We consider in this paper random batch particle methods for efficiently solving the homogeneous Landau equation in plasma physics. The methods are stochastic variations of the particle methods proposed by Carrillo et al. [J. Comput. Phys.: X 7: 100066, 2020] using the random batch strategy. The collisions only take place inside the small but randomly selected batches so that the computational cost is reduced to $\mathcal{O}(N)$ per time step. Meanwhile, our methods can preserve the conservation of mass, momentum, energy and the decay of entropy. Several numerical examples are performed to validate our methods.

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) [47] for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with ${\rm ReLU}^2$ activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bound on the approximation error of deep ${\rm ReLU}^2$ network in $C^1$ norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ${\rm ReLU}^2$ network, both of which are of independent interest.

In this work, we have proposed a generative model, called VAE-KRnet, for density estimation or approximation, which combines the canonical variational autoencoder (VAE) with our recently developed flow-based generative model, called KRnet. VAE is used as a dimension reduction technique to capture the latent space, and KRnet is used to model the distribution of the latent variable. Using a linear model between the data and the latent variable, we show that VAE-KRnet can be more effective and robust than the canonical VAE. VAE-KRnet can be used as a density model to approximate either data distribution or an arbitrary probability density function (PDF) known up to a constant. VAE-KRnet is flexible in terms of dimensionality. When the number of dimensions is relatively small, KRnet can effectively approximate the distribution in terms of the original random variable. For high-dimensional cases, we may use VAE-KRnet to incorporate dimension reduction. One important application of VAE-KRnet is the variational Bayes for the approximation of the posterior distribution. The variational Bayes approaches are usually based on the minimization of the Kullback-Leibler (KL) divergence between the model and the posterior. For high-dimensional distributions, it is very challenging to construct an accurate density model due to the curse of dimensionality, where extra assumptions are often introduced for efficiency. For instance, the classical mean-field approach assumes mutual independence between dimensions, which often yields an underestimated variance due to oversimplification. To alleviate this issue, we include into the loss the maximization of the mutual information between the latent random variable and the original random variable, which helps keep more information from the region of low density such that the estimation of variance is improved. Numerical experiments have been presented to demonstrate the effectiveness of our model.

The discontinuous Galerkin finite element method (DG-FEM) is a high-precision numerical simulation method widely used in various disciplines. In this paper, we derive the auxiliary ordinary differential equation complex frequency-shifted multi-axial perfectly matched layer (AODE CFS-MPML) in an unsplit format and combine it with any high-order adaptive DG-FEM based on an unstructured mesh to simulate seismic wave propagation. To improve the computational efficiency, we implement Message Passing Interface (MPI) parallelization for the simulation. Comparisons of the numerical simulation results with the analytical solutions verify the accuracy and effectiveness of our method. The results of numerical experiments also confirm the stability and effectiveness of the AODE CFS-MPML.

Blood pressure (BP) has been identified as one of the main factors in cardiovascular disease and other related diseases. Then how to accurately and conveniently measure BP is important to monitor BP and to prevent hypertension. This paper proposes an efficient BP measurement model by integrating a fluid-structure interaction model with the photoplethysmogram (PPG) signal and developing a data-driven computational approach to fit two optimization parameters in the proposed model for each individual. The developed BP model has been validated on a public BP dataset and has shown that the average prediction errors among the root mean square error (RMSE), the mean absolute error (MAE), the systolic blood pressure (SBP) error, and the diastolic blood pressure (DBP) error are all below 5 mmHg for normal BP, stage I, and stage II hypertension groups, and, prediction accuracies of the SBP and the DBP are around 96% among those three groups.

In this work, a direct discontinuous Galerkin (DDG) method with artificial viscosity is developed to solve the compressible Navier-Stokes equations for simulating the transonic or supersonic flow, where the DDG approach is used to discretize viscous and heat fluxes. A strong residual-based artificial viscosity (AV) technique is proposed to be applied in the DDG framework to handle shock waves and layer structures appearing in transonic or supersonic flow, which promotes convergence and robustness. Moreover, the AV term is added to classical BR2 methods for comparison. A number of 2-D and 3-D benchmarks such as airfoils, wings, and a full aircraft are presented to assess the performance of the DDG framework with the strong residual-based AV term for solving the two dimensional and three dimensional Navier-Stokes equations. The proposed framework provides an alternative robust and efficient approach for numerically simulating the multi-dimensional compressible Navier-Stokes equations for transonic or supersonic flow.

This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization problem involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional space. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy functional. We present several numerical examples to show the performance of the proposed method.

In this paper, we consider the numerics of the dispersion-managed Korteweg-de Vries (DM-KdV) equation for describing wave propagations in inhomogeneous media. The DM-KdV equation contains a variable dispersion map with discontinuity, which makes the solution non-smooth in time. We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation, where a necessary constraint on the time step has been identified. Then, two exponential-type dispersion-map integrators up to second order accuracy are derived, which are efficiently incorporated with the Fourier pseudospectral discretization in space, and they can converge regardless of the discontinuity and the step size. Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast & strong dispersion-management regime.

A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) methods on two-dimensional unstructured quadrilateral meshes. Firstly, a modified indicator based on modal energy coefficients is proposed to detect troubled cells, where discontinuities exist. Then, troubled cells are decomposed into nonuniform subcells and each subcell has one solution point. A second-order finite difference shock-capturing scheme based on nonuniform nonlinear weighted (NNW) interpolation is constructed to perform the calculation on troubled cells while smooth cells are calculated by the CPR method. Numerical investigations show that the proposed subcell limiting strategy on unstructured quadrilateral meshes is robust in shock-capturing.

A coupling framework that leverages the advantages of the diffuse and sharp interface immersed boundary (IB) methods is presented for handling the interaction among particles and particles with the static complex geometries of the environment. In the proposed coupling approach, the curvilinear IB method is employed to represent the static complex geometries, a variant of the direct forcing IB method is proposed for simulating particles, and the discrete element method is employed for particle-particle and particle-wall collisions. The proposed approach is validated using several classical benchmark problems, which include flow around a sphere, sedimentation of a sphere, collision of two sedimenting spheres, and collision between a particle and a flat wall, with the present predictions showing an overall good agreement with the results reported in the literature. The capability of the proposed framework is further demonstrated by simulating the interaction between multiple particles and a wall-mounted cylinder, and the particle-laden turbulent flow over periodic hills. The proposed method provides an efficient way to simulate particle-laden turbulent flows in environments with complex boundaries.

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with ReLU$^3$ networks (deep network with activation function max$\{0,x^3\}$) and the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ReLU$^3$ network, which is of immense independent interest.

Objective functions in large-scale machine-learning and artificial intelligence applications often live in high dimensions with strong non-convexity and massive local minima. Gradient-based methods, such as the stochastic gradient method and Adam [15], and gradient-free methods, such as the consensus-based optimization (CBO) method, can be employed to find minima. In this work, based on the CBO method and Adam, we propose a consensus-based global optimization method with adaptive momentum estimation (Adam-CBO). Advantages of the Adam-CBO method include:
• It is capable of finding global minima of non-convex objective functions with high success rates and low costs. This is verified by finding the global minimizer of the 1000 dimensional Rastrigin function with 100% success rate at a cost only growing linearly with respect to the dimensionality.
• It can handle non-differentiable activation functions and thus approximate low-regularity functions with better accuracy. This is confirmed by solving a machine learning task for partial differential equations with low-regularity solutions where the Adam-CBO method provides better results than Adam.
• It is robust in the sense that its convergence is insensitive to the learning rate by a linear stability analysis. This is confirmed by finding the minimizer of a quadratic function.