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A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches
Wei Li, Martin Z. Bazant, Juner Zhu
TL;DR
Machine-learning models for engineering can require large datasets, motivating physics-guided alternatives. This paper develops a neural-network framework for elastic plates governed by FvK equations and compares three loss formulations across four loading cases. All approaches can achieve satisfactory accuracy when properly trained, while energy-based training offers simpler hyperparameter tuning and greater computational efficiency but depends more strongly on sampling.
Problem
Machine-learning accuracy depends on large training datasets, which may be unavailable or costly, motivating physics-guided modeling for elastic-plate mechanics.
Method
A neural network maps spatial coordinates to elastic-plate displacement fields while training incorporates data, FvK PDEs and boundary conditions, or total potential energy.
Results
PDE-based and energy-based networks approximately match finite-element responses with satisfactory accuracy when hyperparameters are properly tuned, while the energy-based model uses fewer hyperparameters and is more computationally efficient.
Takeaways & Limitations
Energy-based training simplifies hyperparameter tuning and improves computational efficiency, whereas PDE-based training is less sampling-dependent and may be mesh-free.
Takeaways & Limitations
The framework inherits classic Kirchhoff plate theory’s limited applicability to moderately thick plates and can show relatively large local deviations for highly nonlinear loading cases.
Abstract
from arXiv · showhide
One of the obstacles hindering the scaling-up of the initial successes of machine learning in practical engineering applications is the dependence of the accuracy on the size of the database that "drives" the algorithms. Incorporating the already-known physical laws into the training process can significantly reduce the size of the required database. In this study, we establish a neural network-based computational framework to characterize the finite deformation of elastic plates, which in classic theories is described by the Föppl--von Kármán (FvK) equations with a set of boundary conditions (BCs). A neural network is constructed by taking the spatial coordinates as the input and the displacement field as the output to approximate the exact solution of the FvK equations. The physical information (PDEs, BCs, and potential energies) is then incorporated into the loss function, and a pseudo dataset is sampled without knowing the exact solution to finally train the neural network. The prediction accuracy of the modeling framework is carefully examined by applying it to four different loading cases: in-plane tension with non-uniformly distributed stretching forces, in-plane central-hole tension, out-of-plane deflection, and buckling under compression. \hl{Three ways of formulating the loss function are compared: 1) purely data-driven, 2) PDE-based, and 3) energy-based. Through the comparison with the finite element simulations, it is found that all the three approaches can characterize the elastic deformation of plates with a satisfactory accuracy if trained properly. Compared with incorporating the PDEs and BCs in the loss, using the total potential energy shows certain advantage in terms of the simplicity of hyperparameter tuning and the computational efficiency.
1 Introduction
The introduction motivates physics-guided machine learning as a response to data dependence and positions this study as a neural-network framework for nonlinear elastic-plate mechanics.
- Motivation: Machine-learning accuracy depends strongly on the quantity and quality of training data, limiting applications where large datasets are unavailable or costly.Physics-based models are less reliant on dataset size because their governing laws are known in advance.
- Motivation: Physics-guided data-driven methods seek to combine machine learning’s handling of high-dimensional systems with physics-based modeling’s reduced data reliance.The paper focuses on artificial neural networks as the data-driven component.
- Related work: Existing physics-guided studies implement physical laws through physics-generated datasets, physics-guided model designs, or training-process constraints.The categories are not mutually exclusive and can be combined in one algorithm.
- Related work: Training-process approaches include PINNs, which impose PDEs and boundary or initial conditions in the loss, and deep Ritz methods, which use the PDE-associated energy functional.These methods can also support inverse identification of unknown physical parameters.
- Study scope: This study develops and evaluates a neural-network framework for elastic plates governed by high-order, highly nonlinear Föppl–von Kármán equations.It compares PDE-and-BC-based and total-potential-energy-based losses across four nonlinear loading cases.
2 Physics to be implemented: theory of elastic plates
The plate model derives equilibrium and boundary conditions from virtual work under Kirchhoff plate assumptions, yielding the Föppl–von Kármán equations and their displacement-based variables.
- Variational formulation: Elastic-plate governing equations can be derived from adjacent equilibrium or from an energy method based on virtual work.The energy method additionally provides a complete description of boundary conditions leading to a unique solution.
- Variational formulation: The principle of virtual displacement states that equilibrium makes the variation of total potential energy vanish: δΠ = 0.The variation operator δ represents changes among admissible configurations.
- Plate kinematics: The theory adopts Kirchhoff hypotheses: transverse normals remain straight and normal to the mid-surface, while plate thickness remains unchanged after deformation.These assumptions define the classical thin-plate kinematics used by the framework.
- Plate kinematics: The displacement field uses in-plane components u and uy and transverse displacement w, from which membrane strains and bending curvatures are obtained.Membrane strains represent in-plane deformation, whereas curvatures arise from transverse bending.
- Forces and boundary conditions: The plate’s internal virtual work combines membrane forces and bending moments, while external virtual work incorporates distributed pressure and mixed traction-displacement boundary conditions.Boundary quantities are expressed in local normal and tangential directions.
- Governing equations: The resulting FvK system contains two second-order in-plane equations and one fourth-order out-of-plane equation, requiring boundary conditions for a unique plate solution.The energy derivation exposes both natural and geometric boundary conditions, with four independent primary variables.
3 Computational framework based on ANN
The computational framework represents plate displacement with a fully connected neural network and trains it using data-driven, PDE-and-BC-based, or total-potential-energy-based losses.
- ANN architecture: The framework approximates the difficult FvK solution with a neural network that maps spatial coordinates to displacement fields.The network takes (x, y) as inputs and predicts (ux, uy, w).
- ANN architecture: The fully connected network contains input, hidden, and output layers, with weights, biases, and nonlinear activation functions defining its recursive transformations.ReLU, logistic sigmoid, and hyperbolic tangent are listed as activation choices; the output layer has no activation function.
- Loss functions: The study compares a purely data-driven loss with losses based on the governing PDEs and boundary conditions or on total potential energy.The purely data-driven formulation requires observed displacement fields from experiments or simulations.
- Loss functions: The PDE-based loss minimizes residuals of the three governing PDEs together with static and kinematic boundary-condition residuals.Loss weights balance stress and displacement boundaries, and minimizing the total loss approximates the exact solution mathematically.
- Loss functions: The energy-based loss minimizes total potential energy, combining internal energy, external work, and a penalty term for kinematic boundary conditions.The penalty vanishes when kinematic boundary conditions are satisfied, while energy integrals are evaluated numerically from sampled domain and boundary points.
- Training dataset: Purely data-driven training requires a sufficiently large observed displacement dataset, whereas the framework addresses this challenge by training without knowing the exact solution.High-order FvK equations make obtaining an exact or satisfactory training dataset difficult even for elastic plates.
4 Applications and validations
The framework is validated against finite-element simulations across in-plane tension, central-hole tension, out-of-plane deflection, and buckling. Physics-guided models generally achieve accurate predictions, while energy-based training offers computational and tuning advantages but remains subject to sampling and modeling limitations.
- Four loading cases validate the neural-network framework against fine-mesh finite-element simulations treated as approximate exact solutions.The cases include non-uniform in-plane tension, central-hole tension, out-of-plane deflection, and compression buckling.
- 4.1 In-plane tension with non-uniformly distributed stretching force: The displacement-only data-driven model predicts displacement well but gives less accurate membrane forces because differentiation amplifies errors.
- 4.1 In-plane tension with non-uniformly distributed stretching force: Energy-based training requires 20 minutes versus 160 minutes for PDE-based training because it avoids high-order derivative calculations.
- 4.1 In-plane tension with non-uniformly distributed stretching force: Reducing sampling substantially lowers energy-based accuracy, whereas the PDE-based model retains relatively high accuracy in the tested case.The energy-based disadvantage is magnified for stress-concentration problems, and sufficiently dense sampling is needed near the hole edge.
- 4.2 In-plane central-hole tension: Central-hole tension is more difficult: both physics-guided models capture stress concentration qualitatively, but the energy-based model more closely matches finite-element membrane-force results.The PDE-based model can potentially reach similar accuracy after extensive hyperparameter tuning.
- 4.3 Out-of-plane deflection of a square plate: The energy-based model satisfactorily predicts out-of-plane deflection and correctly predicts buckling configurations under both simply-supported and clamped boundary conditions.The buckling simulations use a first-mode geometric imperfection with a maximum transverse deviation of 0.01 mm.
5 Discussions
Across four loading cases, the framework was evaluated against FE simulations, while PDE-based and energy-based losses were compared for tuning, sampling, and efficiency trade-offs. The energy-based approach matched FE results satisfactorily but remained sensitive to sampling and the underlying plate-theory and network assumptions.
- Comparison through the four loading conditions: Four loading cases provided a broad accuracy assessment spanning in-plane tension, central-hole tension, out-of-plane deflection, and compression buckling.The central-hole study varied hole aspect ratios and included a three-hole plate.
- Comparison through the four loading conditions: The energy-based model remained stable as the stress concentration factor increased, despite a small difference from FE results in central-hole tension.The three-hole case was used to push the framework toward its computational limit.
- Comparison between the PDE-based and energy-based approaches: The PDE-based loss uses more hyperparameters and can fail to match results when non-optimal values are selected, although proper tuning can yield similar accuracy.This difficulty was observed in the second in-plane example.
- Comparison between the PDE-based and energy-based approaches: The energy-based approach depends more strongly on training-sample size and resolution because numerical integration of total potential energy discretizes the domain.Its sampling strategy is close to meshing, but strict mesh-quality requirements are absent.
- Comparison between the PDE-based and energy-based approaches: At comparable accuracy, the energy-based approach is more computationally efficient, including both convergence time and user time spent tuning hyperparameters.Its loss also avoids some derivative-order and residual-count burdens present in the PDE-based formulation.
- Limitations of the proposed neural network framework: The framework inherits the limited applicability of classical plate theory and the simple network has limited ability to approximate highly non-uniform displacement or strain fields.The study did not use a deeper network because of computational-resource limitations.
- Future extensions of the proposed neural network framework: For systems governed by many PDEs and BCs where minimum potential energy applies, minimizing potential energy may be more effective and efficient than summing PDE and BC residuals.The authors present this as a generalizable conclusion and identify thermodynamic and multiphysics systems as potential extensions.
6 Conclusion
The framework predicts elastic-plate responses with satisfactory accuracy, while PDE-based and energy-based losses offer complementary computational trade-offs. Energy-based training simplifies hyperparameter tuning and improves efficiency, whereas PDE-based training is less sampling-dependent.
- The neural-network framework predicts elastic-plate mechanical responses close to finite-element simulations when hyperparameters are properly tuned.
- Energy-based loss uses fewer hyperparameters and is computationally more efficient than PDE-based loss.
- Energy-based training requires large sampling sizes and fine sampling resolution.
- PDE-based loss is less dependent on sampling and has potential to be mesh-free.
- Purely data-driven training is suggested to use both displacement and membrane-force fields for good accuracy.
- The main difference between energy-based and PDE-based approaches concerns loss weighting and derivative calculation.
- The energy-based framework is expected to inspire modeling of complex engineering systems involving multiple scales and multiple physics.
A.1 Strain components
The appendix derives plate strain components from the Kirchhoff displacement field under a moderate-deformation assumption. Transverse displacement gradients may be large, while in-plane gradients remain small.
- Following the Kirchhoff hypotheses, the displacement field is expressed using in-plane and out-of-plane displacement components.
- The general three-dimensional second-order nonlinear Green strains provide the starting strain description.
- Moderate deformation permits relatively large transverse displacement gradients but assumes small in-plane displacement gradients.
- Under this assumption, second-order Green-strain terms are omitted except those involving transverse-displacement derivatives.
- Substitution of the Kirchhoff displacement field simplifies the three-dimensional strains into the two-dimensional plate-theory strain components.
A.2 Integration by parts
The appendix applies integration by parts to the virtual-strain terms so that the variational formulation is expressed in terms of displacement variations and boundary contributions.
- Virtual strains are computed from virtual displacements and substituted into the variational formulation.
- Integration by parts transforms the first term so that displacement variation appears instead of its gradient.
- The boundary normal is represented by n = n_x e_x + n_y e_y, with n_x and n_y as unit-normal direction cosines.
- The second term requires integration by parts twice.
A.3 Boundary conditions
The appendix transforms boundary quantities from global Cartesian coordinates to local boundary coordinates, then rewrites stress-boundary integrands in the local system. A mixed boundary term vanishes for closed stress boundaries or when M_ns = 0.
- A coordinate transformation maps global Cartesian coordinates (x, y, z) to local coordinates (n, s, r).
- The transformation uses the angle θ between the global x axis and the local n axis.
- Displacements and stresses are related between the global and local coordinate systems.
- Stress-boundary integrands are rewritten using quantities expressed in the local coordinate system.
- Integration by parts is applied to resolve an inconsistency in the boundary formulation.
- The term [M_ns δw]_Γσ vanishes when the stress boundary is closed or M_ns = 0.