In this paper, we present a discontinuity and cusp capturing physics-informed neural network (PINN) to solve Stokes equations with a piecewise-constant viscosity and singular force along an interface. We first reformulate the governing equations in each fluid domain separately and replace the singular force effect with the traction balance equation between solutions in two sides along the interface. Since the pressure is discontinuous and the velocity has discontinuous derivatives across the interface, we hereby use a network consisting of two fully-connected sub-networks that approximate the pressure and velocity, respectively. The two sub-networks share the same primary coordinate input arguments but with different augmented feature inputs. These two augmented inputs provide the interface information, so we assume that a level set function is given and its zero level set indicates the position of the interface. The pressure sub-network uses an indicator function as an augmented input to capture the function discontinuity, while the velocity sub-network uses a cusp-enforced level set function to capture the derivative discontinuities via the traction balance equation. We perform a series of numerical experiments to solve two- and three-dimensional Stokes interface problems and perform an accuracy comparison with the augmented immersed interface methods in literature. Our results indicate that even a shallow network with a moderate number of neurons and sufficient training data points can achieve prediction accuracy comparable to that of immersed interface methods.

Deep neural network is a powerful tool for many tasks. Understanding why it is so successful and providing a mathematical explanation is an important problem and has been one popular research direction in past years. In the literature of mathematical analysis of deep neural networks, a lot of works is dedicated to establishing representation theories. How to make connections between deep neural networks and mathematical algorithms is still under development. In this paper, we give an algorithmic explanation for deep neural networks, especially in their connections with operator splitting. We show that with certain splitting strategies, operator-splitting methods have the same structure as networks. Utilizing this connection and the Potts model for image segmentation, two networks inspired by operator-splitting methods are proposed. The two networks are essentially two operator-splitting algorithms solving the Potts model. Numerical experiments are presented to demonstrate the effectiveness of the proposed networks.

In this paper, we propose and analyze high order energy dissipative time-stepping schemes for time-fractional molecular beam epitaxial (MBE) growth model on the nonuniform mesh. More precisely, $(2−α)$-order, second-order and $(3−α)$-order time-stepping schemes are developed for the time-fractional MBE model based on the well known L1, L2-$1_σ,$ and L2 formulations in discretization of the time-fractional derivative, which are all proved to be unconditional energy dissipation in the sense of a modified discrete nonlocal-energy on the nonuniform mesh. In order to reduce the computational storage, we apply the sum of exponential technique to approximate the history part of the time-fractional derivative. Moreover, the scalar auxiliary variable (SAV) approach is introduced to deal with the nonlinear potential function and the history part of the fractional derivative. Furthermore, only first order method is used to discretize the introduced SAV equation, which will not affect high order accuracy of the unknown thin film height function by using some proper auxiliary variable functions $V (ξ)$. To our knowledge, it is the first time to unconditionally establish the discrete nonlocal-energy dissipation law for the modified L1-, L2-$1_{σ^-},$ and L2-based high-order schemes on the nonuniform mesh, which is essentially important for such time-fractional MBE models with low regular solutions at initial time. Finally, a series of numerical experiments are carried out to verify the accuracy and efficiency of the proposed schemes.

This paper investigates the scattering of a three-dimensional elastic shell scatterer embedded in a layered unbounded structure. The background structure comprises a two-layer lossy media separated by an unbounded rough surface. The shell body is filled with an elastic material, the interior of which is vacuum. Given an incident acoustic point source, our aim is to determine the acoustic and elastic wave fields in the space. This problem is known as a fluid-solid interaction problem (FSIP). In this work, the uniqueness of the FSIP solution is proved by using the integral equation method. Based on the decay properties of Green’s function, the equivalence of boundary integral equation systems and the FSIP is established. Since the system of integral equations containing several integral operators on infinite intervals, we introduce the index theorem of integral operators and analyze the integral operators on infinite intervals to prove the system of integral equations is Fredholm. We then obtain the uniqueness of the system of integral equations and the corresponding existence and uniqueness result of the FSIP.

Additive manufacturing (AM), also known as 3D printing, has emerged as a groundbreaking technology that has transformed the manufacturing industry. Its ability to produce intricate and customized parts with remarkable speed and reduced material waste has revolutionized traditional manufacturing approaches. However, the AM process itself is a complex and multifaceted undertaking, with various parameters that can significantly influence the quality and efficiency of the printed parts. To address this challenge, researchers have explored the integration of machine learning (ML) techniques to optimize the AM process. This paper presents a comprehensive review of process optimization for additive manufacturing based on machine learning, highlighting the recent advancements, methodologies, and challenges in this field.

We propose a Specht triangle discretization for a geometrically nonlinear Kirchhoff plate model with large bending isometry. A combination of an adaptive time-stepping gradient flow and a Newton’s method is employed to solve the ensuing nonlinear minimization problem. $\Gamma$−convergence of the Specht triangle discretization and the unconditional stability of the gradient flow algorithm are proved. We present several numerical examples to demonstrate that our approach significantly enhances both the computational efficiency and accuracy.