cs.CEOct 7, 2026

A mesh-based neural energy method for the simulation of heterogeneous composites

Authors: Pius J. M. Wichmann, Stefan Hildebrand, Sandra Klinge

Organizations: Department of Structural and Computational Mechanics, Technische Universität Berlin, Straße des 17. Juni 135, Berlin, 10623, Berlin, Germany

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Abstract

Modeling heterogeneous materials remains a challenge for physics-informed neural networks such as the deep energy method (DEM). The DEM and its variants, here collectively referred to as the neural energy method (NEM), offer a differentiable variational framework. However, their conventional collocation-based implementation (C-NEM) often suffers from physically inadmissible displacement oscillations, integration errors, and high computational costs from automatic differentiation. This work introduces the mesh-based neural energy method (M-NEM), extending the NEM through a mesh-based discretization of the displacement field. By interpolating nodal displacements via shape functions, the M-NEM imposes a kinematic constraint that suppresses oscillations. Furthermore, the method replaces automatic differentiation with algebraic shape function derivatives for strain computation and employs high-order Gaussian quadrature for accurate energy integration. On a directly comparable benchmark problem, the M-NEM reduces stress errors by up to three orders of magnitude relative to the C-NEM while being one to two orders of magnitude faster. On two further benchmarks involving extreme stiffness contrasts, only the M-NEM converges. A comparative study of neural architectures reveals that radial basis function neural networks (RBFNNs) yield optimal performance within the M-NEM, resolving sharp gradients at material interfaces with higher accuracy than multi-layer perceptrons (MLPs) with random Fourier feature (RFF) mapping and faster convergence than Kolmogorov-Arnold networks (KANs).

Figures & tables

Figure 1: Information flow in the NEM . Gray boxes indicate the components replaced in the M-NEM .
Figure 2: Geometry of the coupon with stiff inclusion.
HyperparameterMLP + RFFKANRBFNN
General parameters   
Number of hidden layers1021
Dimension of hidden layers6432–
Activation / Kernel functionTanhTanhMatérn- 32\frac{3}{2}
Discretization & Encoding   
RFF mapping std. dev.1––
Table 1: Hyperparameter values used for the coupon with stiff inclusion.
Figure 3: Normal stress ( σxx\sigma_{xx} ) solutions for the coupon with stiff inclusion: (a,c,e) the C-NEM and (b,d,f) the M-NEM with different neural network architectures, compared to (g) the FEM reference solution.
Figure 4: Comparison of log-scaled relative stress error (a) and runtime (b) for the C-NEM and M-NEM with different neural network architectures on the coupon with stiff inclusion. Large dots show the median of the 30 training runs.
Figure 5: Normal stress ( σxx\sigma_{xx} ) solutions for C-NEM (a) and M-NEM (b) on the coupon with stiff inclusion with a coarse discretization and MLP + RFF architecture.
Figure 6: Idealized simulation domain of the calcified hydrogel unit cell problem.
Figure 7: Fitting hyperelastic material models to experimental PAM hydrogel data from [ 48 ] .
DescriptionSymbolValueUnit
Shear modulusμ1\mu_{1}1×1041\times 10^{4}Pa
Nonlinearity parameterα1\alpha_{1}1.5031.503–
Shear modulusμ2\mu_{2}7.437.43Pa
Nonlinearity parameterα2\alpha_{2}4.074.07–
Table 2: Material parameters for the PAM hydrogel matrix, derived via a two-term Ogden model fit.
HyperparameterMLP + RFFKANRBFNN
General parameters   
Number of hidden layers1021
Dimension of hidden layers25632–
Activation / Kernel functionTanhTanhMatérn- 32\frac{3}{2}
Discretization & Encoding   
RFF mapping std. dev.1––
Table 3: Hyperparameter values used for the calcified hydrogel unit cell.
Figure 8: Von-Mises stress solutions for the calcified hydrogel unit cell: (a-c) the M-NEM with different neural network architectures, and (d) the FEM reference solution.
Figure 9: Comparison of log-scaled relative stress error (a) and runtime (b) for the M-NEM with different neural network architectures on the calcified hydrogel unit cell. Large dots show the median of the 30 training runs.
Figure 10: Idealized simulation domain of the CFRP .
HyperparameterMLP + RFFKANRBFNN
General parameters   
Number of hidden layers1021
Dimension of hidden layers25632–
Activation / Kernel functionTanhTanhMatérn- 32\frac{3}{2}
Discretization & Encoding   
RFF mapping std. dev.1––
Table 4: Hyperparameter values used for the CFRP .
Figure 11: Shear stress ( σxy\sigma_{xy} ) solutions for the CFRP : (a-c) the M-NEM with different neural network architectures and (d) the FEM reference solution.
Figure 12: Comparison of relative stress error (a) and runtime (b) for the M-NEM with different neural network architectures on the CFRP . Large dots show the median of the 30 training runs.
Figure 13: Evolution of the relative stress error over runtime for the CFRP .
Figure 14: Comparison of relative stress error (a) and runtime (b) for different convergence tolerances for the M-NEM with KAN and RBFNN on the CFRP .

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