, most neural operator methods require expensive training with large data sets, but then allow for solving multiple boundary value problems with the same machine learning model. For the comparative analysis, the maximal relative error values over the whole domain and over all components of the strain tensor are evaluated as accuracy measure. The current state of research shows that none of the methods investigated reaches the accuracy and computational performance of the conventional FEM. In many standard applications, the FEM achieves an accuracy of $$10^{-6}$$10 - 6 , whereas the numerical tests in the present work report a relative error of order of magnitude of $$10^{-4}$$10 - 4 for the neural FEM and $$10^{-2}$$10 - 2 to $$10^{-3}$$10 - 3 for neural operator methods.

An extendable, efficient and explainable Machine Learning approach is proposed to represent cyclic plasticity and replace conventional material models based on the Radial Return Mapping algorithm. High accuracy and stability by means of a limited amount of training data is achieved by implementing physics-informed regularizations and the back stress information. The off-loading of the neural network (NN) is applied to the maximal extent. The proposed model architecture is simpler and more efficient compared to existing solutions from the literature using approximately only half the amount of NN parameters, while representing a complete three-dimensional material model. The validation of the approach is carried out by means of results obtained with the Armstrong–Frederick kinematic hardening model. The mean squared error is assumed as the loss function which stipulates several restrictions: deviatoric character of internal variables, compliance with the flow rule, the differentiation of elastic and plastic steps and the associativity of the flow rule. The latter, however, has a minor impact on the accuracy, which implies the generalizability of the model for a broad spectrum of evolution laws for internal variables. Numerical tests simulating several load cases are presented in detail. The validation shows cyclic stability and deviations in normal directions of less than 2% at peak values which is comparable to the order of measurement inaccuracies.

A general, unified neural network approach as replacement for the finite element method without the need for analytic expressions for material laws is suggested. The complete simulation process from the material characterization to simulations on a structural level takes place in the new neural network framework. The drawback of many conventional analytic expressions of material laws to require large numbers of experiments for parametrization is addressed by an integrated inverse approach. Specifically, an adaptation of the Deep Energy Method is combined with a Constitutive Artificial Neural Network (CANN) and trained on measured displacement fields and prescribed boundary conditions in a coupled procedure. Tests on compressible and incompressible Neo-Hookean solids with up to twelve CANN parameters show high accuracy of the approach and very good generalization of CANNs. A small extent of data is required for robust and reliable training.

An extendable, efficient and explainable Machine Learning approach is proposed to represent cyclic plasticity and replace conventional material models based on the Radial Return Mapping algorithm. High accuracy and stability by means of a limited amount of training data is achieved by implementing physics-informed regularizations and the back stress information. The off-loading of the Neural Network is applied to the maximal extent. The proposed model architecture is simpler and more efficient compared to existing solutions from the literature, while representing a complete three-dimensional material model. The validation of the approach is carried out by means of surrogate data obtained with the Armstrong-Frederick kinematic hardening model. The Mean Squared Error is assumed as the loss function which stipulates several restrictions: deviatoric character of internal variables, compliance with the flow rule, the differentiation of elastic and plastic steps and the associativity of the flow rule. The latter, however, has a minor impact on the accuracy, which implies the generalizability of the model for a broad spectrum of evolution laws for internal variables. Numerical tests simulating several load cases are shown in detail and validated for accuracy and stability.

Formulating constitutive models for plasticity presents significant challenges. In particular, cyclic inelastic analysis is computationally intensive, making machine learning a promising tool for capturing complex material behavior with reduced computational cost and enabling near real-time execution after training. This study builds upon a hybrid data-efficient methodology for modeling the cyclic plasticity of metals, which integrates data-driven training with physics-augmented regularization using a fully connected neural network (FCNN). We propose an enhancement to this FCNN framework aimed at improving its efficiency and enabling it to serve as a replacement for conventional models based on radial return mapping (RRM). By selecting input and target variables analogous to those used in RRM, the network’s behavior becomes interpretable—grounded in explicitly defined quantities rather than opaque internal states. Notably, kinematic hardening is modeled without the use of backstress variables as explicit inputs, allowing the model to rely solely on data obtainable from standard laboratory tests—a key innovation of this study. Through physics-augmented regularization, the data-efficient approach achieves high accuracy and numerical stability with relatively limited training data, significantly improving computational performance. Further efficiency gains are realized by optimizing the network architecture, reducing the number of input variables, and refining the data pre- and post-processing steps. The proposed enhancements result in a 15% reduction in training time while maintaining accuracy on par with the original NN model.

Machine Learning methods belong to the group of most up-to-date approaches for solving partial differential equations. The current work investigates two classes, Neural FEM and Neural Operator Methods, for the use in elastostatics by means of numerical experiments. The Neural Operator methods require expensive training but then allow for solving multiple boundary value problems with the same Machine Learning model. Main differences between the two classes are the computational effort and accuracy. Especially the accuracy requires more research for practical applications.