An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications

Esteban SamaniegoCosmin AnitescuSomdatta GoswamiVien Minh Nguyen-ThanhHongwei GuoKhader HamdiaTimon RabczukXiaoying Zhuang

article2019Computer Methods in Applied Mechanics and Engineering1,974 citations

Proposes a deep learning approach for solving partial differential equations in computational mechanics by minimizing the potential energy of mechanical systems, providing a mesh-free alternative to finite element and collocation-based methods.

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Computational modeling of mechanical systems relies heavily on partial differential equations to predict structural behavior, deformation, and failure. Traditional discretization techniques like the finite element method can face significant implementation overhead, particularly when dealing with complex geometries, coupled physics, or higher-order derivatives. While deep learning has recently emerged as an alternative numerical tool, most early machine learning solvers apply point-wise collocation against the strong form of governing equations, which often struggle with non-smooth behaviors and derivative constraints. The article evaluates the Deep Energy Method, a framework that uses deep neural networks to approximate unknown physical fields by directly minimizing the system's total variational potential energy.

To establish this framework, the article implemented deep feed-forward neural networks across open-source machine learning platforms. Rather than training on pre-existing simulation datasets to build surrogate models, the networks serve directly as the continuous approximation space. The training loss function corresponds to the potential energy computed across integration points in the domain, optimized via first-order gradient descent combined with quasi-Newton solvers. The method was validated across a comprehensive suite of computational mechanics benchmarks, including two- and three-dimensional linear elasticity, elastodynamics, 3D hyperelasticity under finite deformation, phase-field fracture modeling, piezoelectric coupling, and fourth-order Kirchhoff plate bending.

The findings demonstrate that the energy-based approach consistently matches or exceeds the accuracy of traditional point collocation methods while requiring less computational overhead. In linear elastic benchmarks, displacement errors remained between 0.5% and 1.8%, while energy norm errors remained around 3.2% to 5.3%. In fracture modeling, the energy method captured crack profiles with an error of 2.88%, whereas collocation produced an error of 70.6% and required ten times as many training iterations due to sharp gradient transitions. For fourth-order plate bending problems, tailoring the neural network activation functions and integrating autoencoder architectures significantly improved convergence, achieving relative deflection errors as low as 0.001% without requiring the difficult C1 continuity constraints demanded by conventional mesh-based methods.

These results show that the Deep Energy Method offers an effective, mesh-free alternative for complex physical simulations. By embedding physical principles directly into the optimization objective, standard machine learning platforms can solve coupled and high-order partial differential equations with minimal code complexity and automatic handling of natural boundary conditions. However, the resulting discrete optimization problems are non-convex, and the current training times remain slower than optimized traditional solvers. Decision-makers should view this framework as a promising foundational method, with future technical efforts needed to accelerate training pipelines and enhance optimization stability for large-scale industrial use cases.

Bauhaus-Universität Weimar Leibniz Universität Hannover Universidad de Cuenca engineering finite element method function approximation deep energy method computational mechanics partial differential equations

arXiv: 1908.10407 somdattagoswami/DEM-PhaseField

Paper: DGM: A deep learning algorithm for solving partial differential equations, Justin Sirignano et al. (2017). Sirignano and Spiliopoulos introduce the Deep Galerkin Method (DGM), establishing foundational concepts for solving partial differential equations with meshfree deep neural networks that the source adapts to an energy-based formulation.
Paper: Automatic differentiation in machine learning: a survey, Atilim Gunes Baydin et al. (2018). This survey provides essential background on automatic differentiation, which forms the computational backbone for evaluating differential operators and energy functionals in neural network PDE solvers.

Paper: DeepXDE: A Deep Learning Library for Solving Differential Equations, Lu Lu et al. (2019). DeepXDE expands the machine learning framework for solving differential equations into an open-source library supporting automatic differentiation and residual-based adaptive refinement across scientific benchmarks.
Paper: Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next, Salvatore Cuomo et al. (2022). This comprehensive review surveys physics-informed neural networks and collocation-based methodologies, offering a broader perspective on modern scientific machine learning architectures and extensions.
Paper: Fourier Neural Operator for Parametric Partial Differential Equations, Zongyi Li et al. (2020). The Fourier Neural Operator extends neural PDE approximation from single-instance solutions to learning mesh-invariant mappings between continuous function spaces for parametric systems.
Paper: Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Lu Lu et al. (2021). DeepONet generalizes function approximation for differential equations to learning infinite-dimensional nonlinear operators via branch and trunk subnetworks.
Paper: KAN: Kolmogorov-Arnold Networks, Ziming Liu et al. (2025). Kolmogorov-Arnold Networks introduce learnable spline-based activation functions that improve accuracy and parameter efficiency over standard MLPs when solving PDEs and scientific problems.
Paper: Neural means and kernel corrections for operator learning, Yitzchak Shmalo (2026). This text develops hybrid operator learning frameworks combining neural networks with kernel methods to enhance accuracy and representation for forward physical systems.

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Abstract

Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate in applications that have an interest for Computational Mechanics. Most contributions that have decided to explore this possibility have adopted a collocation strategy. In this contribution, we concentrate in mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. As proofs of concept, we deal with several problems and explore the capabilities of the method for applications in engineering.

1 Introduction
2 Mathematical modeling of continuous physical systems
2.1 Partial Differential Equations
2.2 Energy Approach
3 Deep Neural Networks for PDE discretization
4 Solution strategies
4.1 Collocation Methods
4.2 Deep Energy Method
4.2.1 Optimization in machine learning
5 Implementation
6 Applications
6.1 DNN with ReLU activation functions and FE in 1D
6.2 Linear Elasticity problem
6.2.1 Pressurized thick-cylinder
6.2.2 Plate with a circular hole
6.2.3 Hollow sphere under internal pressure
6.2.4 Cube with a spherical hole subject to uniform tension
6.3 Elastodynamics
6.4 Hyperelasticity
6.5 Phase field modeling of fracture
6.5.1 One-dimensional phase field model
6.5.2 Single-edge notched tension example
6.6 Piezoelectricity
6.6.1 Cantilever beam
6.7 Kirchhoff Plate bending
6.7.1 Simply-supported square plate on Winkler foundation
6.7.2 Clamped circular plate
6.7.3 Simply-supported square plate under a sinusoidally distributed load
6.7.4 Simply-supported annular plate
7 Concluding remarks
References

Knowls

Knowl 1 — Deep Energy Method Formulation for Continuum Mechanics
model/method
The Deep Energy Method (DEM) is a physics-informed deep learning framework that solves boundary value problems in continuum mechanics by leveraging the principle of minimum potential energy rather than the strong-form differential equations. In DEM, deep neural networks (DNNs) parameterized by trainable weights and biases $\boldsymbol{p}$ serve as function approximators for the displacement field $\boldsymbol{u}_p(\boldsymbol{x})$ . The loss function $\mathcal{L}(\boldsymbol{p})$ is formulated by directly discretizing the total potential energy functional $\mathcal{E}[\boldsymbol{u}]$ over a set of domain integration points $\{\boldsymbol{x}_i\}_{i=1}^{N_\Omega}$ with corresponding quadrature weights $w_i$ :

$\mathcal{E}[\boldsymbol{u}_p] \approx \mathcal{L}(\boldsymbol{p}) = \sum_{i=1}^{N_\Omega} \Psi(\boldsymbol{\epsilon}(\boldsymbol{u}_p(\boldsymbol{x}_i))) w_i - \mathcal{W}_{ext}(\boldsymbol{u}_p)$

where $\Psi$ is the strain energy density function, $\boldsymbol{\epsilon}(\boldsymbol{u}_p) = \frac{1}{2}(\nabla \boldsymbol{u}_p + \nabla \boldsymbol{u}_p^T)$ is the linearized strain tensor computed via automatic differentiation, and $\mathcal{W}_{ext}$ represents the external potential energy of body forces and surface tractions.

A fundamental advantage of this variational formulation is that homogeneous Neumann boundary conditions (traction-free boundaries) are satisfied naturally as Euler-Lagrange natural conditions without needing penalty terms or boundary collocation points.
Knowl 2 — Exact Satisfaction of Dirichlet Boundary Conditions via Output Transformation
model/method
Because artificial neural network outputs do not satisfy the Kronecker delta property at spatial boundaries, enforcing essential (Dirichlet) boundary conditions strictly within the admissible space $\mathcal{H}$ is accomplished by defining an exact trial solution transformation. Instead of adding penalty terms to the loss function, the raw neural network output vector $\hat{\boldsymbol{u}}(\boldsymbol{x}; \boldsymbol{p})$ is modified algebraically as:

$\boldsymbol{u}_p(\boldsymbol{x}) = \bar{\boldsymbol{u}}(\boldsymbol{x}) + D(\boldsymbol{x}) \hat{\boldsymbol{u}}(\boldsymbol{x}; \boldsymbol{p})$

where $\bar{\boldsymbol{u}}(\boldsymbol{x})$ is a known function satisfying the prescribed non-homogeneous Dirichlet conditions on $\partial \Omega_D$ , and $D(\boldsymbol{x})$ is a continuous distance/boundary-scaling function that vanishes identically on $\partial \Omega_D$ (i.e., $D(\boldsymbol{x}) = 0$ for all $\boldsymbol{x} \in \partial \Omega_D$ ).

For example, in a 2D domain $[0, 1] \times [0, 1]$ with homogeneous displacement on the bottom/left edges and a prescribed vertical displacement $\Delta v$ on the top edge $y=1$ , the displacement field is parameterized as:

$u(x, y) = [x(1 - x)] \hat{u}(x, y; \boldsymbol{p}), \qquad v(x, y) = [y(y - 1)] \hat{v}(x, y; \boldsymbol{p}) + y \Delta v$

This ensures that all admissible displacement trial functions identically satisfy the Dirichlet boundary conditions throughout optimization.
Knowl 3 — Representation of 1D Linear Finite Element Spaces by ReLU Neural Networks
theoretical result
Any one-dimensional continuous piecewise linear finite element approximation $u^h(x) = \sum_{i=0}^n u_i N_i(x)$ on an interval $[0, l]$ with mesh nodes $0 = x_0 < x_1 < \dots < x_n = l$ , element sizes $h_i = x_i - x_{i-1}$ , and nodal values $u_i = u^h(x_i)$ can be exactly expressed as a feed-forward neural network with a single hidden layer and Rectified Linear Unit ( $\text{ReLU}(x) = \max(0, x)$ ) activation functions:

$u^h(x) = u_0 + \sum_{i=0}^{n-1} w_i \text{ReLU}(x - x_i)$

where the hidden layer biases represent the nodal positions $x_i$ , and the linear output weights $w_i$ are explicitly related to the nodal field increments $\Delta_i = u_i - u_{i-1}$ (with $\Delta_0 = 0$ ) by:

$w_i = \frac{\Delta_{i+1}}{h_{i+1}} - \frac{\Delta_i}{h_i}, \quad \text{for } i = 0, 1, \dots, n-1$

Consequently, minimizing the strain energy functional $\mathcal{E}(\boldsymbol{w}, \boldsymbol{x}_p) = \int_0^l \Psi(\epsilon(u^h(x; \boldsymbol{w}, \boldsymbol{x}_p))) dx$ with fixed nodal positions $\boldsymbol{x}_p$ is mathematically identical to standard linear FEM. When the nodal positions $\boldsymbol{x}_p$ (network biases) are optimized simultaneously with weights $\boldsymbol{w}$ , the neural network solves a nonlinear, mesh-free adaptive finite element problem.
Knowl 4 — Deep Energy Method Formulation for Phase Field Modeling of Brittle Fracture
model/method
In the phase field approach to brittle fracture, crack topology is regularized over a finite process zone of length scale $l_0$ tracked by a continuous scalar damage field $\phi \in [0, 1]$ , where $\phi=0$ represents intact material and $\phi=1$ denotes fully broken state. The displacement field $\boldsymbol{u}$ and phase field $\phi$ are determined simultaneously by minimizing the total regularized energy functional $\mathcal{E}[\boldsymbol{u}, \phi] = \mathcal{E}_e + \mathcal{E}_c$ subject to $\boldsymbol{u} = \bar{\boldsymbol{u}}$ on $\partial \Omega_D$ :

$\mathcal{E}_e = \int_\Omega g(\phi) \Psi_0(\boldsymbol{\epsilon}(\boldsymbol{u})) \, d\Omega, \qquad \mathcal{E}_c = \frac{G_c}{2 l_0} \int_\Omega \left( \phi^2 + l_0^2 |\nabla \phi|^2 \right) d\Omega + \int_\Omega g(\phi) \mathcal{H}(\boldsymbol{x}, t) \, d\Omega$

where $g(\phi) = (1-\phi)^2$ is the stress degradation function, $G_c$ is the critical fracture energy release rate, $\Psi_0(\boldsymbol{\epsilon})$ is the undamaged elastic strain energy density, and $\mathcal{H}(\boldsymbol{x}, t) = \max_{s \in [0, t]} \Psi_0^+(\boldsymbol{\epsilon}(\boldsymbol{x}, s))$ is the strain-history functional enforcing crack irreversibility based on the tensile strain energy density $\Psi_0^+$ .

In DEM, both $\boldsymbol{u}$ and $\phi$ are parameterized as deep neural networks and solved via a monolithic optimizer that minimizes the discretized sum of elastic and fracture energy losses over domain quadrature points.
Knowl 5 — Superiority of DEM Over Collocation for Phase Field Derivative Discontinuities
empirical result
When approximating a 1D phase field crack profile $\phi(x) = \exp(-|x-a|/l_0)$ with a sharp derivative discontinuity at crack position $a=25$ (domain $[0, 50]$ , length scale $l_0=1$ ):
1. The Deep Collocation Method (DCM), which minimizes the squared residual of the strong-form equation $\phi''(x) - l_0^{-2}\phi(x) = 0$ , yields a relative $L_2$ prediction error of $70.6\%$ due to its inability to handle the second derivative singularity at the crack tip.
2. The Deep Energy Method (DEM), which minimizes the first-order variational energy functional $\mathcal{I}(\phi) = \frac{1}{2}\int_\Omega (\phi(x) - l_0^2 |\nabla \phi|^2) dx$ , achieves a relative $L_2$ error of $2.88\%$ using the same network architecture (3 hidden layers, 50 neurons per layer) and 8,000 collocation points.
In 2D notched plate crack initialization, DEM satisfies traction-free Neumann conditions naturally, requires fewer hidden layers (3 layers vs 5 layers for DCM), and converges approximately 10 times faster than DCM.
Knowl 6 — Deep Energy Method for Finite Deformation Hyperelasticity
model/method
For nonlinear hyperelastic bodies undergoing finite deformation, the material position $\boldsymbol{X} \in \Omega$ is mapped to the spatial position via $\boldsymbol{\phi}(\boldsymbol{X}) = \boldsymbol{X} + \boldsymbol{u}(\boldsymbol{X})$ , with deformation gradient $\boldsymbol{F} = \text{Grad} \, \boldsymbol{\phi}(\boldsymbol{X})$ . The DEM loss function approximates the total potential energy functional:

$\mathcal{L}(\boldsymbol{p}) = \frac{V_\Omega}{N_\Omega} \sum_{i=1}^{N_\Omega} \left[ \Psi(\boldsymbol{F}(\boldsymbol{X}_i; \boldsymbol{p})) - \boldsymbol{f}_b(\boldsymbol{X}_i) \cdot \boldsymbol{\phi}_p(\boldsymbol{X}_i) \right] - \frac{A_{\partial \Omega_N}}{N_{\partial \Omega_N}} \sum_{j=1}^{N_{\partial \Omega_N}} \bar{\boldsymbol{t}}(\boldsymbol{X}_j) \cdot \boldsymbol{\phi}_p(\boldsymbol{X}_j)$

where $V_\Omega$ and $A_{\partial \Omega_N}$ are the domain volume and loaded boundary surface area, and $\boldsymbol{f}_b, \bar{\boldsymbol{t}}$ are body forces and boundary tractions. For a compressible Neo-Hookean material with Lamé parameters $\lambda$ and $\mu$ , the strain energy density is:

$\Psi(I_1, J) = \frac{1}{2}\lambda [\ln(J)]^2 - \mu \ln(J) + \frac{1}{2}\mu(I_1 - 3)$

where $I_1 = \text{tr}(\boldsymbol{F}^T \boldsymbol{F})$ and $J = \det(\boldsymbol{F})$ .

For a 3D cuboid ( $1.25 \times 1.0 \times 1.0$ ) subjected to $60^\circ$ torsion, body forces, and surface tractions, DEM trained with 64,000 points yields an $L_2$ norm of $0.13210$ and an $H^1$ seminorm of $0.51001$ , closely matching the high-resolution finite element (FEniCS) reference values of $0.13275$ and $0.51407$ .
Knowl 7 — Deep Energy Method Formulation for Piezoelectricity
model/method
Piezoelectric media exhibit electro-mechanical cross-coupling governed by the electric displacement $\boldsymbol{D} = \mathbf{e} : \boldsymbol{\epsilon} + \boldsymbol{\kappa} \cdot \boldsymbol{E}$ and Cauchy stress $\boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\epsilon} - \mathbf{e} \cdot \boldsymbol{E}$ , where $\mathbf{C}, \mathbf{e}, \boldsymbol{\kappa}$ are the fourth-order elasticity, third-order piezoelectric, and second-order dielectric permittivity tensors, $\boldsymbol{\epsilon} = \frac{1}{2}(\nabla \boldsymbol{u} + \nabla \boldsymbol{u}^T)$ , and $\boldsymbol{E} = -\nabla \theta$ is the electric field derived from electric potential $\theta$ .

DEM solves the coupled boundary value problem by minimizing the total electromechanical enthalpy functional $\mathcal{E} = \mathcal{E}_i + \mathcal{W}_{ext}$ :

$\mathcal{E}[\boldsymbol{u}, \theta] = \int_\Omega \left( \frac{1}{2} \boldsymbol{\epsilon} : \mathbf{C} : \boldsymbol{\epsilon} - \boldsymbol{E} \cdot \mathbf{e} : \boldsymbol{\epsilon} - \frac{1}{2} \boldsymbol{E} \cdot \boldsymbol{\kappa} \cdot \boldsymbol{E} \right) d\Omega - \int_{\partial \Omega_N} \boldsymbol{t}_N \cdot \boldsymbol{u} \, dA - \mathcal{W}_{elec}$

A multi-output deep neural network simultaneously approximates displacement $\boldsymbol{u}(\boldsymbol{x})$ and potential $\theta(\boldsymbol{x})$ , enabling simulation of direct piezoelectric effects (mechanical force inducing potential) and converse piezoelectric effects (electric voltage driving structural deformation) in a unified mesh-free model.
Knowl 8 — Variational Advantage of DEM for 4th-Order Kirchhoff Plate Bending
model/method
Transverse deflection $w(x, y)$ of a thin Kirchhoff plate under lateral load $p$ satisfies the 4th-order biharmonic equation $\nabla^4 w = p/D$ , where $D$ is the flexural rigidity. In Deep Collocation (DCM), enforcing governing equations and boundary conditions requires computing 4th-order derivatives via automatic differentiation and explicitly formulating loss terms for bending moments $M_n$ , twisting moments $M_{ns}$ , and shear forces $Q_n$ on free and supported boundaries:

$\mathcal{L}_{DCM}(\boldsymbol{p}) = \text{MSE}\left(\nabla^4 w_p - \frac{p}{D}\right) + \text{MSE}_{\Gamma_1}\left(w_p, \frac{\partial w_p}{\partial n}\right) + \text{MSE}_{\Gamma_2}(w_p, M_n) + \text{MSE}_{\Gamma_3}\left(M_n, \frac{\partial M_{ns}}{\partial s} + Q_n\right)$

In contrast, DEM minimizes the 2nd-order potential energy functional:

$\mathcal{E}[w] = \iint_\Omega \left( \frac{1}{2} \boldsymbol{\kappa}^T \mathbf{M} - p w \right) d\Omega - \int_{S_3} \bar{V}_n w \, dS + \int_{S_2+S_3} \bar{M}_n \frac{\partial w}{\partial n} \, dS$

where $\boldsymbol{\kappa} = [-\partial_{xx}w, -\partial_{yy}w, -2\partial_{xy}w]^T$ is the curvature vector. In DEM, natural boundary conditions (moments and effective shear forces) are automatically satisfied by the energy variation, reducing the boundary loss strictly to essential Dirichlet conditions ( $\mathcal{L}_{DEM} = \text{MAE}_{\mathcal{E}} + \text{MSE}_{\Gamma_1} + \text{MSE}_{\Gamma_2}$ ).
Knowl 9 — Tailored Harmonic Activation Functions and Autoencoder Architectures for Plates
model/method
To enhance accuracy and convergence speed when solving thin plate bending problems via the Deep Energy Method, standard activation functions such as $\tanh(x)$ are replaced by a tailored harmonic activation function:

$f(x) = \sin\left(\frac{\pi x}{2}\right)$

This tailored activation significantly reduces relative maximum deflection error (from $2.8 \times 10^{-5}$ down to $0.2 \times 10^{-5}$ ) and whole-domain deflection error on simply-supported square plates under sinusoidal loading.

Additionally, embedding an autoencoder feature extractor with two encoding layers (e.g., configurations $[3, 1] \times H$ or $[3, 2] \times H$ neurons per hidden layer) into the DNN prior to deflection output provides superior stability, faster optimization, and accurate capture of complex deformation modes in plates on elastic Winkler foundations ( $p_{reaction} = k w$ ) and annular plates containing circular cut-outs.
Knowl 10 — Empirical Accuracy of DEM on Linear Elastic Boundary Value Benchmarks
empirical result
The Deep Energy Method using fully connected networks (3 hidden layers, 30–50 neurons per layer, $\text{ReLU}^2$ activations in hidden layers, and linear output activation) trained with an initial Adam optimizer phase followed by L-BFGS optimization achieves the following relative $L_2$ error levels compared to exact analytical solutions:
1. Pressurized thick cylinder (2D quarter annulus, plane stress, $E = 10^5, \nu = 0.3$ ): $0.5\%$ displacement error, $3.7\%$ strain energy error.
2. Infinite plate with a circular hole under uniaxial tension (2D quarter domain, plane stress, $E = 10^5, \nu = 0.3$ ): $1.8\%$ displacement error, $3.19\%$ strain energy error.
3. Hollow sphere under internal pressure (3D one-eighth symmetry domain, $E = 10^3, \nu = 0.3$ ): $1.05\%$ displacement error, $4.44\%$ strain energy error.
4. Cube with a spherical hole under uniform tension (3D one-eighth domain, $E = 10^3, \nu = 0.3$ ): $5.3\%$ strain energy error.
Knowl 11 — Space-Time Deep Collocation for 1D Elastodynamic Wave Propagation
model/method
Time-dependent hyperbolic initial-boundary value problems, exemplified by the 1D elastodynamic wave equation $u_{tt} = c^2 u_{xx}$ on domain $\Omega = (0, L)$ and time interval $(0, T)$ with propagation speed $c=1$ , are solved using a space-time deep neural network $\hat{u}(x, t; \boldsymbol{p})$ that takes both spatial coordinate $x$ and temporal coordinate $t$ as network inputs.

The network parameters $\boldsymbol{p}$ are optimized by minimizing a composite collocation loss function combining PDE residuals, initial conditions, and boundary conditions:

$\mathcal{L}(\boldsymbol{p}) = \frac{1}{N_{int}} \sum_{i=1}^{N_{int}} \left( \hat{u}_{tt}(x_i^{int}, t_i^{int}) - c^2 \hat{u}_{xx}(x_i^{int}, t_i^{int}) \right)^2 + \frac{1}{N_{init}} \sum_{j=1}^{N_{init}} \left[ (\hat{u}(x_j^{init}, 0) - u_0(x_j^{init}))^2 + (\hat{u}_t(x_j^{init}, 0) - v_0(x_j^{init}))^2 \right] + \frac{1}{N_{bnd}} \sum_{k=1}^{N_{bnd}} \left[ (\hat{u}_x(0, t_k^{bnd}) - \bar{u}(t_k^{bnd}))^2 + (\hat{u}(L, t_k^{bnd}) - g(t_k^{bnd}))^2 \right]$

For a rod with $L=4, T=2$ , zero initial conditions, and sinusoidal boundary traction $u_x(0, t) = -\sin(\pi t)$ for $0 \le t \le 1$ , a space-time DNN (3 hidden layers, 50 neurons each) evaluated with $N_{int}=199^2, N_{init}=199, N_{bnd}=200$ achieves relative $L_2$ errors of $1.422 \times 10^{-3}$ in displacement and $5.955 \times 10^{-3}$ in velocity.
Knowl 12 — Theoretical and Computational Limitations of the Deep Energy Method
limitation
While the Deep Energy Method provides an expressive, mesh-free framework for continuum mechanics, it has distinct theoretical and practical limitations:
1. Non-convex optimization: Even for strictly linear and convex physical problems (such as linear elasticity), parameterizing the trial displacement field with a multi-layer neural network transforms the energy minimization into a non-convex finite-sum optimization problem $\min_{\boldsymbol{p}} \sum_i f_i(\boldsymbol{p})$ , creating topological challenges in the loss landscape and risking convergence to local minima.
2. Derivative accuracy degradation: Because displacement fields are the direct network outputs, derived quantities requiring differentiation (e.g., strain energy density and stress tensors computed via automatic differentiation) consistently exhibit higher relative errors than the primary displacement field (e.g., $3.7\% - 5.3\%$ error in strain energy versus $0.5\% - 1.8\%$ in displacement).
3. Computational training cost: Initial training times for DEM using gradient-descent and quasi-Newton optimizers (Adam/L-BFGS) are substantially higher than direct linear solvers in classical finite element (FEM) or isogeometric analysis (IGA) codes, although inference is rapid once trained.

Coverage note — Specific Python/TensorFlow code syntax listings from Section 5 and intermediate hyperparameter trial steps were omitted as they represent implementation boilerplate rather than standalone scientific contributions.

References

1.T. J. Hughes, The finite element method: linear static and dynamic finite element analysis, Courier Corporation, 2012.
2.A. Huerta, T. Belytschko, S. Fern'andez-M'endez, T. Rabczuk, X. Zhuang, M. Arroyo, Meshfree methods, Encyclopedia of Computational Mechanics Second Edition (2018) 1–38.
3.T. J. Hughes, G. Sangalli, M. Tani, Isogeometric analysis: Mathematical and implementational aspects, with applications, in: Splines and PDEs: From Approximation Theory to Numerical Linear Algebra, Springer, 2018, pp. 237–315.
4.I. Goodfellow, Y. Bengio, A. Courville, Deep learning, MIT press, 2016.
5.M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard, et al., Tensorflow: A system for large-scale machine learning, in: 12th {USENIX} Symposium on Operating Systems Design and Implementation ({OSDI} 16), 2016, pp. 265–283.
6.N. Ketkar, Introduction to pytorch, in: Deep learning with python, Springer, 2017, pp. 195–208.
7.M. S. Alnæs, A. Logg, K. B. Ølgaard, M. E. Rognes, G. N. Wells, Unified form language: A domain-specific language for weak formulations of partial differential equations, ACM Transactions on Mathematical Software (TOMS) 40 (2) (2014) 9.
8.M. Raissi, Deep hidden physics models: Deep learning of nonlinear partial differential equations, The Journal of Machine Learning Research 19 (1) (2018) 932–955.
9.M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations, arXiv preprint arXiv:1711.10561.
10.B. Dacorogna, Introduction to the Calculus of Variations, World Scientific Publishing Company, 2014.
11.J. Baiges, R. Codina, I. Castanar, E. Castillo, A finite element reduced order model based on adaptive mesh refinement and artificial neural networks.
12.T. Kirchdoerfer, M. Ortiz, Data-driven computational mechanics, Computer Methods in Applied Mechanics and Engineering 304 (2016) 81–101.
13.L. Ruthotto, E. Haber, Deep Neural Networks Motivated by Partial Differential Equations, arXiv e-print 1804.04272.
14.E. Weinan, B. Yu, The deep ritz method: A deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics 6 (1) (2018) 1–12.
15.L. Lei, C. Ju, J. Chen, M. I. Jordan, Non-convex finite-sum optimization via scsg methods, in: Advances in Neural Information Processing Systems, 2017, pp. 2348–2358.
16.P. Petersen, M. Raslan, F. Voigtlaender, Topological properties of the set of functions generated by neural networks of fixed size, arXiv preprint arXiv:1806.08459.
17.M. I. Jordan, Dynamical, symplectic and stochastic perspectives on gradient-based optimization, University of California, Berkeley.
18.X. Glorot, Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, in: Proceedings of the thirteenth international conference on artificial intelligence and statistics, 2010, pp. 249–256.
19.M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, et al., Tensorflow: Large-scale machine learning on heterogeneous distributed systems, arXiv preprint arXiv:1603.04467.
20.J. He, L. Li, J. Xu, C. Zheng, Relu deep neural networks and linear finite elements, arXiv preprint arXiv:1807.03973.
21.J. Opschoor, P. Petersen, C. Schwab, Deep relu networks and high-order finite element methods, SAM, ETH Zürich.
22.Y. Jia, C. Anitescu, Y. J. Zhang, T. Rabczuk, An adaptive isogeometric analysis collocation method with a recovery-based error estimator, Computer Methods in Applied Mechanics and Engineering 345 (2019) 52–74.
23.S. P. Timoshenko, J. Goodier, Theory of elasticity, McGraw-hill, 1970.
24.V. Avrutskiy, Enhancing approximation abilities of neural networks by training derivatives, arXiv preprint arXiv:1712.04473.
25.J. Berg, K. Nyström, A unified deep artificial neural network approach to partial differential equations in complex geometries, Neurocomputing 317 (2018) 28–41.
26.P. L. Gould, Y. Feng, Introduction to linear elasticity, Springer, 1994.
27.D. Schillinger, J. A. Evans, F. Frischmann, R. R. Hiemstra, M.-C. Hsu, T. J. Hughes, A collocated c0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics, International Journal for Numerical Methods in Engineering 102 (3-4) (2015) 576–631.
28.M. A. Scott, R. N. Simpson, J. A. Evans, S. Lipton, S. P. Bordas, T. J. Hughes, T. W. Sederberg, Isogeometric boundary element analysis using unstructured t-splines, Computer Methods in Applied Mechanics and Engineering 254 (2013) 197–221.
29.A. Logg, K.-A. Mardal, G. Wells, Automated solution of differential equations by the finite element method: The FEniCS book, Vol. 84, Springer Science & Business Media, 2012.
30.B. Bourdin, G. Francfort, J.-J. Marigo, Numerical experiments in revisited brittle fracture, Journal of the Mechanics and Physics of Solids 48 (4) (2000) 797–826. doi:10.1016/S0022-5096(99)00028-9.
31.A. Griffith, The Phenomena of Rupture and Flow in Solids, Philisophical Transactions of the Royal Society of London 221 (Series A) (1921) 163–198.
32.M. J. Borden, T. J. Hughes, C. M. Landis, C. V. Verhoosel, A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework, Computer Methods in Applied Mechanics and Engineering 273 (2014) 100–118. doi:10.1016/J.CMA.2014.01.016.
33.C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations, International Journal for Numerical Methods in Engineering 83 (10) (2010) 1273–1311. doi:10.1002/nme.2861.
34.C. Miehe, M. Hofacker, F. Welschinger, A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, Computer Methods in Applied Mechanics and Engineering 199 (45-48) (2010) 2765–2778. doi:10.1016/J.CMA.2010.04.011.
35.M. J. Borden, C. V. Verhoosel, M. A. Scott, T. J. Hughes, C. M. Landis, A phase-field description of dynamic brittle fracture, Computer Methods in Applied Mechanics and Engineering 217-220 (2012) 77–95. doi:10.1016/J.CMA.2012.01.008.
36.S. P. Timoshenko, S. Woinowsky-Krieger, Theory of plates and shells, McGraw-hill, 1959.
37.V. P. Nguyen, C. Anitescu, S. P. Bordas, T. Rabczuk, Isogeometric analysis: an overview and computer implementation aspects, Mathematics and Computers in Simulation 117 (2015) 89–116.

Citation

MLA

Samaniego, E., et al. “An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications”. Computer Methods in Applied Mechanics and Engineering, vol. 362, 2020, p. 112790, https://doi.org/10.1016/j.cma.2019.112790.

APA

Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V. M., Guo, H., Hamdia, K., Zhuang, X., & Rabczuk, T. (2020). An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 362, 112790. https://doi.org/10.1016/j.cma.2019.112790

Chicago

Samaniego, E., C. Anitescu, S. Goswami, et al. 2020. “An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications”. Computer Methods in Applied Mechanics and Engineering 362: 112790. https://doi.org/10.1016/j.cma.2019.112790.

Harvard

Samaniego, E. et al. (2020) “An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications”, Computer Methods in Applied Mechanics and Engineering, 362, p. 112790. Available at: https://doi.org/10.1016/j.cma.2019.112790.

Vancouver

1. Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh VM, Guo H, Hamdia K, Zhuang X, Rabczuk T (2020) An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering 362:112790

BibTeX

@article{Samaniego_2020, title={An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications}, volume={362}, ISSN={0045-7825}, url={http://dx.doi.org/10.1016/j.cma.2019.112790}, DOI={10.1016/j.cma.2019.112790}, journal={Computer Methods in Applied Mechanics and Engineering}, publisher={Elsevier BV}, author={Samaniego, E. and Anitescu, C. and Goswami, S. and Nguyen-Thanh, V.M. and Guo, H. and Hamdia, K. and Zhuang, X. and Rabczuk, T.}, year={2020}, month=Apr, pages={112790} }

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An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications

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Knowls

Knowl 1 — Deep Energy Method Formulation for Continuum Mechanics

Knowl 2 — Exact Satisfaction of Dirichlet Boundary Conditions via Output Transformation

Knowl 3 — Representation of 1D Linear Finite Element Spaces by ReLU Neural Networks

Knowl 4 — Deep Energy Method Formulation for Phase Field Modeling of Brittle Fracture

Knowl 5 — Superiority of DEM Over Collocation for Phase Field Derivative Discontinuities

Knowl 6 — Deep Energy Method for Finite Deformation Hyperelasticity

Knowl 7 — Deep Energy Method Formulation for Piezoelectricity

Knowl 8 — Variational Advantage of DEM for 4th-Order Kirchhoff Plate Bending

Knowl 9 — Tailored Harmonic Activation Functions and Autoencoder Architectures for Plates

Knowl 10 — Empirical Accuracy of DEM on Linear Elastic Boundary Value Benchmarks

Knowl 11 — Space-Time Deep Collocation for 1D Elastodynamic Wave Propagation

Knowl 12 — Theoretical and Computational Limitations of the Deep Energy Method

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