## 2 Materials and Methods
### FEM Use Cases  
For illustration, we base our evaluation on three classic use cases from structural mechanics. We consider the (1) tensile load, (2) bending load and (3) compressive load:  
1. Elongation of a plate with a perforation;
2. Bending of a beam;
3. Compression of a block with four perforations. See Table 1 and Figure 2. We utilize an isotropic elasto-plastic rate-independent material model (i.e., a perfectly plastic material). The kinematic relations for our 2D plane strain use cases are defined by the total strain components \(\varepsilon_{xx}=\frac{\partial u_{x}}{\partial x}\), \(\varepsilon_{yy}=\frac{\partial u_{y}}{\partial y}\), \(\varepsilon_{xy}=\frac{1}{2}(\frac{\partial u_{x}}{\partial y}+\frac{\partial u _{y}}{\partial x})\), \(\varepsilon_{zz}=0\) with displacements \(u_{x}\) and \(u_{y}\) and deviatoric strain components \(e_{xx}=\varepsilon_{xx}-\frac{\varepsilon_{yy}}{3}\), \(e_{yy}=\varepsilon_{yy}-\frac{\varepsilon_{yy}}{3}\), \(e_{xy}=\varepsilon_{xy}\) and \(e_{zz}=-\frac{\varepsilon_{yy}}{3}\). Since there is no volumetric plastic strain in the von Mises yield function, the volumetric strain can be expressed as  
Figure 1: Principle of our surrogate model approach: all \(N\) nodes (i.e., their coordinates), together with the respective generalization variable, are sequentially entered into a surrogate model, which then sequentially predicts the outcome of the respective coordinates (i.e., the displacements, strains, and stresses of the respective node). \(\varepsilon_{vol}=\text{trace}(\varepsilon)\) s.t. \(\varepsilon_{vol}=\varepsilon_{xx}+\varepsilon_{yy}\). The deviatoric stress components are defined by \(s_{xx}=\sigma_{xx}-(\frac{\sigma_{xx}+\sigma_{yy}+\sigma_{zz}}{3})\), \(s_{yy}=\sigma_{yy}-(\frac{\sigma_{xx}+\sigma_{yy}+\sigma_{zz}}{3})\), \(s_{xy}=\sigma_{xy}\) and \(s_{zz}=\sigma_{zz}-(\frac{\sigma_{xx}+\sigma_{yy}+\sigma_{zz}}{3})\), where \(\sigma_{ij}\) (\(i,j\in\{x,y\}\)) are the components of the Cauchy stress tensor. The plastic strain components are defined by \(\varepsilon_{xx}^{pl}=\varepsilon^{pl}\frac{3}{2}\frac{s_{xx}}{q}\), \(\varepsilon_{yy}^{pl}=\varepsilon^{pl}\frac{3}{2}\frac{s_{yy}}{q}\), \(\varepsilon_{xy}^{pl}=\varepsilon^{pl}\frac{3}{2}\frac{s_{xy}}{q}\) and \(\varepsilon_{zz}^{pl}=\varepsilon^{pl}\frac{3}{2}\frac{s_{zz}}{q}\) with equivalent plastic strain of the von Mises model as \(\varepsilon^{pl}=\bar{\varepsilon}-\frac{\sigma_{Y}}{3q}\geq 0\), where \(\sigma_{Y}\) is the yield stress and \(\mu\) the second Lame parameter. The total equivalent strain is defined by \(\bar{\varepsilon}=\sqrt{\frac{2}{3}\sum_{i,j\in\{x,y\}}e_{ij}e_{ij}}\) with deviatoric strain components \(e_{ij}\). The decomposition of the strain is \(\varepsilon_{ij}=\varepsilon_{ij}^{el}+\varepsilon_{ij}^{pl}\) with elastic component \(\varepsilon_{ij}^{el}\) and plastic component \(\varepsilon_{ij}^{pl}\) of the respective strain matrices. The equivalent stress is defined by \(q=\sqrt{\frac{3}{2}s_{ij}s_{ij}}\). In our PINN approach, we utilize the definitions of the total strain components, deviatoric strain and stress components and plastic strain components in the respective regularization term. We use quarter symmetry in use cases 1 and 3 to make efficient use of computational resources. Additional information regarding the variation of parameters in the simulations is presented in Table 2, where simulations marked in bold are used for the test and evaluation of the surrogate models and are not in the training dataset. Conversely, simulations not marked in bold represent the training dataset and are not in the test dataset. In use cases exhibiting varied geometry parameters (i.e., elongation of a plate and compression of a block use cases), the mesh is also different in each simulation. Thus, we train and evaluate the surrogate models on use cases with different meshes (i.e., in each simulation, the node coordinates differ). The first use case, a perforated steel strip under tensile load, is similar to the nonlinear solid mechanics use case of [21,22].