This paper introduces a neural network approach for solving two-dimensional traveltime tomography (TT) problems based on the eikonal equation. The mathematical problem of TT is to recover the slowness field of a medium based on the boundary measurement of the traveltimes of waves going through the medium. This inverse map is high-dimensional and nonlinear. For the circular tomography geometry, a perturbative analysis shows that the forward map can be approximated by a vectorized convolution operator in the angular direction. Motivated by this and filtered back-projection, we propose an effective neural network architecture for the inverse map using the recently proposed BCR-Net, with weights learned from training datasets. Numerical results demonstrate the efficiency of the proposed neural networks.

In this paper, we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two weak formulations. For the strong formulation, the solution is directly parameterized with a neural network and optimized by minimizing the PDE residual. It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the $L^1$ sense. The weak formulations are derived following (Brenier in Examples of hidden convexity in nonlinear PDEs, 2020) which characterizes the very weak solutions of QPME. Specifically speaking, the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations. Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions. This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network-based methods, which we hope can provide some useful experience for future investigations.

As one of the main governing equations in kinetic theory, the Boltzmann equation is widely utilized in aerospace, microscopic flow, etc. Its high-resolution simulation is crucial in these related areas. However, due to the high dimensionality of the Boltzmann equation, high-resolution simulations are often difficult to achieve numerically. The moment method which was first proposed in Grad (Commun Pure Appl Math 2(4):331–407, 1949) is among the popular numerical methods to achieve efficient high-resolution simulations. We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation, which effectively reduces the dimensionality of the problem. However, one of the main challenges is that it leads to an unclosed moment system, and closure is needed to obtain a closed moment system. It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory. Other than the traditional human designs of closures, the machine learning-based approach has attracted much attention lately in Han et al. (Proc Natl Acad Sci USA 116(44):21983–21991, 2019) and Huang et al. (J Non-Equilib Thermodyn 46(4):355–370, 2021). In this work, we propose a machine learning-based method to derive a moment closure model for the Boltzmann–BGK equation. In particular, the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances, i.e., the Galilean invariance, reflecting invariance, and scaling invariance, inherited from the original Boltzmann–BGK equation and playing an important role in the correct simulation of the Boltzmann equation. Numerical simulations on the 1D–1D examples including the smooth and discontinuous initial condition problems, Sod shock tube problem, the shock structure problems, and the 1D–3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.

Green’s function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green’s function is difficult to compute. The troubles arise in the following threefold. Firstly, compared with the original PDE, the dimension of Green’s function is doubled, making it impossible to be handled by traditional mesh-based methods. Secondly, Green’s function usually contains singularities which increase in the difficulty to get a good approximation. Lastly, the computational domain may be very complex or even unbounded. To override these problems, we develop a new framework for computing Green’s function leveraging the fundamental solution, boundary integral method and neural networks with reasonably high accuracy in this paper. We focus on Green’s function of Poisson and Helmholtz equations in bounded domains, unbounded domains. We also consider Poisson equation and Helmholtz equations in domains with interfaces. Extensive numerical experiments illustrate the efficiency and accuracy of our method for solving Green’s function. In addition, Green’s function provides the operator from the source term and boundary condition to the PDE solution. We apply Green’s function to solve PDEs with different sources, and obtain reasonably high-precision solutions, which shows the good generalization ability of our method. However, the requirements for explicit fundamental solutions to remove the singularity of Green’s function hinder the application of our method in more complex PDEs, such as variable coefficient equations, which will be investigated in our future work.

One of the key problems in isogeometric analysis(IGA) is domain parameterization, i.e., constructing a map between a parametric domain and a computational domain. As a preliminary step of domain parameterization, the mapping between the boundaries of the parametric domain and the computational domain should be established. The boundary correspondence strongly affects the quality of domain parameterization and thus subsequent numerical analysis. Currently, boundary correspondence is generally determined manually and only one approach based on optimal mass transport discusses automatic generation of boundary correspondence. In this article, we propose a deep neural network based approach to generate boundary correspondence for 2D simply connected computational domains. Given the boundary polygon of a planar computational domain, the main problem is to pick four corner vertices on the input boundary in order to subdivide the boundary into four segments which correspond to the four sides of the parametric domain. We synthesize a dataset with corner correspondence and train a fully convolutional network to predict the likelihood of each boundary vertex to be one of the corner vertices, and thus to locate four corner vertices with locally maximum likelihood. We evaluate our method on two types of datasets: MPEG-7 dataset and CAD model dataset. The experiment results demonstrate that our algorithm is faster by several orders of magnitude, and at the same time achieves smaller average angular distortion, more uniform area distortion and higher success rate, compared to the traditional optimization-based method. Furthermore, our neural network exhibits good generalization ability on new datasets.

The traditional pipeline for non-rigid registration is to iteratively update the correspondence and alignment such that the transformed source surface aligns well with the target surface. Among the pipeline, the correspondence construction and iterative manner are key to the results, while existing strategies might result in local optima. In this paper, we adopt the widely used deformation graph-based representation, while replacing some key modules with neural learning-based strategies. Specifically, we design a neural network to predict the correspondence and its reliability confidence rather than the strategies like nearest neighbor search and pair rejection. Besides, we adopt the GRU-based recurrent network for iterative refinement, which is more robust than the traditional strategy. The model is trained in a self-supervised manner and thus can be used for arbitrary datasets without ground-truth. Extensive experiments demonstrate that our proposed method outperforms the state-of-the-art methods by a large margin.