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Yanli Wang

Publications and source records attributed to Yanli Wang.

At least 55 records · Page 3Linked to original sources
arXiv2023

A spectral method for a Fokker-Planck equation in neuroscience with applications in neural networks with learning rules

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.

math.NA↗
arXiv2023

Solving Boltzmann equation with neural sparse representation

We consider the neural sparse representation to solve Boltzmann equation with BGK and quadratic collision model, where a network-based ansatz that can approximate the distribution function with extremely high efficiency is proposed. Precisely, fully connected neural networks are employed in the time and spatial space so as to avoid the discretization in space and time. The different low-rank representations are utilized in the microscopic velocity for the BGK and quadratic collision model, resulting in a significant reduction in the degree of freedom. We approximate the discrete velocity distribution in the BGK model using the canonical polyadic decomposition. For the quadratic collision model, a data-driven, SVD-based linear basis is built based on the BGK solution. All these will significantly improve the efficiency of the network when solving Boltzmann equation. Moreover, the specially designed adaptive-weight loss function is proposed with the strategies as multi-scale input and Maxwellian splitting applied to further enhance the approximation efficiency and speed up the learning process. Several numerical experiments, including 1D wave and Sod problems and 2D wave problem, demonstrate the effectiveness of these neural sparse representation methods.

math.NA↗
arXiv2023

Diffusion Models in Bioinformatics: A New Wave of Deep Learning Revolution in Action

Denoising diffusion models have emerged as one of the most powerful generative models in recent years. They have achieved remarkable success in many fields, such as computer vision, natural language processing (NLP), and bioinformatics. Although there are a few excellent reviews on diffusion models and their applications in computer vision and NLP, there is a lack of an overview of their applications in bioinformatics. This review aims to provide a rather thorough overview of the applications of diffusion models in bioinformatics to aid their further development in bioinformatics and computational biology. We start with an introduction of the key concepts and theoretical foundations of three cornerstone diffusion modeling frameworks (denoising diffusion probabilistic models, noise-conditioned scoring networks, and stochastic differential equations), followed by a comprehensive description of diffusion models employed in the different domains of bioinformatics, including cryo-EM data enhancement, single-cell data analysis, protein design and generation, drug and small molecule design, and protein-ligand interaction. The review is concluded with a summary of the potential new development and applications of diffusion models in bioinformatics.

cs.LG↗
arXiv2022

Highly efficient energy-conserving moment method for the multi-dimensional Vlasov-Maxwell system

We present an energy-conserving numerical scheme to solve the Vlasov-Maxwell (VM) system based on the regularized moment method proposed in [Z. Cai, Y. Fan, and R. Li. CPAM, 2014]. The globally hyperbolic moment system is deduced for the multi-dimensional VM system under the framework of the Hermite expansions, where the expansion center and the scaling factor are set as the macroscopic velocity and local temperature, respectively. Thus, the effect of the Lorentz force term could be reduced into several ODEs about the macroscopic velocity and the moment coefficients of higher order, which could significantly reduce the computational cost of the whole system. An energy-conserving numerical scheme is proposed to solve the moment equations and the Maxwell equations, where only a linear equation system needs to be solved. Several numerical examples such as the two-stream instability, Weibel instability, and the two-dimensional Orszag Tang vortex problem are studied to validate the efficiency and excellent energy-preserving property of the numerical scheme.

math.NA↗
arXiv2022

An asymptotic-preserving IMEX method for nonlinear radiative transfer equation

We present an asymptotic preserving method for the radiative transfer equations in the framework of PN method. An implicit and explicit method is proposed to solve the P N system based on the order analysis of the expansion coefficients of the distribution function. The order of each coefficient expanded in the Knudsen number is found through the process of Maxwellian iteration, and the coefficients of high order are treated explicitly while that of low order are treated implicitly in each equation of P N system. Energy inequality is proved for this numerical scheme. Several numerical examples validate this new AP scheme in both optical thick and thin regions.

math.NA↗
arXiv2022

Learning invariance preserving moment closure model for Boltzmann-BGK equation

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 by Grad 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. and Huang et al. 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.

math.NA↗
arXiv2022

Hermite spectral method for multi-species Boltzmann equation

We introduce a numerical scheme for the full multi-species Boltzmann equation based on Hermite spectral method. With the proper choice of expansion centers for different species, a practical algorithm is derived to evaluate the complicated multi-species binary collision operator. New collision models are built by combining the quadratic collision model and the simple BGK collision model under the framework of the Hermite spectral method, which enables us to balance the computational cost and accuracy. Several numerical experiments are implemented to validate the dramatic efficiency of this new Hermite spectral method. Moreover, we can handle the problems with as many as 100 species, which is far beyond the capability of the state-of-art algorithms.

math.NA↗
arXiv2021

Numerical Solver for the Boltzmann Equation With Self-Adaptive Collision Operators

We use the Burnett spectral method to solve the Boltzmann equation whose collision term is modeled by separate treatments for the low-frequency part and high-frequency part of the solution. For the low-frequency part representing the sketch of the distribution function, the binary collision is applied, while for the high-frequency part representing the finer details, the BGK approximation is applied. The parameter controlling the ratio of the high-frequency part and the low-frequency part is selected adaptively on every grid cell at every time step. This self-adaptation is based on an error indicator describing the difference between the model collision term and the original binary collision term. The indicator is derived by controlling the quadratic terms in the modeling error with linear operators. Our numerical experiments show that such an error indicator is effective and computationally affordable.

math.NA↗
arXiv2021

$Q$-tensor gradient flow with quasi-entropy and discretizations preserving physical constraints

We propose and analyze numerical schemes for the gradient flow of $Q$-tensor with the quasi-entropy. The quasi-entropy is a strictly convex, rotationally invariant elementary function, giving a singular potential constraining the eigenvalues of $Q$ within the physical range $(-1/3,2/3)$. Compared with the potential derived from the Bingham distribution, the quasi-entropy has the same asymptotic behavior and underlying physics. Meanwhile, it is very easy to evaluate because of its simple expression. For the elastic energy, we include all the rotationally invariant terms. The numerical schemes for the gradient flow are built on the nice properties of the quasi-entropy. The first-order time discretization is uniquely solvable, keeping the physical constraints and energy dissipation, which are all independent of the time step. The second-order time discretization keeps the first two properties unconditionally, and the third with an $O(1)$ restriction on the time step. These results also hold when we further incorporate a second-order discretization in space. Error estimates are also established for time discretization and full discretization. Numerical examples about defect patterns are presented to validate the theoretical results.

math.NA↗
arXiv2021

Optimal Variable Speed Limit Control Strategy on Freeway Segments under Fog Conditions

Fog is a critical external factor that threatens traffic safety on freeways. Variable speed limit (VSL) control can effectively harmonize vehicle speed and improve safety. However, most existing weather-related VSL controllers are limited to adapt to the dynamic traffic environment. This study developed optimal VSL control strategy under fog conditions with fully consideration of factors that affect traffic safety risks. The crash risk under fog conditions was estimated using a crash risk prediction model based on Bayesian logistic regression. The traffic flow with VSL control was simulated by a modified cell transmission model (MCTM). The optimal factors of VSL control were obtained by solving an optimization problem that coordinated safety and mobility with the help of the genetic algorithm. An example of I-405 in California, USA was designed to simulate and evaluate the effects of the proposed VSL control strategy. The optimal VSL control factors under fog conditions were compared with sunny conditions, and different placements of VSL signs were evaluated. Results showed that the optimal VSL control strategy under fog conditions changed the speed limit more cautiously. The VSL control under fog conditions in this study effectively reduced crash risks without significantly increasing travel time, which is up to 37.15% reduction of risks and only 0.48% increase of total travel time. The proposed VSL control strategy is expected to be of great use in the development of VSL systems to enhance freeway safety under fog conditions.

eess.SY↗
arXiv2021

Hermite spectral method for Fokker-Planck-Landau equation modeling collisional plasma

We propose an Hermite spectral method for the Fokker-Planck-Landau (FPL) equation. Both the distribution functions and the collision terms are approximated by series expansions of the Hermite functions. To handle the complexity of the quadratic FPL collision operator, a reduced collision model is built by adopting the quadratic collision operator for the lower-order terms and the diffusive Fokker-Planck operator for the higher-order terms in the Hermite expansion of the reduced collision operator. The numerical scheme is split into three steps according to the Strang splitting, where different expansion centers are employed for different numerical steps to take advantage of the Hermite functions. The standard normalized Hermite basis [36] is adopted during the convection and collision steps to utilize the precalculated coefficients of the quadratic collision terms, while the one constituted by the local macroscopic velocity and temperature is utilized for the acceleration step, by which the effect of the external force can be simplified to an ODE. Projections between different expansion centers are achieved by an algorithm proposed in [29]. Several numerical examples are studied to test and validate our new method

physics.comp-ph↗
arXiv2020

A Two-fluid Model for Plasma with Prandtl Number Correction

A two-fluid model is derived from the plasma kinetic equations using the moment model reduction method. The moment method we adopt was recently developed with a globally hyperbolic regularization where the moment model attained is locally well-posed in time. Based on the hyperbolic moment model with well-posedness, the Maxwellian iteration method is utilized to get the closure relations for the resulted two-fluid model. By taking the Shakhov collision operator in the Maxwellian iteration, the two-fluid model inherits the correct Prandtl number from the plasma kinetic equations. The new model is formally the same as the five-moment two-fluid model except for the closure relations, where the pressure tensor is anisotropic and the heat flux is presented. This provides the model the capacity to depict problems with anisotropic pressure tensor and large heat flux.

physics.plasm-ph↗
arXiv2020

A hybrid moment method for multi-scale kinetic equations based on maximum entropy principle

We propose a hybrid moment method for the multi-scale kinetic equations in the framework of the regularized moment method [7]. In this method, the fourth order moment system is chosen as the governing equations in the fluid region. When transiting from the fluid regime to the kinetic regime, the maximum entropy principle is adopted to reconstruct the kinetic distribution function, so that the information in the fluid region can be utilized thoroughly. Moreover, only one uniform set of numerical scheme is needed for both the fluid and kinetic regions. Numerical tests show the high efficiency of this new hybrid method.

physics.comp-ph↗
arXiv2019

Regularized 13-Moment Equations for Inverse Power Law Models

We propose a systematic methodology to derive the regularized thirteen-moment equations in the rarefied gas dynamics for a general class of linearized collision models. Detailed expressions of the moment equations are written down for all inverse power law models as well as the hard-sphere model. By linear analysis, we show that the equations are stable near the equilibrium. The models are tested for shock structure problems to show its capability to capture the correct flow structure in strong nonequilibirum.

physics.flu-dyn↗
arXiv2019

Approximation of the Boltzmann Collision Operator Based on Hermite Spectral Method

Based on the Hermite expansion of the distribution function, we introduce a Galerkin spectral method for the spatially homogeneous Boltzmann equation with the realistic inverse-power-law models. A practical algorithm is proposed to evaluate the coefficients in the spectral method with high accuracy, and these coefficients are also used to construct new computationally affordable collision models. Numerical experiments show that our method captures the low-order moments very efficiently.

math.NA↗
arXiv2019

Numerical Simulation of Microflows using Hermite Spectral Methods

We propose a Hermite spectral method for the spatially inhomogeneous Boltzmann equation. For the inverse-power-law model, we generalize an approximate quadratic collision operator defined in the normalized and dimensionless setting to an operator for arbitrary distribution functions. An efficient algorithm with a fast transform is introduced to discretize this new collision operator. The method is tested for one-dimensional benchmark microflow problems.

math.SP↗