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Accelerated search and design of stretchable graphene kirigami using machine learning

Paul Z. Hanakata, Ekin D. Cubuk, David K. Campbell, Harold S. Park

arXiv:1808.06111v1physics.comp-phcond-mat.dis-nn

TL;DR

Finding effective kirigami cut patterns is difficult because possible configurations grow exponentially with system size. This paper uses machine learning to predict graphene kirigami yield properties and search for stretchable designs; its CNN predictions approach MD precision, and the search finds high-yield-strain configurations.

  • Problem

    Locating cuts to achieve a target kirigami performance is unresolved, while the number of possible cut configurations grows exponentially with system size.

    Source: p. 1

  • Method

    The paper trains NN and CNN models to approximate graphene kirigami yield strain and stress, then uses a CNN for inverse design across an approximately 4,000,000-configuration space.

    Source: pp. 1–2

  • Results

    CNN-f16-f32-f64-h64 achieved R2 = 0.91 and RMSE = 0.054, close to MD precision of 0.046, while fewer than 10 generations found configurations with yield strains ≥1.0.

    Source: pp. 3–4

  • Takeaways & Limitations

    The highest-stretchability designs use closely spaced alternating edge cuts, which allow larger rotations and out-of-plane deflection through thinner connecting ribbons.

    Source: p. 4

    A degree-3 polynomial expansion for a 30 × 80 input would require approximately 10^9 parameters, limiting that approach for this system size.

    Source: LINEAR MODEL · p. 9

Abstract

from arXiv · show

Making kirigami-inspired cuts into a sheet has been shown to be an effective way of designing stretchable materials with metamorphic properties where the 2D shape can transform into complex 3D shapes. However, finding the optimal solutions is not straightforward as the number of possible cutting patterns grows exponentially with system size. Here, we report on how machine learning (ML) can be used to approximate the target properties, such as yield stress and yield strain, as a function of cutting pattern. Our approach enables the rapid discovery of kirigami designs that yield extreme stretchability as verified by molecular dynamics (MD) simulations. We find that convolutional neural networks (CNN), commonly used for classification in vision tasks, can be applied for regression to achieve an accuracy close to the precision of the MD simulations. This approach can then be used to search for optimal designs that maximize elastic stretchability with only 1000 training samples in a large design space of $\sim 4\times10^6$ candidate designs. This example demonstrates the power and potential of ML in finding optimal kirigami designs at a fraction of iterations that would be required of a purely MD or experiment-based approach, where no prior knowledge of the governing physics is known or available.

MOLECULAR DYNAMICS METHODS

The molecular-dynamics setup models graphene kirigami and applies tensile loading to measure its mechanical response. The enumerated cut patterns define a design space, with reflection symmetry reducing the configurations requiring simulation.

MOLECULAR DYNAMICS METHODS

  • The graphene sheet was relaxed at 4.2 K, then stretched by moving its edges at a strain rate of 0.01/ps until fracture.The sheet was first relaxed by conjugate-gradient minimization and 50 ps of NVT relaxation; the simulations used non-periodic conditions in all directions.Source: MOLECULAR DYNAMICS METHODS · p. 6
  • The sheet contains 2,400 rectangular graphene unit cells, or 9,600 carbon atoms, and each allowed cut spans 3 × 16 unit cells.The sheet is approximately 100 × 200 Å; the 15 allowed cut locations each correspond to a 3 × 16-unit-cell cut.Source: MOLECULAR DYNAMICS METHODS · p. 6
  • Of 32,768 possible cut configurations, 29,791 are not detached, and reflection symmetry means only about one quarter need MD simulation.The system is non-periodic, so translation symmetry is broken while reflection symmetry remains.Source: MOLECULAR DYNAMICS METHODS · p. 6

MACHINE LEARNING PERFORMANCE AND MOLECULAR DYNAMICS PRECISION

The paper evaluates prediction errors against molecular-dynamics variability. It defines MD precision from repeated initial conditions and adjusts the yield-strain estimate for the greater variability at higher cut densities.

MACHINE LEARNING PERFORMANCE AND MOLECULAR DYNAMICS PRECISION

  • RMSE compares model predictions with MD test values across the test dataset.The passage defines mtest as the number of test samples, with ytest as their true MD values and ypred as model predictions.Source: MACHINE LEARNING PERFORMANCE AND… · p. 6
  • Thermal fluctuations at 4.2 K make MD yield strain and stress vary, and the paper calls this variation the MD precision.The paper treats this variability as irreducible error and approximates it using RMSD across observations.Source: MACHINE LEARNING PERFORMANCE AND… · p. 6
  • For yield strain, the summed RMSD uses cut densities from 5/15 to 12/15 because variability is more pronounced at higher cut density.The passage excludes 13/15–15/15 cut densities because those structures are fully detached.Source: MACHINE LEARNING PERFORMANCE AND… · p. 6

Hyperparameters for training

The paper specifies the layer computation and training setup for its neural-network models. It defines the hidden-layer width and the symbols used to describe layer operations and CNN architecture.

Hyperparameters for training

  • TensorFlow r1.8 built the NN and CNN models, while scikit-learn was used for ordinary least-squares regression.TensorFlow ran on four CPUs and one NVIDIA Tesla K40m GPU.Source: Hyperparameters for training · p. 7
  • The number of neurons in hidden layer l is denoted h, and each layer computes activations from its weights, bias, and prior activations using ReLU.The passage identifies g as ReLU, a(l) as activations, b(l) as bias, and W(l) as weights.Source: Hyperparameters for training · p. 7
  • Both NN and CNN models used a learning rate of 0.0001, batch size 200, and a maximum of 300 epochs.For the search algorithm, CNN-f16-f32-f64-h64 used a learning rate of 0.001 and batch size 100.Source: Hyperparameters for training · p. 7

Convolutional Neural Networks Architecture

The CNN architecture processes 30 × 80 binary images with convolution and pooling before a fully connected layer. The selected deep CNN performs well on held-out data and outperforms the NN models in the reported comparison.

Convolutional Neural Networks Architecture

  • The CNN takes a 30 × 80 grayscale 0/1 grid image and uses 3 × 3 convolutions, ReLU, and 2 × 2 max-pooling before a fully connected layer.The fully connected layer size ranges from 4 to 2,024; the final layer omits ReLU for regression.Source: Convolutional Neural Networks… · p. 7
  • The 29,791 samples were split 80%/10%/10% for training, validation, and testing, and the authors report no overfitting problem.The training R2 was slightly larger than validation or test R2 across the models.Source: Convolutional Neural Networks… · p. 7
  • CNN models outperform NN models, and increasing neuron count generally increases model accuracy in the reported comparisons.Figure 5 compares R2 and RMSE for yield strain and yield stress across neuron counts or fully connected layer sizes and grid counts.Source: Figure 5 · p. 7
  • For CNN-f16-f32-f64-h64, R2 was 0.93, 0.91, and 0.92 on training, validation, and test data, respectively.The corresponding RMSE values were 0.046, 0.056, and 0.052.Source: Convolutional Neural Networks… · p. 7

ROTATIONS IN KIRIGAMI

The paper relates kirigami cut placement to elastic extension through ribbon rotation. Alternating edge cuts and closer spacing are associated with greater yield strain and stretchability.

ROTATIONS IN KIRIGAMI

  • Figure 6 plots test-set true versus predicted yield strain and yield stress for the CNN-f16-f32-f64-h64 model.The red line marks y = x.Source: Figure 6 · p. 8
  • Configuration II can extend to ≈7cx versus ≈3cx for configuration I, and its yield strain is almost twice as large.With no bond breaking, the extra extension is attributed to one fewer connecting node; alternating edge cuts effectively transform the membrane into a quasi-1D structure.Source: ROTATIONS IN KIRIGAMI · pp. 7–8, Figure 7 · p. 8
  • Closer-packed alternating edge cuts increase stretchability from 1.10 to 1.68 as thinner ribbons twist more readily around nodes.The comparison is between configurations II and III; sparse cuts in III leave thicker ribbons around the nodes.Source: ROTATIONS IN KIRIGAMI · p. 8, Figure 7 · p. 8

LINEAR MODEL

The linear model represents yield properties as functions of cut-pattern features and uses fitted coefficients to suggest spatial design rules. Its polynomial form becomes parameter-intensive for finer grids, while the paper presents CNNs as a more scalable alternative.

LINEAR MODEL

  • The expansion converts cut-pattern inputs into polynomial features, with binary inputs reducing the infinite series to 2^n terms before symmetry and locality reductions.The paper describes this as a feature transformation; nearby interactions are commonly retained in Ising and tight-binding models.Source: LINEAR MODEL · p. 8
  • Yield strain fits with a degree-3 polynomial at R2 = 0.76 and RMSE = 0.084; yield stress fits with degree 2 at R2 = 0.93 and RMSE = 4.1 GPa.The fits use ordinary least squares on a 15-element input vector, with 1 for no-cut and -1 for cut.Source: LINEAR MODEL · p. 8
  • For yield strain, negative first-order βi favor more cuts, while βij suggest same-type cells along y and opposite types adjacent along x.The proposed arrangement is a line of cuts along y alternating in x, analogous to springs with different constants connected in series.Source: LINEAR MODEL · p. 9, Figure 8 · p. 9
  • For yield stress, positive first-order βi indicate cuts lower stress; βij suggest arrays along x alternating in y, like springs connected in parallel.The figure maps learned first- and second-order parameters for yield stress; β12 = −1.5 while β16 = +1.5.Source: LINEAR MODEL · p. 9, Figure 9 · p. 9
  • At 2,400 features, a degree-3 expansion requires ∼109 parameters, and parameter growth makes finer grids or larger systems inefficient.The paper contrasts this with CNNs, which retain 2D local motifs through learned filters rather than flattening patterns into 1D.Source: LINEAR MODEL · p. 9
  • Flattening cut patterns loses local spatial information, while CNN filters learn 2D motifs and are presented as a more appropriate, scalable model.The paper notes that series expansions can recover neighbor interactions but become computationally inefficient as the number of cells grows.Source: LINEAR MODEL · p. 9
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