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A new family of Constitutive Artificial Neural Networks towards automated model discovery

Kevin Linka, Ellen Kuhl

arXiv:2210.02202v2cs.LGcond-mat.mtrl-sci

TL;DR

Classical Neural Networks can fit constitutive data but may violate physical constraints, extrapolate poorly, and overfit sparse datasets. This paper designs Constitutive Artificial Neural Networks that encode mechanics and thermodynamics into the network and learn both constitutive models and parameters. The resulting networks describe and predict benchmark rubber behavior while selecting physically structured model components, although insufficiently rich data can yield non-unique solutions.

  • Problem

    Classical Neural Networks can fit constitutive data but may ignore underlying physics, fail outside the training regime, and overfit when data are sparse.

  • Method

    The paper reverse-engineers network inputs, outputs, activation functions, and architecture to encode constitutive constraints and polyconvexity.

  • Results

    Constitutive Artificial Neural Networks describe and predict benchmark rubber behavior robustly and simultaneously learn constitutive models and material parameters.

  • Takeaways & Limitations

    The networks provide a physically constrained alternative to user-defined model selection by combining model discovery with parameter learning.

  • Takeaways & Limitations

    Insufficiently rich data can produce non-unique network solutions, with different activation-function subsets representing the stress response.

Abstract

from arXiv · show

For more than 100 years, chemical, physical, and material scientists have proposed competing constitutive models to best characterize the behavior of natural and man-made materials in response to mechanical loading. Now, computer science offers a universal solution: Neural Networks. Neural Networks are powerful function approximators that can learn constitutive relations from large data without any knowledge of the underlying physics. However, classical Neural Networks ignore a century of research in constitutive modeling, violate thermodynamic considerations, and fail to predict the behavior outside the training regime. Here we design a new family of Constitutive Artificial Neural Networks that inherently satisfy common kinematic, thermodynamic, and physic constraints and, at the same time, constrain the design space of admissible functions to create robust approximators, even in the presence of sparse data. We revisit the non-linear field theories of mechanics and reverse-engineer the network input to account for material objectivity, symmetry, and incompressibility; the network output to enforce thermodynamic consistency; the activation functions to implement physically reasonable restrictions; and the network architecture to ensure polyconvexity. We demonstrate that this new class of models is a generalization of the classical neo Hooke, Blatz Ko, Mooney Rivlin, Yeoh, and Demiray models and that the network weights have a clear physical interpretation. When trained with classical benchmark data for rubber, our network autonomously selects the best constitutive model and learns its parameters. Our findings suggests that Constitutive Artificial Neural Networks have the potential to induce a paradigm shift in constitutive modeling, from user-defined model selection to automated model discovery. Our source code, data, and examples are available at https://github.com/LivingMatterLab/CANN.

1 Motivation

Classical Neural Networks can fit constitutive data but provide little physical insight and may ignore established mechanics. The paper proposes Constitutive Artificial Neural Networks that embed physical knowledge into the model design.

  • Motivation: Classical Neural Networks were historically used as black-box regression operators for constitutive modeling.Their lack of transparency limited momentum in the constitutive-modeling community.
  • Motivation: Physics-Informed Neural Networks add physics equations to the loss, whereas Constitutive Artificial Neural Networks hardwire constraints into inputs, outputs, and architecture.The two strategies differ in how physical knowledge enters the network.
  • Motivation: Existing neural approaches can reproduce constitutive behavior, but their models and parameters often lack intuitive interpretation.This limitation is identified across approaches including thermodynamics-based and post-processing-corrected networks.
  • Motivation: The proposed Constitutive Artificial Neural Networks are designed as generalizations of widely used constitutive models with physically interpretable parameters.The manuscript reverse-engineers network components from continuum mechanics and constitutive modeling.

2 Kinematics

The kinematic framework represents deformation through the deformation gradient and derived Cauchy–Green tensors and invariants. Material assumptions such as isotropy and incompressibility reduce the variables needed to characterize constitutive behavior.

  • Deformation measures: The deformation map sends material particles X to their deformed positions x, and its gradient F describes local deformation.The Jacobian J is associated with the deformation gradient.
  • Deformation measures: The right and left Cauchy–Green tensors are defined as C = F^t · F and b = F · F^t.They provide deformation measures independent of the particular particle mapping representation.
  • Invariants: The principal invariants I1, I2, and I3 are constructed from the Cauchy–Green tensors and satisfy I1 = 3, I2 = 3, and I3 = 1 in the undeformed state.I3 equals J^2.
  • Incompressibility: Near incompressibility decomposes F into volumetric and isochoric parts and uses the invariants Ī1, Ī2, and J.The isochoric deformation gradient has determinant one.
  • Incompressibility: For perfect incompressibility, J = 1 and the constitutive description reduces to the two invariants I1 and I2.The principal and isochoric first and second invariants coincide.
  • Material symmetry: Transverse isotropy introduces a fourth invariant I4 = n · F^t · F · n = λ^2 to represent a pronounced material direction.The structural direction n has unit length in the reference configuration.

3 Constitutive equations

The constitutive formulation constrains neural-network inputs, outputs, activations, and architecture using mechanics and thermodynamics. These choices reduce admissible functions while enforcing objectivity, thermodynamic consistency, physical restrictions, and polyconvexity.

  • Motivation: Learning P(F) directly can produce overly generic functions, violate thermodynamic and physical constraints, and overfit when data are limited.The constitutive design instead seeks to constrain the admissible function space.
  • Robustness: These constraints are intended to reduce overfitting risk and improve robustness when available data are sparse.The restricted parameterization is part of the constitutive-network design.
  • Thermodynamics: Thermodynamic consistency is enforced by learning the free energy ψ(F) and deriving the Piola stress P = ∂ψ/∂F.For hyperelastic materials, this satisfies the dissipation condition with zero dissipation.
  • Objectivity and symmetry: Material objectivity replaces deformation-gradient inputs with the right Cauchy–Green tensor C, while isotropy further represents response through invariants.These transformations make the constitutive response independent of rigid rotations and reference-frame changes.
  • Incompressibility: Perfect incompressibility reduces the network input from three invariants to only I1 and I2.This follows from the constraint I3 = J^2 = 1.
  • Physical restrictions: Physical restrictions constrain free energy to be non-negative, zero in the reference configuration, and divergent under extreme compression or expansion.The network uses invariant deviations and compatible activation functions to encode these conditions.
  • Polyconvexity: Polyconvexity motivates additive free-energy subfunctions ψ(F) = ψ1(I1) + ψ2(I2) + ψ3(I3), guiding the choice of activation functions.Power, exponential, and logarithmic forms are identified as possible polyconvex subfunctions.

4 Classical Neural Networks

Classical Neural Networks map deformation-gradient components to stress components using trainable weights, biases, activation functions, and layered architectures. Although flexible, they can violate constitutive constraints, overfit sparse data, and extrapolate poorly.

  • Limitations: Classical Neural Networks may violate thermodynamic consistency, material objectivity, symmetry, incompressibility, constitutive restrictions, and polyconvexity.Their high degrees of freedom also create risks of overfitting with sparse data and poor extrapolation outside the training regime.
  • Neural Network input and output: Classical Neural Networks approximate the nine components of Piola stress P(F) from the nine components of deformation gradient F.The network treats constitutive modeling as a tensor-to-tensor function-approximation problem.
  • Neural Network architecture: A fully connected feed-forward network passes information forward through hidden layers, with weighted connections and biases between successive layers.Each hidden-layer node receives weighted information from all nodes in the preceding layer and adds a bias before applying an activation function.
  • Activation functions: Activation functions transform weighted nodal inputs into signals for subsequent layers and may be continuous or discontinuous, linear or nonlinear, and bounded or unbounded.Common examples include identity, logistic, hyperbolic tangent, ReLU, ELU, and soft plus functions.
  • Neural Network architecture: For a simple tanh network, each hidden layer adds nested nonlinear transformations, producing a complex stress–deformation relation that is difficult to invert explicitly.The example’s final output approximates P1, but its weights and biases have no clear physical interpretation.
  • Activation functions: Constitutive restrictions rule out several common final-layer activation functions because they can be discontinuous, constant, or bounded under extreme deformation.The physically reasonable restrictions require stress and energy to increase appropriately and become unbounded for extreme deformations.

5 Constitutive Artificial Neural Networks

Constitutive Artificial Neural Networks are designed to encode thermodynamic, kinematic, material, and polyconvexity constraints directly into their inputs, outputs, activations, and architecture. Their structured free-energy representation generalizes established constitutive models while giving network weights physical meaning.

  • Network design: The networks approximate scalar free energy from deformation invariants and derive stress from that energy to enforce thermodynamic consistency.The Piola stress is obtained by differentiating the learned free energy with respect to the deformation gradient.
  • Network design: Inputs use deformation invariants, while selected invariant-specific connections avoid mixed terms to support material constraints and polyconvexity.For perfect incompressibility, the input can reduce to I1 and I2; the architecture decouples individual invariants.
  • Network design: Custom activation functions combine linear, quadratic, cubic, exponential, and logarithmic forms to represent physically reasonable constitutive terms.The functions are designed to be monotonic, smooth, zero at the origin, and unbounded for extreme deformations.
  • Network design: The eight-term incompressible example combines linear, quadratic, linear-exponential, and quadratic-exponential contributions from I1 and I2.Its redundant parameterization can be reduced to twelve network parameters, which scale the corresponding energy contributions.
  • Model interpretation: The resulting stress relation is a generalization of popular incompressible hyperelastic models, and its weights correspond to well-defined physical parameters.The authors demonstrate recovery of popular constitutive functions, including the neo Hooke, Blatz Ko, Mooney Rivlin, Yeoh, and Demiray families.
  • Model interpretation: With sparse data, the networks can interpolate and extrapolate within the validity range of their thermodynamic assumptions while reducing black-box behavior and overfitting risk.The constrained degrees of freedom and physically interpretable parameters are presented as practical advantages over classical neural networks.

6 Special homogeneous deformation modes

The paper evaluates the constitutive representation through three perfectly incompressible homogeneous deformation modes: uniaxial tension, equibiaxial tension, and pure shear. Their invariant–stretch and stress–stretch relations expose how individual network terms contribute to the predicted stresses.

  • Special homogeneous deformation modes: The three special modes are perfectly incompressible uniaxial tension, equibiaxial tension, and pure shear, each with a specified diagonal deformation gradient.Their deformation gradients are respectively diag {λ, λ^-1/2, λ^-1/2}, diag {λ, λ, λ^-2}, and diag {λ, 1, λ^-1}.
  • Uniaxial tension: For uniaxial tension, one stretch is λ and the two transverse stretches are λ^-1/2, producing P = diag {P1, 0, 0}.The transverse stress conditions determine the pressure before obtaining P1 as a function of λ.
  • Equibiaxial tension: For equibiaxial tension, two stretches equal λ and the third equals λ^-2, producing F = diag {λ, λ, λ^-2} and P = diag {P1, P2, 0}.The zero third-direction stress condition determines pressure, after which P1 and P2 are written in terms of λ.
  • Pure shear: For pure shear, the stretches are λ, 1, and λ^-1, producing F = diag {λ, 1, λ^-1} and P = diag {P1, P2, 0}.The incompressibility relation supplies the third stretch, and the stress expressions are obtained using the invariant derivatives.
  • Stress–stretch relations: Figure 7 decomposes stress–stretch responses into eight linear, quadratic, linear-exponential, and quadratic-exponential contributions from I1 and I2.The curves correspond to the eight nodes of the incompressible network and are shown for all three deformation modes.

7 Results

The study compares classical and Constitutive Artificial Neural Networks on benchmark rubber data across single-mode and multi-mode training. Classical networks interpolate well but struggle with extrapolation, sparse data, and physical interpretability, whereas the constitutive architecture is designed to address these issues.

  • Experimental design: The experiments systematically compare fully connected classical and Constitutive Artificial Neural Networks using benchmark rubber stress-stretch data.The benchmarks cover single-mode uniaxial tension and multi-mode uniaxial tension, equibiaxial tension, and pure shear.
  • Classical Neural Networks: Classical networks interpolate nonlinear rubber behavior well within 1 ≤ λ ≤ 8 but predict poorly for λ > 8.This behavior held across six networks varying in depth and breadth.
  • Classical Neural Networks: Classical networks robustly interpolate the six single-mode datasets, but sparse gum-stock data produce oscillations between 2.4 ≤ λ ≤ 6.4.That dataset contains only seven points, while the network has nθ = 25 parameters.
  • Classical Neural Networks: Classical networks robustly fit six multi-mode training sets, including S-shaped curves and pronounced stretch stiffening, but provide no physical insight.The loss decreased by four orders of magnitude within less than 20,000 epochs.
  • Constitutive Artificial Neural Networks: Constitutive Artificial Neural Networks fit six single-mode datasets well, with loss decreasing by four orders of magnitude within less than 10,000 epochs.Their input preprocessing uses invariants and post-processing generates stress from the learned free energy.

Constitutive Artificial Neural Networks generate non-unique solutions for insufficiently rich

With insufficiently rich data, Constitutive Artificial Neural Networks can fit the data equally well through different subsets of active weights. This produces non-unique stress representations and undermines direct identification of constitutive parameters.

  • Non-unique solutions: Six random initializations all reduced the loss by more than four orders of magnitude and interpolated the training data equally well.The comparison used a two-layer, eight-node network with twelve weights and custom-designed activation functions.
  • Non-unique solutions: Different trained networks activated different subsets of terms, with remaining output weights training to zero.Examples include [I1 −3]2 and [I2 −3] versus [I1 −3], [exp([I2 −3]) −1], and [exp([I2 −3]2) −1].
  • Non-unique solutions: The best weight selection was non-unique, making the intended correlation between network weights and physically interpretable constitutive parameters ambiguous.The authors identify insufficiently rich data as the relevant condition for this non-uniqueness.

Constitutive Artificial Neural Networks are a natural generalization of existing constitutive

Constitutive Artificial Neural Networks recover stress responses across multiple deformation modes while their active invariant-based terms correspond to established constitutive models. The learned contributions therefore connect network structure with recognizable material-model forms.

  • Multi-mode constitutive behavior: The Constitutive Artificial Neural Network learns free energy from invariants while training separately on uniaxial tension, equibiaxial tension, and pure shear data.The network uses two layers, eight nodes, twelve weights, and custom-designed activation functions.
  • Model correspondence: A dominant [I1 −2] term for 20° pure shear correlates with the Blatz Ko model.The comparison identifies the dominant green contribution in the learned stress representation.
  • Model correspondence: Interacting [I1 −1] and [I1 −2] terms for 20° and 50° pure shear correlate with the Mooney Rivlin model.The correspondence is based on the learned invariant-term contributions.
  • Model correspondence: A dominant [exp([I1 −3])] term for 50° uniaxial and equibiaxial tension correlates with the Demiray model.The result illustrates how different deformation-mode datasets activate different constitutive terms.

Constitutive Artificial Neural Networks identify a single unique model and parameter set

Constitutive Artificial Neural Networks can select active free-energy terms and map their weights to physically meaningful parameters. With multi-mode data, the network trains robustly and uniquely identifies both the constitutive model and its parameters.

  • Model and parameter identification: The network learns free energy as a function of invariants ψ(I1, I2) while simultaneously training on multiple deformation modes.The multi-mode example uses a two-layer, eight-node network with twelve weights and custom-designed activation functions.
  • Physical interpretation: Five network weights map to physically meaningful parameters: µ1 = 0.2370 MPa, a1 = 0.0582 MPa, a2 = 0.0013 MPa, b1 = 0.0387, and b2 = 0.0022.The parameters include a shear modulus, stiffness-like parameters, and unit-less exponential coefficients.
  • Model and parameter identification: Multi-mode data from uniaxial tension, biaxial tension, and pure shear yield robust and unique training outcomes.The authors contrast this with non-uniqueness observed when the data are insufficiently rich.
  • Model and parameter identification: Training autonomously selects weights that activate relevant free-energy terms while driving remaining weights to zero.This allows the network to identify a free-energy function and material parameters from a broad spectrum of alternatives.
  • Physical interpretation: A four-weight example maps to µ1 = 0.2830 MPa, µ2 = 0.0141 MPa, a1 = 0.0434 MPa, and b1 = 0.0541.These weights are likewise interpreted as shear moduli, a stiffness-like parameter, and an exponential coefficient.

8 Discussion

Constitutive Artificial Neural Networks learn constitutive models and their parameters together, addressing the physical opacity of classical Neural Networks. Their behavior can depend on initialization, producing multiple stress responses for the same architecture.

  • Six initializations of a two-layer, eight-node network with 12 weights produce six Piola stress functions against the same uniaxial rubber data.The figure uses stretch λ on the horizontal axis and color-coded node contributions to show the differing stress functions.
  • Constitutive Artificial Neural Networks simultaneously learn both the constitutive model and its material parameters.This replaces the conventional sequence of selecting a model and then fitting its parameters.
  • Classical Neural Networks approximate stress from deformation data while ignoring the underlying physics of constitutive relations.The constitutive relation is tensor-valued, mapping deformation measures to stress.

Constitutive Artificial Neural Networks include kinematical, thermodynamical, and physical

The proposed networks encode constitutive structure through their inputs, outputs, activation functions, and architecture while learning robustly from rubber data. They can recover interpretable models and parameters, although richer extensions may reduce sparsity of the weights.

  • Constitutive Artificial Neural Networks include kinematical, thermodynamical, and physical: The feed-forward network with two layers, eight nodes, and twelve weights represents uniaxial tension, equibiaxial tension, and pure shear data using custom activation functions.The figures compare predicted Piola stress P(λ) with rubber data at 20° and 50°.
  • Constitutive Artificial Neural Networks include kinematical, thermodynamical, and physical: The network uses strain invariants as input, free energy as output, and derives Piola stress as P = ∂ψ/∂F.For incompressible isotropic materials, its eight subfunctions combine into a free-energy approximation generalizing several classical models.
  • Constitutive Artificial Neural Networks include kinematical, thermodynamical, and physical: Classical Neural Networks interpolate benchmark data but fail to extrapolate and tend to overfit sparse data.Both network types reduce training error in fewer than 10,000 epochs, but only the constitutive design incorporates prior physical knowledge.
  • Constitutive Artificial Neural Networks include kinematical, thermodynamical, and physical: Constitutive Artificial Neural Networks remain descriptive and predictive without overfitting, with most weights becoming zero and subsets of activation functions selected.With insufficiently rich data, the overall free-energy function is robust but the individual I1 and I2 contributions are non-unique; multi-mode data removes this non-uniqueness.
  • Constitutive Artificial Neural Networks include kinematical, thermodynamical, and physical: The networks discover three-term rubber models resembling Mooney Rivlin and Demiray forms, while non-zero weights represent shear moduli, stiffness-like parameters, and exponential coefficients.The authors state that autonomous model and parameter selection removes the need for the user to choose a constitutive model.
  • Constitutive Artificial Neural Networks include kinematical, thermodynamical, and physical: The demonstrated setting is limited to perfectly incompressible isotropic materials, while future extensions may target compressible, anisotropic, and history-dependent materials.For more complex networks, the authors foresee that most weights might no longer train to zero.

9 Conclusion

Constitutive Artificial Neural Networks impose kinematical, thermodynamical, and physical constraints by design while restricting admissible functions for robust training. Unlike classical Neural Networks, they can describe, predict, and explain data while reducing overfitting risk.

  • Constitutive Artificial Neural Networks satisfy kinematical, thermodynamical, and physical constraints by design while constraining the admissible function space.The conclusion contrasts this design with classical Neural Networks' weaker ability to describe, predict, and explain data.
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