In this paper, we propose a machine learning approach via model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator representation with regularization from data. For linear PDEs, we use a DNN to parameterize the Green’s function and obtain the neural operator to approximate the solution according to the Green’s method. To train the DNN, the empirical risk consists of the mean squared loss with the least square formulation or the variational formulation of the governing equation and boundary conditions. For complicated problems, the empirical risk also includes a few labels, which are computed on coarse grid points with cheap computation cost and significantly improves the model accuracy. Intuitively, the labeled dataset works as a regularization in addition to the model constraints. The MOD-Net solves a family of PDEs rather than a specific one and is much more efficient than original neural operator because few expensive labels are required. We numerically show MOD-Net is very efficient in solving Poisson equation and one-dimensional radiative transfer equation. For nonlinear PDEs, the nonlinear MOD-Net can be similarly used as an ansatz for solving nonlinear PDEs, exemplified by solving several nonlinear PDE problems, such as the Burgers equation.

Partial differential equations (PDEs) play a dominant role in the mathematical modeling of many complex dynamical processes. Solving these PDEs often requires prohibitively high computational costs, especially when multiple evaluations must be made for different parameters or conditions. After training, neural operators can provide PDEs solutions significantly faster than traditional PDE solvers. In this work, invariance properties and computational complexity of two neural operators are examined for transport PDE of a scalar quantity. Neural operator based on graph kernel network (GKN) operates on graph-structured data to incorporate nonlocal dependencies. Here we propose a modified formulation of GKN to achieve frame invariance. Vector cloud neural network (VCNN) is an alternate neural operator with embedded frame invariance which operates on point cloud data. GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN. However, GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN.

In this paper, a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory (HWENO) scheme is designed for hyperbolic conservation laws. The main idea of this scheme is derived from our previous work [J. Comput. Phys., 446 (2021) 110653], in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the boundaries. However, in this paper, only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order moments. In addition, an extra modification procedure is used to modify those first order moments in the troubled-cells, which leads to an improvement of stability and an enhancement of resolution near discontinuities. To obtain the same order of accuracy, the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the general WENO scheme. Moreover, the linear weights are not unique and are independent of the node position, and the CFL number can still be 0.6 whether for the one or two dimensional case, which has to be 0.2 in the two dimensional case for other HWENO schemes. Extensive numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.

In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and thus our approach relies on modelling the target solutions with the temporal normalizing flows. The temporal normalizing flow is then trained based on the TFP loss function, without requiring any labeled data. Being a machine learning scheme, the proposed approach is mesh-free and can be easily applied to high dimensional problems. We present a variety of test problems to show the effectiveness of the learning approach.

We introduce an improved bond-based peridynamic (BPD) model for simulating brittle fracture in particle reinforced composites based on a micromodulus correction approach. In the peridynamic (PD) constitutive model of particle reinforced composites, three kinds of interactive bond forces are considered, and precise definition of mechanical properties for PD bonds is essential for the fracture analysis in particle reinforced composites. A new micromodulus model of PD bonds for particle reinforced composites is proposed based on the equivalence between the elastic strain energy density of classical continuum mechanics and peridynamic model and the harmonic average approach. The damage of particle reinforced composites is defined locally at the level of pairwise bond, and the critical stretch criterion is described as a function of fracture energy based on the composite failure theory. The algorithm procedure for the improved BPD model based on the finite element/discontinuous Galerkin finite element method is brought forward in detail. Several numerical examples are performed to test the feasibility and effectiveness of the proposed model and algorithm in analysis of elastic deformation, crack nucleation and propagation in particle reinforced composites. Additionally, the impact of distribution, shape and size of particles on the fractures of composite materials are also investigated. Numerical results demonstrate that the improved BPD model can effectively be used to analyze the fracture in particle reinforced composites.

The performances of the Color-Gradient (CG) and the Shan-Chen (SC) multicomponent Lattice Boltzmann models are quantitatively compared side-by-side on multiple physical flow problems where breakup, coalescence and contraction of fluid ligaments are important. The flow problems are relevant to microfluidic applications, jetting of microdroplets as seen in inkjet printing, as well as emulsion dynamics. A significantly wider range of parameters is shown to be accessible for CG in terms of density-ratio, viscosity-ratio and surface tension values. Numerical stability for a high density ratio $\mathcal{O}(1000)$ is required for simulating the drop formation process during inkjet printing which we show here to be achievable using the CG model but not using the SC model. Our results show that the CG model is a suitable choice for challenging simulations of droplet formation, due to a combination of both numerical stability and physical accuracy. We also present a novel approach to incorporate repulsion forces between interfaces for CG, with possible applications to the study of stabilized emulsions. Specifically, we show that the CG model can produce similar results to a known multirange potentials extension of the SC model for modelling a disjoining pressure, opening up its use for the study of dense stabilized emulsions.

In [A NURBS-enhanced finite volume solver for steady Euler equations, X. C. Meng, G. H. Hu, J. Comput. Phys., Vol. 359, pp. 77–92], a NURBS-enhanced finite volume method was developed to solve the steady Euler equations, in which the desired high order numerical accuracy was obtained for the equations imposed in the domain with a curved boundary. In this paper, the method is significantly improved in the following ways: (i) a simple and efficient point inversion technique is designed to compute the parameter values of points lying on a NURBS curve, (ii) with this new point inversion technique, the $h$-adaptive NURBS-enhanced finite volume method is introduced for the steady Euler equations in a complex domain, and (iii) a goal-oriented a posteriori error indicator is designed to further improve the efficiency of the algorithm towards accurately calculating a given quantity of interest. Numerical results obtained from a variety of numerical experiments with different flow configurations successfully show the effectiveness and robustness of the proposed method.

The paper presents a holomorphic operator function approach for the transmission eigenvalue problem of elastic waves using the discontinuous Galerkin method. To use the abstract approximation theory for holomorphic operator functions, we rewrite the elastic transmission eigenvalue problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for the discontinuous Galerkin method is proved following the abstract theory of the holomorphic Fredholm operator. The spectral indicator method is employed to compute the transmission eigenvalues. Extensive numerical examples are presented to validate the theory.

In this paper, we present a Runge-Kutta Discontinuous Galerkin (RKDG) method for solving the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations under the Lagrangian framework. The fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation are discretized using a DG method based on linear Taylor expansions. By using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD, an exactly divergence-free numerical magnetic field can be obtained. The nodal velocities and the corresponding numerical fluxes are explicitly calculated by solving multidirectional approximate Riemann problems. Two kinds of limiter are proposed to inhibit the non-physical oscillation around the shock wave, and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the rotor problems. This Lagrangian RKDG method conserves mass, momentum, and total energy. Several numerical tests are presented to demonstrate the accuracy and robustness of the proposed scheme.

To include the bound electron effects in particle-in-cell (PIC) simulation, we propose a model in which the response of the dipole components of partially ionized ions to external electromagnetic fields can be included. Instead of treating the macroion particle as a single particle without an internal structure, the ions are considered to have a structure composed of a central nucleus and a bounded electron cloud in our model. The two parts experience the interactions of both the external electromagnetic fields and the internal Coulomb fields. In this way, the laser scattering effects by a partially ionized medium can be modeled properly in the PIC simulation. The laser propagation in a neutral medium and the Bragg scattering of the laser in crystal structure have been simulated with a PIC code modified based on our model as the benchmark. Our model may find applications to study some interesting problems, such as the x-ray laser-driven wakefield acceleration in crystals, the x-ray laser-driven high energy density physics, and intense laser propagation in partially ionized nonlinear optical materials, etc.