Fig. 1 ( a ) Illustration of the spatial patterns obtained from our Cahn–Hilliard simulations where each row corresponds to the time evolution in a single simulation for c 0 = 0.5 (case 1), c 0 = 0.63 (case 2), and c 0 = 0.75 (case 3) shown in the... More

Fig. 2 ( a ) A schematic of our ML metamodels that are used to predict change in strain energy Δ Ψ at a fixed level of applied displacement from each material property distribution. ( b ) A schematic of transfer learning whereby a model trained on one dataset (in this case a low fidelity ... More

Fig. 3 FID with respect to the number of epochs for the StyleGAN2-ADA, WGAN-CP, and WGAN-GP ML-based generative models. In the right panel, we include examples of output patterns as model training proceeds to visualize the relationship between a lower FID value and improved resemblance to the real... More

Fig. 4 Visualization of the ML-based and procedural generative model results in order of increasing FID. For each pattern type, we show a comparison of strain energy Δ Ψ at d  =   0.001 for real and generated patterns with low fidelity data for: ( a ) StyleGAN2-ADA patterns, ( b ) WGAN-G... More

Fig. 5 Metamodel performance with respect to the size of the training dataset. Note that “dataset size” refers to the combined number of unique real and generated synthetic patterns. For a dataset of 16 , 000 real patterns, R 2 is 0.9992. For a dataset of 1000 real and 15 , 000 ... More

Fig. 6 Qualitative interpretation of R 2 scores for transfer learning evaluation. True versus predicted strain energy values of high fidelity test data are plotted for three different metamodels trained with 1000 high fidelity real data points. ( a ) Metamodel weights are initialized randomly (i.... More

Fig. 1 Suction device Cutometer and its probe, which has circular cavity, can apply negative pressure on skin ( a ). A schematic indicates how negative pressure is applied inside the probe cavity, and optical system enables to measure the deformation of skin as the apex height ( b ). Cutometer pro... More

Fig. 2 Deformation of skin with suction device Cutometer is predicted by the FE model. A probe is a rigid body and fixed during simulation, while skin is a deformable body cosidered hyperelastic for the instant loading scenario. Constant negative pressure is applied to deform the skin, and then, t... More

Fig. 3 Overview on the procedure to solve inverse problem using Bayesian inference and GP as the forward function: GP, which substitutes the forward function of finite element analysis, needs to be trained first ( a ). Once the GP is trained, and the hyperparameters are optimized, the GP replaces ... More

Fig. 4 Evaluation of the GP as a function of training points by means of RMSE. There are 15 GPs, one per each finite element model and boundary conditions. RMSE is separately plotted with respect to applied pressure. More

Fig. 5 GP surrogate is further tested in terms of standardized residuals and quantile-quantile plots. Standardized residuals with respect to predictive value (GP predictive mean) reside in the range [ − 3 ,   3 ] ( a ). The distribution of residuals against theoretical residuals ( b ... More

Fig. 6 Sensitivity analysis using GP surrogate is done in terms of global Sobol index with respect to the applied pressure. Each input parameter has five Sobol indices because there are different models denoted in the legend above the bar graphs. More

Fig. 7 Quantification of the performance of Bayesian inference to solve inverse problem of determining parameter values x from skin height values y . 50 height values y * are obtained from evaluating the finite element (FE) model, which is taken as the ground truth for the ... More

Fig. 8 Bayesian inference for five additional test cases. The goal of the Bayesian inference problem is to learn the posterior distribution over the parameters x = ( μ ,   k 1 ,   k 2 ,   κ ,   θ ) given observations y of the maximum height of skin in the Cu... More

Fig. 9 Bayesian inference for the five test cases as before but considering three different prior distributions over noise variance: Gaussian, inverse gamma, and exponential distributions. Comparing to Fig. 8 , which shows inference assuming a constant noise variance for the 15 observations y , ... More

Fig. 1 Left: Problem setup for 2D test cases: we consider a 50 mm × 50 mm tumor domain, with initial conditions defined as a tangent hill, such that ϕ   = 0.5 inside the tumor and 0 outside. Right: values of diffusion rate D and proliferation rate ρ considered in the 2D simulation study.... More