We introduce a new approach for solving forward systems of differential equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the challenge of applying PINNs to forward dynamical systems and demonstrates improved accuracy through its application to neuron models. Specifically, we apply operator splitting to decompose the original neuron model into sub-problems that are then solved using PINNs. Moreover, we develop an $L^1$ scheme for discretizing fractional derivatives in fractional neuron models, leading to improved accuracy and efficiency. The results of this study highlight the potential of splitting PINNs in solving both integer- and fractional-order neuron models, as well as other similar systems in computational science and engineering.

Parametric dynamical systems are widely used to model physical systems, but their numerical simulation can be computationally demanding due to nonlinearity, long-time simulation, and multi-query requirements. Model reduction methods aim to reduce computation complexity and improve simulation efficiency. However, traditional model reduction methods are inefficient for parametric dynamical systems with nonlinear structures. To address this challenge, we propose an adaptive method based on local dynamic mode decomposition (DMD) to construct an efficient and reliable surrogate model. We propose an improved greedy algorithm to generate the atoms set $\Theta$ based on a sequence of relatively small training sets, which could reduce the effect of large training set. At each enrichment step, we construct a local sub-surrogate model using the Taylor expansion and DMD, resulting in the ability to predict the state at any time without solving the original dynamical system. Moreover, our method provides the best approximation almost everywhere over the parameter domain with certain smoothness assumptions, thanks to the gradient information. At last, three concrete examples are presented to illustrate the effectiveness of the proposed method.

In this work, we consider the Fokker-Planck equation of the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model for neuron networks. Due to the firing events of neurons at the microscopic level, this Fokker-Planck equation contains dynamic boundary conditions involving specific internal points. To efficiently solve this problem and explore the properties of the unknown, we construct a flexible numerical scheme for the Fokker-Planck equation in the framework of spectral methods that can accurately handle the dynamic boundary condition. This numerical scheme is stable with suitable choices of test function spaces, and asymptotic preserving, and it is easily extendable to variant models with multiple time scales. We also present extensive numerical examples to verify the scheme properties, including order of convergence and time efficiency, and explore unique properties of the model, including blow-up phenomena for the NNLIF model and learning and discriminative properties for the NNLIF model with learning rules.

In this paper, a direct arbitrary Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme is proposed for simulating compressible multi-material flows on the adaptive quadrilateral meshes. Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of the fluid mixture. The coupled system is discretized in the reference element and we use a kind of Taylor expansion basis functions to construct the interpolation polynomials of the variables. We show the property that the material derivatives of the basis functions in the DG discretization are equal to zero, with which the scheme is simplified. In addition, the mesh velocity in the ALE framework is obtained by using the adaptive mesh method from [H.Z. Tang and T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. NUMER. ANAL]. This adaptive mesh method can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduces the numerical dissipation near the material interfaces in the simulations. With the help of this adaptive mesh method, the resolution of the solution near the target regions can be greatly improved and the computational efficiency of the simulation is increased. Our scheme can be applied in the simulations for the gas and water media efficiently, and it is more concise compared to some other methods such as the indirect ALE methods. Several examples including the gas-water flow problem are presented to demonstrate the efficiency of our scheme, and the results show that our scheme can capture the wave structures sharply with high robustness.

Images captured under insufficient light conditions often suffer from noticeable degradation of visibility, brightness and contrast. Existing methods pose limitations on enhancing low-visibility images, especially for diverse low-light conditions. In this paper, we first propose a new variational model for estimating the illumination map based on fractional-order differential. Once the illumination map is obtained, we directly inject the well-constructed illumination map into a general image restoration model, whose regularization terms can be viewed as an adaptive mapping. Since the regularization term in the restoration part can be arbitrary, one can model the regularization term by using different off-the-shelf denoisers and do not need to explicitly design various priors on the reflectance component. Because of flexibility of the model, the desired enhanced results can be solved efficiently by techniques like the plug-and-play inspired algorithm. Numerical experiments based on three public datasets demonstrate that our proposed method outperforms other competing methods, including deep learning approaches, under three commonly used metrics in terms of visual quality and image quality assessment.

The monotonicity of discrete Laplacian implies discrete maximum principle, which in general does not hold for high order schemes. The $Q^2$ spectral element method has been proven monotone on a uniform rectangular mesh. In this paper we prove the monotonicity of the $Q^2$ spectral element method on quasi-uniform rectangular meshes under certain mesh constraints. In particular, we propose a relaxed Lorenz’s condition for proving monotonicity.

In this paper, we extend the unified gas kinetic particle (UGKP) method to the frequency-dependent radiative transfer equation with both absorption-emission and scattering processes. The extended UGKP method could capture the diffusion and free transport limit and provide a smooth transition in the physical and frequency space in the regime between the above two limits. The proposed scheme has the properties of asymptotic-preserving and regime-adaptive, which make it an accurate and efficient scheme in the simulation of multiscale photon transport problems. In the UGKP formulation of flux construction and distribution closure, the coefficients of the non-equilibrium free stream distribution and near-equilibrium Planck expansion are independent of the time step. Therefore, even with a large CFL number, the UGKP can preserve a physically consistent ratio of the non-equilibrium and the near-equilibrium proportion. The methodology of scheme construction is a coupled evolution of the macroscopic energy equation and the microscopic radiant intensity equation, where the numerical flux in the macroscopic energy equation and the closure in the microscopic radiant intensity equation are constructed based on the integral solution. Both numerical dissipation and computational complexity are well controlled, especially in the optically thick regime. 2D multi-thread code on a general unstructured mesh has been developed. Several numerical tests have been simulated to verify the numerical scheme and code, covering a wide range of flow regimes. The numerical scheme and code we developed are highly demanded and widely applicable in high-energy engineering applications.

In this paper, we revisit some nonoverlapping domain decomposition methods for solving diffusion problems with discontinuous coefficients. We discover some interesting phenomena, that is, the Dirichlet-Neumann algorithm and Robin-Robin algorithms may make full use of the ratio of coefficients in some special cases. Detailedly, in the case of two subdomains, we find that their convergence rates are $\mathcal{O}(ν_1/ν_2)$ if $ν_1 < ν_2,$ where $ν_1, \ ν_2$ are coefficients of two subdomains. Moreover, in the case of many subdomains with red-black partition, the condition number bounds of Dirichlet-Neumann algorithm and Robin-Robin algorithm are $1+\epsilon(1+{\rm log}(H/h))^2$ and $C+\epsilon(1+ {\rm log}(H/h))^2,$ respectively, where $\epsilon$ equals ${\rm min}\{ν_R/ν_B,ν_B/ν_R\}$ and $ν_R,ν_B$ are the coefficients of red and black domains. By contrast, Neumann-Neumann algorithm and Dirichlet-Dirichlet algorithm could not obtain such good convergence results in these cases. Finally, numerical experiments are preformed to confirm our findings.

This paper is devoted to constructing and analyzing a new upwind finite volume element method for anisotropic convection-diffusion-reaction problems on general quadrilateral meshes. We prove the coercivity, and establish the optimal error estimates in $H^1$ and $L^2$ norm respectively. The novelty is the discretization of convection term, which takes the two terms Taylor expansion. This scheme has not only optimal first-order accuracy in $H^1$ norm, but also optimal second-order accuracy in $L^2$ norm, both for dominant diffusion and dominant convection. Numerical experiments confirm the theoretical results.