Constitutive parameterized deep energy method for solid mechanics problems with random material parameters

Zhangyong Liang; Jihong Zhu; Huanhuan Gao

cs.LGMar 27, 2026

Constitutive parameterized deep energy method for solid mechanics problems with random material parameters

Authors: Zhangyong Liang, Jihong Zhu, Huanhuan Gao

Organizations: National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China · Key Laboratory of CNC Equipment Reliability, Ministry of Education\National Key Laboratory of Automotive Chassis Integration and Bionics, School of Mechanical and Aerospace Engineering, Jilin University, Renmin Street 5988, 130025, Changchun, China · School of Mechanical and Aerospace Engineering, Jilin University, Renmin Street 5988, 130025, Changchun, China

Abstract

In practical structural design and solid mechanics simulations, material properties inherently exhibit random variations within bounded intervals. However, evaluating mechanical responses under continuous material uncertainty remains a persistent challenge. Traditional numerical approaches, such as the Finite Element Method (FEM), incur prohibitive computational costs as they require repeated mesh discretization and equation solving for every parametric realization. Similarly, data-driven surrogate models depend heavily on massive, high-fidelity datasets, while standard physics-informed frameworks (e.g., the Deep Energy Method) strictly demand complete retraining from scratch whenever material parameters change. To bridge this critical gap, we propose the Constitutive Parameterized Deep Energy Method (CPDEM). In this purely physics-driven framework, the strain energy density functional is reformulated by encoding a latent representation of stochastic constitutive parameters. By embedding material parameters directly into the neural network alongside spatial coordinates, CPDEM transforms conventional spatial collocation points into parameter-aware material points. Trained in an unsupervised manner via expected energy minimization over the parameter domain, the pre-trained model continuously learns the solution manifold. Consequently, it enables zero-shot, real-time inference of displacement fields for unknown material parameters without requiring any dataset generation or model retraining. The proposed method is rigorously validated across diverse benchmarks, including linear elasticity, finite-strain hyperelasticity, and complex highly nonlinear contact mechanics. To the best of our knowledge, CPDEM represents the first purely physics-driven deep learning paradigm capable of simultaneously and efficiently handling continuous multi-parameter variations in solid mechanics.

Figures & tables

Fig. 1: Material-parameter sampling and evaluation at unseen in-range parameters

(E,\nu)

for a fixed mechanical setting. Circles indicate training samples; the diamond denotes an unseen query. The depicted responses are schematic.

Fig. 2: Mechanical response family under varying material parameters. Vectors

\bm{\eta}_{1}

\bm{\eta}_{2}

, and

\bm{\eta}^{*}

specify spatially uniform properties on the same domain

\Omega

with fixed loading and boundaries

\Gamma_{D}

and

\Gamma_{N}

. Interior curves depict schematic responses.

Fig. 3: CPDEM architecture. The material encoder

g_{\theta_{p}}

maps the constitutive vector to latent features, and the coordinate encoder

g_{\theta_{c}}

maps the collocation coordinates to spatial features. Their concatenated outputs are supplied to the fusion network

g_{\theta_{g}}

, which predicts the displacement field.

Fig. 4: Material sampling and energy averaging in CPDEM. Spatial quadrature gives each material energy; their average trains one shared network. Points and field lines are schematic.

Fig. 5: Parameter coverage and error bounds for unseen parameters. (a) Distance to the nearest training parameter,

r\leq h_{\mathcal{P}_{N}}

. (b) Schematic error terms from exact-solution variation, training error, and learned-solution variation.

Problem	$\dim(\bm{\eta})$	Components of $\bm{\eta}$	$g_{\theta_{p}}$	$g_{\theta_{c}}$	$g_{\theta_{g}}$	Output dim.
2-D linear-elastic beam	2	$[E,\nu]$	$[2,16,16]$	$[2,16,16]$	$[32,32,2]$	2
2-D Neo-Hookean beam	2	$[E,\nu]$	$[2,20]$	$[2,20]$	$[40,20,2]$	2
3-D Neo-Hookean beam	2	$[E,\nu]$	$[2,30]$	$[3,30]$	$[60,30,3]$	3
3-D Neo-Hookean cuboid (torsion)	2	$[E,\nu]$	$[2,30]$	$[3,30]$	$[60,30,3]$	3
2-D Plate-with-hole $J_{2}$ plasticity	3	$[E,\nu,H]$	$[3,32,32]$	$[3,32,32]$	$[64,64,3]$	3
2-D viscoelastic creep	4	$[E_{\infty},E_{1},\xi,\nu]$	$[4,32,32]$	$[3,32,32]$	$[64,64,2]$	2

Table 1: CPDEM architecture implementation details.

Fig. 7: CPDEM and FEM displacement fields for the 2D linear elastic cantilever. Rows correspond to $(E,\nu)=(100,0.25)$ , $(223,0.30)$ , and $(500,0.35)$ . Columns show FEM displacement magnitude, CPDEM prediction, and pointwise absolute error.

Fig. 8: Training energy histories for the 2D Neo-Hookean cantilever. (a) Total potential energy; (b) internal energy; (c) external potential energy. The gray dashed line marks the Adam-to-L-BFGS transition.

Fig. 9: Training error histories for a plane-stress linear-elastic cantilever. (a) mean point error; (b) maximum point error; (c) loaded-end squared-error share. All three material parameter sets are pretraining samples.

Fig. 10: CPDEM and FEM displacement fields for the 2D Neo-Hookean cantilever. Rows correspond to $(E,\nu)=(800,0.25)$ , $(979,0.30)$ , and $(1200,0.35)$ . FEM displacement magnitude, CPDEM prediction, and pointwise absolute error.

Fig. 11: Three-dimensional Neo-Hookean bending beam. The

L=4\,\mathrm{m}

and

H=W=1\,\mathrm{m}

beam is clamped on the left and loaded by

\bar{t}_{y}=-5\,\mathrm{Pa}

on the right.

Fig. 12: CPDEM and FEM displacement fields for the 3-D Neo-Hookean cantilever. Rows correspond to $(E,\nu)=(800,0.25)$ , $(979,0.30)$ , and $(1200,0.35)$ . FEM displacement magnitude, CPDEM prediction, and pointwise absolute error.

Fig. 13: Three-dimensional hyperelastic cuboid under torsion. The left face is fixed. The right face is prescribed the $60^{\circ}$ twist in Eq. ( 46 ). Lateral tractions and body force are specified in the text.

Fig. 14: FEM and CPDEM displacement fields for the 3-D hyperelastic cuboid under torsion. Rows correspond to $(E,\nu)=(8\times 10^{5},0.25)$ , $(9.99862\times 10^{5},0.30)$ , and $(1.2\times 10^{6},0.35)$ . FEM reference, CPDEM prediction, pointwise absolute error, and deformed configuration. Each column uses a common color scale across the displayed cases.

Fig. 15: Plate with a hole: Geometry and applied displacements.

Fig. 16: Cyclic plate with a hole at peak load. Rows correspond to $(E,\nu,H)=(800,0.25,700)$ , $(1000,0.30,500)$ , and $(1200,0.35,300)$ . FEM and CPDEM displacement magnitudes, absolute error, and equivalent plastic strain.

Fig. 17: CPDEM and FEM cyclic displacement histories at interior points P1–P4:

u_{x}

(top) and

u_{y}

(bottom).

Fig. 18: CPDEM and FEM histories at P1–P4 under cyclic loading. Plastic strain (top) and von Mises stress (bottom).

Fig. 19: Viscoelastic creep of a 2-D cantilever beam. The left edge is clamped and the right edge is subjected to a horizontal, uniformly distributed tensile pressure

p=0.4\,\mathrm{Pa}

held constant for

t\geq 0

. The geometry is

L=10\,\mathrm{m}

and

h=1\,\mathrm{m}

Fig. 20: CPDEM training dynamics for viscoelastic creep at $t=10\,\mathrm{s}$ . The three rows correspond to the material parameter sets in Eq. ( 57 ). The insets show the horizontal and vertical displacement fields at the displayed snapshot.

Fig. 21: Viscoelastic creep state at $t=10\,\mathrm{s}$ : CPDEM and FEM reference. Rows correspond to the material parameter sets in Eq. ( 57 ). FEM reference, CPDEM prediction, and horizontal displacement absolute error.

Fig. 22: CPDEM and FEM creep histories at P1–P4 under constant traction. Horizontal displacement (top) and cumulative displacement increment (bottom), with zero vertical displacement at P1–P4.

Fig. 23: Stress relaxation of a 2-D cantilever beam. The left edge is clamped and the right edge is prescribed the elongation

\Delta L=0.1\,\mathrm{m}

t=0

. The geometry is

L=10\,\mathrm{m}

and

h=1\,\mathrm{m}

Fig. 24: CPDEM training energy histories for stress relaxation at $t=10\,\mathrm{s}$ . Panels (a)–(c) correspond to $(E_{\infty},E_{1},\xi,\nu)=(5,10,50,0.35)$ , $(2,10,50,0.25)$ , and $(2,10,10,0.25)$ . Insets show predicted $u_{x}$ and $u_{y}$ , and FEM $u_{y}$ .

Fig. 25: CPDEM and FEM displacement fields for stress relaxation at $t=10\,\mathrm{s}$ with $\Delta L=1\,\mathrm{m}$ . Rows correspond to $(E_{\infty},E_{1},\xi,\nu)=(5,10,50,0.35)$ , $(2,10,50,0.25)$ , and $(2,10,10,0.25)$ . FEM, CPDEM prediction, and absolute error.

Fig. 26: CPDEM and FEM relaxation histories at P1–P4: axial displacement (top) and displacement increment (bottom).

Fig. 27: Viscoelastic creep buckling of a quarter circular shell. (a) Geometry and external pressure; (b) sampling points and monitored outer surface; (c) outer-surface deformation at successive times.

Fig. 28: CPDEM training energy histories for viscoelastic creep buckling. Panels (a)–(c) correspond to $(E_{\infty},E_{1},\xi,\nu)=(4,7,20,0.35)$ , $(4,10,20,0.35)$ , and $(4,10,20,0.25)$ . Insets show $|\bm{u}|$ , $u_{r}$ , and $u_{\theta}$ .

Fig. 29: CPDEM and reference displacement fields for viscoelastic creep buckling. Rows correspond to $(E_{\infty},E_{1},\xi,\nu)=(4,7,20,0.35)$ , $(4,10,20,0.35)$ , and $(4,10,20,0.25)$ . Columns show the reference, prediction, and absolute error.

Fig. 30: CPDEM and FEM displacement histories at four monitoring points on the 2-D circle.

u_{x}

(top) and

u_{y}

(bottom).

Physics-Informed ML

Physics-informed machine learning combines data-driven models with physical laws to solve, predict, and infer scientific systems.

See all 833 papers

A mesh-based neural energy method for the simulation of heterogeneous composites
Oct 7, 2026Pius J. M. Wichmann, Stefan Hildebrand, Sandra KlingeNeural PDE Solvers Structural Mechanics
Department of Structural and Computational Mechanics, Technische Universität Berlin, Straße des 17. Juni 135, Berlin, 10623, Berlin, Germany
Physics-Informed Coarsening for Multigrid Graph Neural Surrogates
May 29, 2026Amir Bazzi, David Cardinaux, Ramy Nemer +3Graph Neural Networks Neural PDE Solvers
1CEMEF, Mines Paris – PSL, Sophia Antipolis, France · 2Aubert et Duval, France · 3Transvalor, Mougins, France.
A Hybrid Conditional Diffusion-DeepONet Framework for High-Fidelity Stress Prediction in Hyperelastic Materials
Mar 18, 2026Purna Vindhya Kota, Meer Mehran Rashid, Somdatta Goswami +1Neural Surrogate Modeling Hyperelasticity
Department of Civil and Systems Engineering, Johns Hopkins University, 3400 N. Charles Street, Baltimore, 21218, MD, USA

Oct 7, 2026J/K move · Enter open · S save

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

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

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

Neural PDE Solvers Structural Mechanics

Open paper AskPDF arXiv

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).

Abstract

Physics-Informed ML

Explore similar work

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

Physics-Informed Coarsening for Multigrid Graph Neural Surrogates

A Hybrid Conditional Diffusion-DeepONet Framework for High-Fidelity Stress Prediction in Hyperelastic Materials