We for the first time leverage deep learning approaches to solve forward and inverse problems of two-dimensional laminar flows for power-law fluids. We propose a deep-learning framework, called Power-Law-Fluid-Net (PL-Net). We develop a surrogate model to solve the forward problems of the power-law fluids, and solve the inverse problems utilizing only a small set of measurement data under the assumption that boundary conditions (BCs) can be partially known. In the design of the methods, we incorporate the hard boundary condition constraints to accelerate the iteration of stochastic gradient descent methods for minimizing loss functions. For the forward problems, by incorporating the constitutive parameters into the input variables of neural networks, the PL-Net serves as a surrogate model for simulating the pressure-driven flows inside pipes having cross sections of varying shapes. We investigate the influences of the BC type, activation function type, and number of collocation points on the accuracy of numerical solutions. For the inverse problems, the PL-Net infers the physical quantities or constitutive parameters from a small number of measurements of flow field variables. The BCs of the inverse problems can even be partially known. We demonstrate the effects of BC type, number of sensors, and noise level on accuracy of inferred quantities. Computational examples indicate the high accuracy of the PL-Net in tackling both the forward and inverse problems of the power-law fluids.

Building an accurate constitutive model for soft materials is essential for better understanding its rate-dependent deformation characteristics and improving the design of soft material devices. To establish a concise constitutive model with few parameters and clear physical meaning, a variable-order fractional model is proposed to accurately describe and predict the rate-dependent mechanical behavior of soft materials. In this work, the discrete variable-order fractional operator enables the predicted stress response to be entirely consistent with the whole stress history and the fractional order’s path-dependent values. The proposed model is further implemented in a numerical form and applied to predict several typical soft materials’ tensile and compressive deformation behavior. Our research indicates that the proposed variable-order fractional constitutive model is capable of predicting the nonlinear rate-dependent mechanical behavior of soft materials with high accuracy and more convinced reliability in comparison with the existing fractional models, where the fractional order contains a constant initial order to depict the initial elastic response and a linear variable-order function to account for the strain hardening behavior after acrossing the yield point.