Bilayer membranes self-assembled from amphiphilic molecules are ubiquitous in biological and soft matter systems. The elastic properties of bilayer membranes are essential in determining the shape and structure of bilayers. A novel method to calculate the elastic moduli of the self-assembled bilayers within the framework of the self-consistent field theory is developed based on an asymptotic expansion of the order parameters in terms of the bilayer curvature. In particular, the asymptotic expansion method is used to derive analytic expressions of the elastic moduli, which allows us to design more efficient numerical schemes. The efficiency of the proposed method is illustrated by a model system composed of flexible amphiphilic chains dissolved in hydrophilic polymeric solvents.

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Motivated by the weighted averaging method for training neural networks, we study the time-fractional gradient descent (TFGD) method based on the time-fractional gradient flow and explore the influence of memory dependence on neural network training. The TFGD algorithm in this paper is studied via theoretical derivations and neural network training experiments. Compared with the common gradient descent (GD) algorithm, the optimization effect of the time-fractional gradient descent algorithm is significant when the value of fractional $\alpha$ is close to 1, under the condition of appropriate learning rate $\eta$. The comparison is extended to experiments on the MNIST dataset with various learning rates. It is verified that the TFGD has potential advantages when the fractional $\alpha$ nears 0.95∼0.99. This suggests that the memory dependence can improve training performance of neural networks.

The Doi–Onsager molecular theory is capable of providing a rather accurate description of the local behavior of molecules; however, its computation is extremely time-consuming, since some higher-dimensional variables are typically involved. Therefore, establishing a computable reduced model that can capture essential physical properties is an important issue. In this work, we derived a reduced Q-tensor hydrodynamic theory that described smectic phases with density variations from the Doi–Onsager molecular theory using the Bingham closure approximation. The coefficients in the tensor model were derived from those in the molecular model. The energy dissipation law was inherited from the tensor model. Some special cases for the model were also discussed.

In this article, we consider the Qian–Sheng model in the Landau–de Gennes framework describing nematic liquid crystal flows when the inertial effect is neglected. By taking the limit of elastic constant to zero (also called the uniaxial limit) and utilizing the so-called Hilbert expansion method, we provide a rigorous derivation from the non-inertial Qian–Sheng model to the Ericksen–Leslie model.

Starting from a dynamic tensor model about two second-order tensors, we derive the frame hydrodynamics for the biaxial nematic phase using the Hilbert expansion. The coefficients in the frame model are derived from those in the tensor model. The energy dissipation of the tensor model is maintained in the frame model. The model is reduced to the Ericksen--Leslie model if the biaxial bulk energy minimum of the tensor model is reduced to a uniaxial one.