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Thermodynamically Consistent Machine Learning for Fracture Prediction in Heterogeneous Materials

Muhamad Alim, Xiaolong He, Mei-Na Tsao, Chi-Hua Yu, J. S. Chen

arXiv:2610.20614v1physics.comp-phcond-mat.mtrl-scics.CEcs.LG

TL;DR

Fracture simulations are computationally expensive, and existing machine-learning fracture models can act as black boxes with substantial data needs. The paper introduces a geometry-conditioned, thermodynamics-informed latent-dynamics framework; across three numerical examples, it predicts fracture evolution more accurately than AE-LSTM in approximately 70%–75% of testing samples.

  • Problem

    Physics-based fracture simulations are time-consuming, while many machine-learning fracture models are black boxes and require large training datasets to generalize reliably.

    Source: §1 · pp. 1–2

  • Method

    The framework combines geometry-conditioned autoencoders with pGFINNs to learn thermodynamically consistent fracture dynamics in a latent space.

    Source: §1 · p. 3, §2.2 · p. 4

  • Results

    In approximately 70%–75% of testing samples, tLaSDI-pGFINN predicts fracture evolution more accurately than AE-LSTM.

    Source: §6 · p. 29

  • Takeaways & Limitations

    Across three numerical examples with varied loading, inclusion configurations, and crack initiation locations, the framework shows good agreement with reference solutions.

    Source: §6 · p. 29

    The geometry autoencoder is unnecessary in the single-cavity example because its geometry is simple and invariant across samples.

    Source: §5.1 · p. 15

Abstract

from arXiv · show

Understanding fracture behavior is critical for assessing the structural integrity and longevity. Highly time-consuming physics-based fracture simulations motivate the development of machine-learning-based fracture evolution prediction with surrogate models. However, most existing machine-learning models for fracture prediction operate as black boxes and require large training datasets to generalize reliably. In this work, Thermodynamically Consistent Latent Dynamics Identification (tLaSDI) for parametric systems is introduced to predict fracture patterns and evolution in heterogeneous materials. The model learns the thermodynamic state variable in a lower-dimensional latent space using parametric GENERIC Formalism-Informed Neural Networks (pGFINNs) that enforce key thermodynamic principles, such as total energy conservation and entropy generation. This model embeds physics into the neural network architecture using a hard-constraint method that inherently complies with the underlying thermodynamic principles. The proposed tLaSDI-pGFINNs is further enhanced by a Geometry-Conditioned Autoencoder in learning the geometry configuration for effective geometry-guided learning. The energy and entropy evolution histories demonstrate that the model preserves the thermodynamic constraints on unseen testing samples. The results further show that tLaSDI-pGFINNs can more accurately capture both fracture patterns and the subsequent fracture evolution than the purely data-driven Autoencoder Long Short-Term Memory (AE-LSTM) model.

1. Introduction

Fracture in heterogeneous materials is difficult to model because discontinuous properties produce rough deformation fields and configuration-dependent evolution, while conventional methods face computational or accuracy limitations. This work proposes a geometry-conditioned, thermodynamics-informed latent-dynamics framework for fracture prediction.

1. Introduction

  • Fracture prediction in heterogeneous materials remains challenging because discontinuous properties create rough deformation fields and configuration-dependent evolution.FEM approaches can require re-meshing, high mesh resolution near cracks, or mesh-dependent cohesive solutions; XFEM remains ineffective for three-dimensional complex patterns.Source: §1 · p. 1
  • Existing fracture machine-learning approaches include models that predict damage fields or patterns without predicting damage evolution in some settings.The cited CNN approach predicts a damage field and final peak load, but not damage evolution in composite specimens.Source: §1 · pp. 1–2
  • Soft physics constraints add governing-equation residuals to the loss, whereas hard constraints embed physics in the network architecture.The cited GENERIC Neural ODE preserves thermodynamic structure by construction; the cited penalty-based model can allow negative entropy production and fail to generalize in extrapolation.Source: §1 · p. 2
  • Applying thermodynamics-informed latent-space frameworks to fracture evolution in heterogeneous materials remains limited.The work builds on tLaSDI and parametric GFINNs for low-dimensional, parameter-dependent latent dynamics.Source: §1 · p. 3
  • The framework combines geometry-conditioned convolutional autoencoders with pGFINNs for thermodynamically consistent fracture evolution in latent space.It is tested on fracture problems varying boundary conditions, inclusion geometries and spatial distributions, and crack initiation locations.Source: §1 · p. 3

2. Thermodynamics-Informed Learning Framework

GENERIC represents dynamics through reversible and dissipative processes structured to preserve energy and produce non-decreasing entropy. pGFINNs parameterize the learned thermodynamic components and construct operator matrices to satisfy the required degeneracy conditions.

2.1. GENERIC Formalism

  • GENERIC separates reversible, nondissipative dynamics from dissipative processes using energy, entropy, a Poisson matrix, and a friction matrix.Its degeneracy conditions ensure energy conservation and non-decreasing entropy.Source: §2.1 · pp. 3–4
  • GENERIC trajectories conserve energy and have non-decreasing entropy, combining Hamiltonian and generalized gradient-flow behavior.When the friction matrix is zero, the dynamics reduce to Hamiltonian equations; when the Poisson matrix is zero and entropy is negative free energy, they reduce to a generalized gradient flow.Source: §2.1 · p. 4

2.2. Parametric Generic Formalism Informed Neural Networks

  • In pGFINNs, the Poisson matrix and friction matrix are constructed to satisfy entropy and energy degeneracy, respectively.The Poisson matrix construction uses a skew-symmetric factor and a matrix-valued function whose product with the entropy gradient is zero; the friction matrix construction similarly enforces zero product with the energy gradient.Source: §2.2 · p. 4
  • The networks for entropy, energy, and the matrix factors are parameterized by μ to represent parameter-dependent dynamics while preserving thermodynamic structure.The framework extends GFINNs, which do not inherently support governing operators with parametric variability.Source: §2.2 · pp. 4–5

3. Thermodynamics-Based Latent Dynamics Framework for Fracture Prediction

The framework combines latent-space parametric dynamics with geometry-conditioned decoding to predict fracture in heterogeneous materials. Its formulation links training losses to an approximation-error bound, while geometry conditioning improves damage-field reconstruction near material interfaces.

3.1. tLaSDI-pGFINN Formulation

  • tLaSDI-pGFINN encodes high-dimensional states into latent space, predicts future latent states autoregressively using system parameters, and decodes them back to physical states.Its training combines integration, reconstruction, Jacobian, and modeling losses to connect latent dynamics with the full-order system.Source: §3.1 · pp. 5–7

3.2. Error Analysis

  • The approximation-error estimate bounds total error by integration, reconstruction, Jacobian, and modeling terms, providing a theoretical foundation for the training losses.The bound assumes consistent initialization and a bounded, Lipschitz-continuous decoder Jacobian, with bounded pGFINN predictions.Source: §3.2 · pp. 7–9

3.3. Geometry Conditioned Autoenccoder

  • The Geometry-Conditioned Autoencoder injects geometry information into the decoder with FiLM modulation to guide reconstruction of geometry-sensitive damage fields.Its geometry and damage encoders produce compact latent representations used by the modulated damage decoder.Source: §3.3 · pp. 9–10
  • Geometry-conditioned reconstruction more clearly captures inclusion interfaces and horizontal crack propagation than reconstruction without geometry conditioning.The comparison includes ground truth, geometry-conditioned predictions, and predictions without geometry conditioning.Source: Figure 4 · p. 11, §3.3 · p. 11

4. Data Generation and Numerical Setup

The section describes thermodynamically consistent phase-field data generation and the AE-LSTM baseline used for comparison. It outlines the energy terms and fracture formulation, then explains recurrent and latent-state prediction.

4.1. Data Generation using Thermodynamically Consistent Phase-Field Model

  • The phase-field potential energy accounts for elastic energy, fracture-surface energy, and external work.The elastic-energy density and fracture-surface terms are defined in the accompanying formulation.Source: §4.1 · p. 11
  • Elastic energy is split into tensile and compressive parts; the tensile part is degraded by (1−d)^2, with k ≪ 1 maintaining well-posedness.The degradation applies to tensile elastic energy, while k is a small positive number.Source: §4.1 · p. 11
  • The standard model obtains u and d by minimizing Π(u,d), while cited modified procedures relate d to u through δΠ(u,d) = 0 with γ = 1.Source: §4.1 · p. 11
  • For irreversible fracture, the formulation employs H(x,t), and solves u and d in a staggered manner.Source: §4.1 · p. 12

4.2. Baseline Model

  • The AE-LSTM baseline uses an LSTM to learn input-output relationships by capturing temporal dependencies in sequential data.The study uses it to compare prediction performance with the thermodynamics-based model.Source: §4.2 · p. 12
  • A standard RNN updates hidden states from inputs and prior hidden states, but can fail to learn long-range dependencies as sequences lengthen.Vanishing gradients can prevent meaningful updates for earlier inputs, while exploding gradients can destabilize training.Source: §4.2 · p. 12
  • LSTM gates regulate what cell-state information is forgotten, added to memory, and passed to the next sequence.The forget gate controls retention or removal, the input gate controls new memory, and the output gate selects internal memory for the next sequence.Source: §4.2 · pp. 12–13
  • In AE-LSTM inference, the encoder maps the physical state to a latent vector, the LSTM predicts the next latent state, and the decoder returns a predicted fracture field.The autoregressive loop re-encodes each predicted field to generate the latent representation for the next prediction step.Source: §4.2 · p. 13

5. Numerical Results

The numerical results evaluate tLaSDI-pGFINN on fracture-prediction examples, including multiple-inclusion tests with varying crack initiation locations, using error metrics and thermodynamic behavior to assess predictions. They also report prediction time and the role of geometry conditioning in crack reconstruction.

5. Numerical Results

  • The section evaluates prediction accuracy using MSE and mean relative L2 error, averaged across testing samples.MSE summarizes squared differences over spatial pixels and time steps; relative L2 error measures error relative to the reference solution magnitude.Source: §5 · pp. 13–14

5.1. Single Cavity Fracture Evolution under Varying External Tractions

  • In the single-cavity test, tLaSDI-pGFINNs predicts 70% of testing samples more accurately than AE-LSTM.The comparison uses autoregressive rollout from the initial input; the reported MSE and RelL2 are averages over testing samples.Source: §5.1 · p. 15, Table 1 · p. 15, Figure 10 · p. 16
  • During fracture evolution, tLaSDI-pGFINN keeps the energy time rate on the order of 10^-7 at all time steps, conserving energy while producing non-decreasing entropy.For unseen testing samples, energy remains nearly constant and entropy increases monotonically. Sample 38's relatively small entropy rate from t=0.2 to 0.6 accords with crack initiation only after t=0.553, while sample 31's higher rate from t=0.0 to 0.4 accompanies initiation at t=0.277.Source: §5.1 · p. 18, §5.2 · p. 23

5.2. Single-Inclusion Fracture with Geometry Variation

  • For single-inclusion geometry variation, tLaSDI-pGFINN with Geometry Conditioner outperforms AE-LSTM and predicts sharper crack patterns, especially when geometry affects crack initiation.It remains relatively accurate in an extrapolation case with an inclusion near the boundary, while AE-LSTM fails to reproduce the fully developed crack pattern.Source: §5.2 · pp. 20–21, Table 3 · p. 20
  • For the single-inclusion example, tLaSDI-pGFINN requires approximately 5 seconds per test case versus approximately 36 minutes (2160 seconds) for a full-order simulation, a speedup of approximately 432 times.The geometry-conditioned architecture increases training time for both models, while the passage reports higher prediction accuracy for tLaSDI-pGFINN.Source: §5.2 · pp. 23–24, Table 4 · p. 24

5.3. Multiple-Inclusion Fracture with Varying Crack Initiation Locations

  • On multiple-inclusion testing samples, tLaSDI-pGFINN consistently achieves lower MSE and relative L2 errors than AE-LSTM.It also captures complete crack evolution more accurately across varied crack-initiation locations, where AE-LSTM can miss fully developed cracks or produce inaccurate patterns.Source: §5.3 · pp. 25–26, Table 5 · p. 25, Figure 28 · p. 25
  • In the multiple-inclusion example, energy and entropy histories show thermodynamic constraints remain satisfied, with entropy continuously increasing during irreversible fracture.The reported crack-pattern comparisons include cracks near inclusions and farther away, where AE-LSTM can fail to capture the final developed evolution.Source: §5.3 · p. 26, Figure 34 · p. 29
  • tLaSDI-pGFINN achieves higher fracture-prediction accuracy than AE-LSTM and requires approximately 7 seconds per test case.Incorporating the geometry autoencoder architecture and larger spatial dimensions increases training time for both models.Source: §5.3 · p. 29
  • Because crack paths are strongly affected by interactions with inclusions, the geometry conditioner provides spatial and interface information to improve crack reconstruction.Source: §5.3 · p. 29

6. Conclusions

The study combines thermodynamically constrained latent dynamics with a Geometry-Conditioned Autoencoder to predict fracture evolution in heterogeneous materials. Across three numerical examples, it reports stronger prediction of fracture patterns and evolution than AE-LSTM, with thermodynamic laws preserved on unseen samples.

6. Conclusions

  • The framework combines a Geometry-Conditioned Autoencoder with latent-space pGFINN dynamics that enforce GENERIC symmetry and degeneracy conditions.The geometry-conditioned autoencoder is intended to capture material interface features governing crack initiation and propagation.Source: §6 · p. 29
  • Across three numerical examples, tLaSDI-pGFINN predicts fracture evolution more accurately than AE-LSTM in approximately 70%–75% of testing samples.The examples vary loading conditions, inclusion locations and diameters, and crack initiation locations.Source: §6 · p. 29
  • On unseen testing samples, energy conservation and entropy dissipation remain satisfied throughout prediction.The framework is also reported to outperform AE-LSTM on complex crack topologies and delayed crack initiation.Source: §6 · p. 29
  • The authors conclude that the framework captures fracture patterns close to ground truth and produces more accurate, stable, and physically consistent predictions than models without embedded physical principles.Source: §6 · p. 29
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