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On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling

Jan Niklas Fuhg, Nikolaos Bouklas

arXiv:2109.11028v1cs.CE

TL;DR

Data-driven constitutive models face challenges from ANN convergence and exact-fit limitations, while GPR has a big-data scaling challenge. This paper combines physics-informed isotropic and anisotropic surrogates with invariant-space sampling, and reports accurate predictions beyond training domains.

  • Problem

    ANNs lack convergence guarantees and can fail to exactly represent training points, while GPR scales poorly with large datasets.

    Source: §1 · p. 2

  • Method

    The paper trains local approximate GPR surrogates using invariant-based stress-generator mappings for isotropic and anisotropic materials, with space-filling sampling in invariant space.

    Source: §1 · p. 3, §7 · p. 22

  • Results

    The physics-informed technique vastly outperforms classical component mapping and accurately captures stress paths far outside the training domain.

    Source: §7 · p. 22

  • Takeaways & Limitations

    The proposed sampling technique generates evenly spread invariant-space samples for isotropic and anisotropic materials from a bounded deformation-gradient domain.

    Source: §7 · p. 22

    The isotropic coefficient-recovery matrix can be severely ill-conditioned depending on the number of unique principal strains.

    Source: §3.2 · p. 7

Abstract

from arXiv · show

Data-driven constitutive modeling is an emerging field in computational solid mechanics with the prospect of significantly relieving the computational costs of hierarchical computational methods. Traditionally, these surrogates have been trained using datasets which map strain inputs to stress outputs directly. Data-driven constitutive models for elastic and inelastic materials have commonly been developed based on artificial neural networks (ANNs), which recently enabled the incorporation of physical laws in the construction of these models. However, ANNs do not offer convergence guarantees and are reliant on user-specified parameters. In contrast to ANNs, Gaussian process regression (GPR) is based on nonparametric modeling principles as well as on fundamental statistical knowledge and hence allows for strict convergence guarantees. GPR however has the major disadvantage that it scales poorly as datasets get large. In this work we present a physics-informed data-driven constitutive modeling approach for isostropic and anisotropic materials based on probabilistic machine learning that can be used in the big data context. The trained GPR surrogates are able to respect physical principles such as material frame indifference, material symmetry, thermodynamic consistency, stress-free undeformed configuration, and the local balance of angular momentum. Furthermore, this paper presents the first sampling approach that directly generates space-filling points in the invariant space corresponding to bounded domain of the gradient deformation tensor. Overall, the presented approach is tested on synthetic data from isotropic and anisotropic constitutive laws and shows surprising accuracy even far beyond the limits of the training domain, indicating that the resulting surrogates can efficiently generalize as they incorporate knowledge about the underlying physics.

1 Introduction

The introduction motivates physics-informed constitutive surrogates and presents a GPR-based approach for isotropic and anisotropic materials that incorporates physical constraints and invariant-space sampling.

1 Introduction

  • Physics-guided constitutive models seek surrogates that obey physical conditions rather than relying solely on raw stress-strain data.The introduction lists objectivity, material symmetry, thermodynamic consistency, stress-free reference configuration, and angular-momentum balance among the relevant principles.Source: §1 · p. 2
  • ANNs lack convergence guarantees and depend substantially on user-specified parameters, while mean-error minimization can prevent exact representation of training points.The latter is a concern for preserving the stress-free undeformed configuration.Source: §1 · p. 2
  • GPR offers strict convergence guarantees and probabilistic mean and variance outputs, while local approximate GPR is proposed to address its big-data challenge.The introduction also cites GPR applications to data correction, adaptive modeling, and viscoelastic surrogates.Source: §1 · pp. 2–3
  • The proposed framework extends physics-informed GPR constitutive modeling to anisotropic materials and big-data settings, while incorporating five physical constraints and invariant-space sampling.The constraints include stress-free undeformed configuration, frame indifference, material symmetry, thermodynamic consistency, and local balance of angular momentum.Source: §1 · p. 3

2 Physics-based constraints for data-driven mappings

The section sets out physical requirements for hyperelastic mappings and motivates representing stress through invariant-dependent coefficients and stress generators. This construction supplies several physical properties by design, with exact-inference methods preserving additional properties at training points.

2 Physics-based constraints for data-driven mappings

  • Material symmetry requires the strain energy to respect the material symmetry group, while frame indifference requires observer-independent constitutive behavior.The section expresses these requirements through transformations of the strain energy and stress.Source: §2 · p. 4
  • The stress is represented as a linear combination of integrity-basis generators with coefficients depending on invariants, yielding symmetry and material-frame indifference by construction.For standard anisotropic cases, the section notes that suitable integrity bases are known.Source: §2 · p. 5
  • A dataset of strain inputs and stress outputs can be transformed into invariant inputs and corresponding stress-generator coefficients for surrogate training.The surrogate’s predicted coefficients are combined with the generators to approximate stress.Source: §2 · p. 5
  • With exact inference, choosing an invariant input for the undeformed configuration preserves the stress-free state, and thermodynamic consistency holds at training points.The section also states that symmetric generators ensure local angular-momentum balance.Source: §2 · p. 5

3 Physics-informed mapping for hyperelastic materials

The section contrasts classical component-wise mappings with physics-informed mappings based on stress generators and invariants. It develops the latter for isotropic and transversely isotropic materials, including how coefficients and consistent tangents are obtained.

3.1 Classical mapping approach

  • The classical approach maps upper-triangular components of the symmetric right Cauchy-Green tensor to those of the symmetric second Piola-Kirchhoff stress.It can use other symmetric strain and stress measures, and accommodates anisotropy present in the data.Source: §3.1 · p. 6

3.2 Physics-informed mapping approach for isotropic materials

  • For isotropic materials, the stress is decomposed into three generators, enabling a mapping from three invariants to three scalar coefficients instead of a six-to-six component mapping.The resulting stress is symmetric and satisfies material symmetry and frame indifference; the consistent tangent uses derivatives of the coefficients with respect to the invariants.Source: §3.2 · pp. 6–7

3.3 Physics-informed mapping approach for transversly isotropic materials

  • For transversely isotropic materials, stress uses six generators and maps five invariants to six coefficients, preserving symmetry, material symmetry, thermodynamic consistency, and frame indifference.The six coefficients can be estimated by least squares because the corresponding system is underdetermined; the consistent tangent depends on coefficient derivatives.Source: §3.3 · pp. 7–8
  • Orthotropic materials are not explicitly treated, though the section notes that their seven invariants and seven generators could support a similar coefficient-recovery approach.For the isotropic coefficient-recovery system, conditioning can be severe depending on the number of unique principal strains.Source: §3.2 · p. 7, §3.3 · p. 8

4 Space-filling sampling approach in invariant space

This section develops simulated-annealing methods for space-filling sampling in bounded, physically attainable invariant domains, extending the approach from principal invariants to anisotropic pseudo-invariants. For anisotropic samples, it varies tensor rotations while preserving primary invariants and numerically recovers a corresponding right Cauchy–Green tensor.

4 Space-filling sampling approach in invariant space

  • LHS samples of deformation gradients map to unevenly distributed invariant points, and the non-injective mapping can produce duplicate invariant inputs.These gaps can leave regions with low sample density and reduce the information gained from stress-response experiments.Source: §4 · p. 9, Figure 2 · p. 9
  • The method uses simulated annealing to generate evenly distributed physical samples in a predetermined principal-invariant region, requiring proposed positions to remain inside the bounded deformation-gradient domain and correspond to physical points.The move criterion decreases distance to the nearest neighbor of the remaining dataset relative to the sample’s previous position.Source: §4 · pp. 10, 12
  • After 7,500 annealing steps, 200 samples in the 17.5% input domain form a space-filling distribution in principal invariant space.The initial sample positions were not evenly spread; the example used T = 1 and α = 0.9995.Source: §4 · p. 13, Figure 5 · p. 14

4.1 Extension to anisotropic materials

  • Even when 500 samples are well spread in primary invariant space, their pseudo-invariants are clustered rather than space-filling.The method cannot directly test reachability of the full quintuple (I1, I2, I3, I4, I5), and a convex hull does not adequately identify where pseudo-invariant points can reside.Source: §4.1 · pp. 13–14
  • The anisotropic extension varies right Cauchy-Green tensor rotations to spread pseudo-invariants while preserving the primary invariants and ensuring reachable invariant combinations.A simulated-annealing search over three rotation angles changes pseudo-invariants without changing the already sampled principal invariants.Source: §4.1 · pp. 14–16
  • The resulting simulated-annealing procedure generates evenly spaced samples in pseudo-invariant space from fixed primary-invariant samples.The final positions are shown in Figure 7c after 10,000 iteration steps.Source: §4.1 · p. 14, Figure 7 · p. 15
  • The anisotropic sampler perturbs a candidate using a random direction and distance, then accepts invariant samples when their nearest-neighbor distance exceeds the current spacing.It constructs a test tensor using rotations and derives I4 and I5 from that tensor.Source: §4.1 · p. 16
  • The approach numerically solves a constrained nonlinear system to recover a right Cauchy–Green tensor for a sampled anisotropic invariant quintuple.The system matches I1–I5; bounds on the tensor come from the training deformation-gradient domain, and reachable quintuples are guaranteed by Algorithm 3.Source: §4.1 · p. 17
  • The sampling approach is not specific to particular constitutive responses and can generate samples for multiple problems.A real tensor solution can be obtained with a nonlinear solver when the input invariant quintuple is reachable.Source: §4.1 · p. 17

5 Gaussian process regression

The section presents Gaussian process regression (GPR) for constitutive surrogates and describes local approximate GPR as a way to make it tractable for larger datasets. It outlines the model’s covariance structure, parameter fitting, and use of nearby inducing points.

5 Gaussian process regression

  • GPR represents the input-output relationship as a realization of a Gaussian process, with a mean, output covariance structure, and independent Gaussian-process components.The inputs and outputs are represented as vectors, and output vectors are recombined for regression.Source: §5 · p. 17
  • The model uses Matérn 3/2 autocorrelation, assumes uncorrelated outputs, and fits the remaining parameters using restricted maximum likelihood.Under the uncorrelated-output assumption, the matrix A is set to the identity.Source: §5 · pp. 17–18

Local approximate Gaussian process regression

  • Local approximate GPR makes larger datasets tractable by fitting a local model from n ≪ N inducing points near each input of interest.The approach relies on nearby points having more influence on a prediction than distant points.Source: Local approximate Gaussian process… · p. 18
  • Each local-GPR prediction requires retraining its parameters, but with around 100 local data points it can be performed basically in real time using an efficient optimization algorithm.The authors report that this formulation accurately predicts complex constitutive relationships based on data.Source: Local approximate Gaussian process… · p. 18

6 Numerical tests

Across isotropic and transversely isotropic tests, physics-informed invariant-space surrogates outperform classical mapping, with space-filling samples improving results further. They also generalize more accurately beyond the 17.5% training domain.

6 Numerical tests

  • Tests use a 17.5% deformation-gradient training domain and 20,000 Latin-hypercube test points; duplicate invariant inputs are removed because they add no information and can ill-condition GPR covariance.Removing duplicates leaves classical-mapping datasets no larger than those used for physics-informed mappings.Source: §6 · pp. 18–19

6.1 Isotropic example

  • Invariant-space isotropic surrogates perform around a factor of 10 better than classical mapping despite using far fewer training points, with space-filling samples improving further.Invariant-space datasets contain 180–360 samples after duplicates are removed.Source: §6.1 · pp. 19–20, Figure 8 · p. 19
  • With 275 isotropic invariant-space training points, space-filling sampling yields uniformly lower errors, especially near test-domain boundaries where ordinary invariant samples are uneven.The comparison evaluates 20,000 test points and contrasts 2,500 deformation-gradient samples projected to 275 invariant-space samples with 278 space-filling samples.Source: §6.1 · p. 20, Figure 9 · p. 20
  • Beyond the 17.5% training domain, physics-informed isotropic models are significantly more accurate than classical mapping, and space-filling sampling improves results further.All approaches capture the response within the training region, while classical mapping produces non-physical results beyond it.Source: §6.1 · p. 20, Figure 10 · p. 21

6.2 Transversely isotropic example

  • For transversely isotropic tests, physics-informed models follow the true stress response far outside the training domain, and space-filling samples outperform deformation-gradient samples.The comparison uses 20,000 test points; the models capture responses within the training domain, while classical mapping becomes unreliable outside it.Source: §6.2 · pp. 21–22, Figure 11 · p. 22

7 Discussion and outlook

The discussion presents a physics-informed hyperelastic surrogate built from stress generators and invariant inputs, then reports improved performance over classical mapping and accurate stress-path prediction beyond training. It also proposes invariant-space space-filling sampling and identifies further generalization study as future work.

7 Discussion and outlook

  • The method represents second Piola–Kirchhoff stress as a linear combination of irreducible stress generators and maps input invariants to the combination's scalar coefficients.The surrogate uses local approximate Gaussian process regression, a nonparametric model with convergence guarantees.Source: §7 · p. 22
  • The trained models inherently capture stress-free undeformed configuration, angular-momentum balance, frame indifference, material symmetry, and thermodynamic consistency.Source: §7 · p. 22
  • The approach outperforms classical mapping and accurately captures stress paths far outside training; the proposed invariant-space sampling generates evenly spread samples from bounded deformation-gradient domains.The authors identify further study of out-of-domain generalization as future work.Source: §7 · p. 22
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