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Learning Constitutive Relations using Symmetric Positive Definite Neural Networks
Kailai Xu, Daniel Z. Huang, Eric Darve
TL;DR
Neural-network constitutive models can become numerically unstable, motivating a structure that respects key constitutive constraints. The paper introduces SPD-NN, which predicts a Cholesky factor of tangent stiffness and computes stress incrementally. Across hyperelastic, elasto-plastic, and multiscale problems, it reports improved numerical stability and robustness, while requiring full-field training data.
Problem
Neural-network constitutive models can exhibit numerical instabilities, while the numerical performance of physics-constrained alternatives in conservation equations remained insufficiently studied.
Method
SPD-NN predicts the lower-triangular Cholesky factor of a tangent stiffness matrix and uses an incremental constitutive form, with direct or indirect training depending on available data.
Results
SPD-NN showed better numerical stability than two other neural-network architectures across one-dimensional and two-dimensional hyperelastic, elasto-plastic, and multiscale material problems.
Takeaways & Limitations
The architecture provides a robust constitutive-modeling approach for finite element simulations involving hyperelastic, elasto-plastic, and multiscale fiber-reinforced materials.
Takeaways & Limitations
Training requires full-field strain-stress or displacement data, which may be unavailable when measurements are limited to surfaces of three-dimensional solids.
Abstract
from arXiv · showhide
We present the Cholesky-factored symmetric positive definite neural network (SPD-NN) for modeling constitutive relations in dynamical equations. Instead of directly predicting the stress, the SPD-NN trains a neural network to predict the Cholesky factor of a tangent stiffness matrix, based on which the stress is calculated in the incremental form. As a result of the special structure, SPD-NN weakly imposes convexity on the strain energy function, satisfies time consistency for path-dependent materials, and therefore improves numerical stability, especially when the SPD-NN is used in finite element simulations. Depending on the types of available data, we propose two training methods, namely direct training for strain and stress pairs and indirect training for loads and displacement pairs. We demonstrate the effectiveness of SPD-NN on hyperelastic, elasto-plastic, and multiscale fiber-reinforced plate problems from solid mechanics. The generality and robustness of the SPD-NN make it a promising tool for a wide range of constitutive modeling applications.
1. Introduction
The paper addresses numerical instability in neural-network constitutive models by introducing SPD-NN, which constrains tangent stiffness through a Cholesky factor. The approach targets weak strain-energy convexity and time consistency, and is demonstrated across several solid-mechanics applications.
- Motivation: Unconstrained neural-network constitutive models can produce numerical instabilities when embedded in conservation equations.Earlier physics-constrained networks improved validation accuracy, but their performance in the resulting conservation equations had not been studied.
- Contribution: SPD-NN focuses on numerical stability in hybrid models that combine conservation equations with neural-network constitutive relations.The architecture uses customized output layers to enforce its target constitutive properties.
- Architecture: The network predicts a tangent stiffness matrix constrained to be symmetric positive definite, which weakly enforces convexity of the strain-energy function.The SPD matrix is represented through a predicted lower-triangular Cholesky factor.
- Architecture: The incremental constitutive form is designed to preserve time consistency for path-dependent materials such as elastoplastic materials.The network maps strain and related quantities to a Cholesky factor rather than directly learning the nonlinear strain-stress map.
- Applications: The approach is evaluated with direct and indirect training data across hyperelastic, elasto-plastic, and multiscale material problems.The paper also presents a software library combining finite element methods with neural networks.
2. Background
The background formulates solid-mechanics governing equations and constitutive constraints relevant to stable data-driven modeling. It emphasizes convexity through symmetric positive definiteness, time consistency, and the scope assumptions underlying the paper’s material models.
- Governing Equations: The governing equations require constitutive relations that map deformation history to stress, alongside mass, displacement, stress, and body-force variables.The finite element formulation uses displacement discretization, internal and external forces, and generalized-α temporal integration.
- Governing Equations: The method covers both infinitesimal and finite deformations, with finite-deformation examples and infinitesimal-deformation examples reported separately.Finite-strain formulations use deformation-gradient and Piola–Kirchhoff stress descriptions.
- Constitutive Constraints: For hyperelastic materials, convex strain energy corresponds to a symmetric positive definite tangent stiffness matrix and supports material and numerical stability.In one dimension, this assumption is equivalent to a monotonically increasing strain-stress relation.
- Constitutive Constraints: Elasto-plastic constitutive relations are rate-independent and path-dependent, transitioning from elastic to permanent plastic deformation at yield.Plastic states remain on the yield surface, whereas elastic states satisfy a negative yield function.
- Constitutive Constraints: Time consistency requires stress increments to vanish as strain increments vanish and is crucial for convergence as the time step tends to zero.The tangent stiffness is independent of strain or stress rate in the rate-independent formulation.
- Constitutive Constraints: The paper identifies symmetric positive definiteness and time consistency as essential properties for data-driven constitutive models.The SPD property leads to a nonsingular stiffness matrix and is reported as crucial for numerical stability.
- Scope: Strain-softening is explicitly outside the scope of the paper’s positive-definiteness assumption.The text notes that exceptions to the convexity assumption exist.
3. Methodology
The methodology replaces direct stress prediction with an incremental SPD-NN architecture that predicts a Cholesky factor of the tangent stiffness matrix. This structure enforces physical properties intended to improve constitutive-model behavior in numerical solvers.
- Training strategy: SPD-NN learns constitutive relations through direct or indirect training procedures selected according to the available data.The paper distinguishes training from strain–stress data and training from other available measurements.
- Architecture: Unlike σ-NN and Δσ-NN architectures, SPD-NN outputs a lower triangular Cholesky factor Lθ of the tangent stiffness matrix Hθ.Stress is then computed using the incremental constitutive form.
- Physical constraints: The Cholesky construction guarantees a symmetric positive semidefinite tangent stiffness matrix, matching a property of commonly used constitutive relations.This structural constraint is the principal physical distinction from direct stress or stress-increment prediction.
- Physical constraints: SPD-NN satisfies the time-consistency condition for both path-dependent and path-independent constitutive relations.The cited comparison states that direct stress and stress-increment networks fail this condition for path-dependent materials.
- Material dependence: For nonlinear elastic materials, the tangent stiffness depends on the current strain, while linear elastic materials use a constant parametric stiffness tensor.The current-step strain is required to determine the stress at that step.
• Elasto-Plasticity
For elasto-plastic materials, SPD-NN blends elastic and plastic constitutive forms with a differentiable transition function while using structured tangent stiffness representations. The formulation incorporates an estimated yield strength and acknowledges additional history requirements for strongly hysteretic materials.
- Elastic–plastic transition: Elasto-plastic constitutive behavior transitions between elastic and plastic regimes, so the model uses separate approximations for each region.The elastic and plastic forms are combined because the transition point is not known exactly.
- Elastic–plastic transition: The differentiable transition function D(σn, ˜σY) blends the elastic and plastic forms with values between 0 and 1.Differentiability is needed to evaluate the gradient during indirect-data training.
- Elastic–plastic transition: The estimated yield strength ˜σY need not be accurate, but it should be smaller than the actual yield strength σY.The plastic form can degenerate to the linear elastic form, whereas the reverse is not true.
- History dependence: For strongly hysteretic materials, more than one time step of historical data might be needed because the plastic-form history used here spans only one time step.The paper frames this as a possible requirement rather than a universal condition.
- Orthotropic representation: For the orthotropic materials considered, the tangent stiffness is block diagonal and its Cholesky factor is a corresponding structured lower triangular matrix.The neural network outputs only the nonzero entries of this factor.
The Drucker’s postulate requires
The paper develops direct and indirect training procedures for neural-network constitutive relations, including sequential residual evaluation for full-field data and stress-field initialization using quadratic elements.
- The proposed constitutive-relations framework supports separate direct and indirect training methods according to the available data.Direct data contain strain–stress pairs, whereas indirect data contain deformation–load full-field sequences.
- Direct data are obtained from strain–stress or strain–stress-increment pairs, including measurements and numerical simulation results.The data are organized as N sequences observed at n time snapshots.
- Indirect training uses full-field deformation–load sequences and computes acceleration and stress quantities for a residual-based minimization problem.The stress predictions and back-propagated gradients must be evaluated sequentially across time steps.
- Indirect-data training resembles recurrent neural-network training because stress states are propagated sequentially through time.This sequential computation makes parallelization difficult and exposes training to exploding and vanishing gradients.
- 3.3. Initialization for Indirect Data Training: The indirect-training initialization addresses underdetermined stress recovery by using quadratic displacement elements and linear stress fields.The resulting stress-field reconstruction is solved by least squares because the discretized system becomes overconstrained.
- 3.3. Initialization for Indirect Data Training: Although the least-square stress approximation can have errors larger than 100%, it remains qualitatively correct and sufficient for obtaining a good initial guess.The initialized neural network is then pre-trained using the direct-data technique.
4. Applications
The applications evaluate SPD-NN on one-dimensional elasto-plastic trusses and two-dimensional hyperelastic, elasto-plastic, and multiscale fiber-reinforced plates. Across these tests, SPD-NN provides stable finite-element predictions and approximates reference constitutive responses, while also accelerating multiscale simulations.
- 1D truss: The 1D truss data capture plastic loading, strain hardening, and elastic unloading under multiple loading conditions.The truss has four elements, with one loading condition held out for testing.
- 1D truss: Although all architectures fit strain-stress curves well in the NN test, most σ-NN and ∆σ-NN models become numerically unstable in the NN-FEM test.The paper attributes this instability to violated SPD constraints and accumulated errors under slight extrapolation from the training conditions.
- 1D truss: SPD-NN produces stable NN-FEM predictions whose displacement trajectories and strain-stress curves overlap the exact results.The result is reported for the held-out finite-element loading scenario.
- 2D plates: For the plate applications, SPD-NN predictions agree reasonably with reference displacement and stress fields, including hyperelastic, elasto-plastic, and multiscale cases.Both direct and indirect training are used where the available data require them.
- Multiscale fiber-reinforced plate: SPD-NN homogenized simulations reduce fiber-resolved computational time from around 24 hours to a few minutes, subject to serial-benchmark and resolution caveats.The paper notes that parallelization and fiber-resolution choices affect this comparison.
5. Discussion on Neural Networks
The discussion examines why neural networks are used to approximate Cholesky factors and how training choices affect accuracy, scaling, optimization, and reliability. It also highlights sensitivity to initialization and the slower cost of indirect training.
- Function approximators: Neural networks are used to approximate complex, nonsmooth constitutive relations with uneven data distributions and relatively high input and output dimensions.The discussion contrasts these requirements with limitations of traditional basis functions.
- Optimization method: The implementation uses L-BFGS-B with line search rather than stochastic gradient methods to optimize the constitutive-model loss functions.The method attempts to enforce the Wolfe conditions during line search.
- Data scaling: Scaling inputs and outputs to similar magnitudes helps train SPD-NNs faster, reduces overfitting, and improves predictions.The scaling introduces strain and stress reference quantities and scales the tangent stiffness matrix.
- Data scaling: The elasto-plastic SPD-NN uses a Cholesky-factor form and scales inputs such as strain and stress histories to comparable magnitudes.The strain reference is chosen on the order of the estimated Young’s modulus.
- Local minima: Neural-network optimization exhibits local minima: models with minimal training losses perform well, whereas median-loss models are less satisfactory.Pre-training for indirect data reduces initialization concerns, while direct training is about four times faster than indirect training.
6. Conclusion
Data-driven constitutive models couple neural networks with conservation equations, but training the current SPD-NN approach requires full-field strain–stress or displacement data. SPD-NN improves numerical robustness through its structured tangent-stiffness output, while incomplete-data settings remain open for future work.
- SPD-NN embeds neural-network material models in conservation equations to form coupled hybrid systems.
- SPD-NN outputs the Cholesky factor of the tangent stiffness matrix instead of stress or stress increments, weakly imposing strain-energy convexity and preserving time consistency.
- SPD-NN exhibits better numerical stability than two other neural-network architectures in tested one-dimensional and two-dimensional solid-mechanics problems.
- Current training requires full-field strain–stress pairs or displacement measurements, whereas incomplete-data constitutive modeling remains for future investigation.Surface-only measurements on a 3D solid body are given as an example of incomplete data.