Using dropout based active learning and surrogate models in the inverse viscoelastic parameter identification of human brain tissue

Figures

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  FIGURE 1. Rheological scheme of the implemented generalized Maxwell model with one Maxwell element. Each nonlinear spring represents a hyperelastic one-term Ogden model characterized by the shear modulus μ and the nonlinearity α. η denotes the viscosity of the dashpot element.

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  FIGURE 2. Experimental data of a human brain tissue specimen from the frontal cortex.

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  FIGURE 3. (A) Signal flow through an LSTM cell. σ denotes the sigmoid activation function. (B) Recurrent neural network architecture consisting of three layers with long short-term memory (LSTM) cells and two dropout layers in between. The input vector x contains the constitutive parameters as well as strain and time while the output h is the nominal stress P and shear stress τ. The internal cell state is stored in the state vector a and the subscript n denotes the timestep.

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  FIGURE 4. Correlation between the variance obtained by Monte Carlo dropout sampling and the mean absolute error (MAE) where both are normalized by their standard deviation over all samples. Results are shown for 100 points in the material parameter space and the same time series input (shear and stretch) that were created using Poisson Disk sampling.

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  FIGURE 5. Comparison between the value as well as the standard deviation of the determination coefficient R 2 for adding new points to the training data set when they are selected randomly or via active learning, e.g., those with the highest estimated variance.

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  FIGURE 6. Comparison of metamodel and finite element simulation output in terms of nominal and shear stress for the same time series input (stretch and shear stress) as well as identical material parameters.

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  FIGURE 7. Parameter identification results for experimental data of human brain tissue from the frontal cortex. The finite element simulation output is shown for the final identified parameter set with α ∞ = −16, μ ∞ = 162 Pa, η 1 = 13,949 Pa⋅s, α 1 = −18, μ 1 = 398 Pa.

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  FIGURE 8. History of parameter values during the optimization using the hybrid approach with the surrogate and finite element model combined (hybrid) as well as only the finite element model. Function evaluations contain all model evaluations including finite difference approximations of the gradient as well as failed optimization steps. The cpu times are measured using the perf_counter function of the Python module time.