Mosquito borne diseases pose a significant health risk for humans. In North America, Culex mosquitoes are a major vector for several diseases including West Nile Virus and St. Louis Encephalitis. In ...many instances, models used to predict the spread of mosquito borne disease rely on a quantification of mosquito abundance. In our work, we present a novel age-structured partial differential equation model for simulating Culex mosquito abundance. The model is constructed using a system of two dimensional coupled advection reaction equations, in which the first dimension represents the age of the mosquitoes within a growth-stage population and the second dimension is time. We form six mosquito growth-stage populations by subdividing the mosquito life cycle into six stages: egg, larvae/pupae, and four adult gonotrophic cycles. Each growth-stage population is coupled through the boundary conditions on the age of the mosquito, which advances the population through its life cycle. The model also includes a population of diapausing adults represented using an ordinary differential equation. The solution curves for each equation provide the distribution of mosquitoes over time for each growth-stage population. This model provides information on the relative abundance of mosquitoes as well as the abundance of mosquitoes at specific ages. We simulate mosquito abundance for the Greater Ontario Area and compare the simulated adult abundance to mosquito trap count data. The model produces mosquito abundance patterns similar to those observed in trap count data.
•Age-structured model for Culex mosquito abundance.•Explicit tracking of age of mosquitoes within growth stages.•Continuous diapausing function allowing for multiple year simulations.•Simulated abundance patterns are reasonable as compared to mosquito trap count data.
This article develops a decentralized actuator fault compensation scheme for hypersonic vehicles, using a model transform based backstepping control design. Different from the actuator dynamics ...represented by second-order ordinary differential equations (ODE) studied in the existing literature, a new actuator dynamics is studied in this article, modeled by a fourth-order partial differential equation (PDE), i.e., Euler-Bernoulli beam (EBB) equation. An invertible model transformation is introduced to transform the EBB PDE-ODE cascade system into a heat-like PDE-ODE cascade system and a new target system is constructed from a two-step backstepping procedure. A novel decentralized fault-tolerant control scheme is proposed for the heat-like PDE-ODE cascade system and the well-posedness of the system solutions is analyzed. The closed-loop system stability and the well-posedness property of the original EBB PDE-ODE cascade system are ensured. Simulation results illustrate the effectiveness of the proposed fault compensation scheme.
In this paper, the initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation over an open bounded domain
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are considered. Based on an ...appropriate maximum principle that is formulated and proved in the paper, too, some a priory estimates for the solution and then its uniqueness are established. To show the existence of the solution, first a formal solution is constructed using the Fourier method of the separation of the variables. The time-dependent components of the solution are given in terms of the multinomial Mittag-Leffler function. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional multi-term diffusion equation that turns out to be a classical solution under some additional conditions. Another important consequence from the maximum principle is a continuously dependence of the solution on the problem data (initial and boundary conditions and a source function) that – together with the uniqueness and existence results – makes the problem under consideration to a well-posed problem in the Hadamard sense.
A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network’s loss function. The PINN approach has shown great ...success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal coordinates. However, the PINN’s accuracy suffers significantly for strongly non-linear and higher-order time-varying partial differential equations such as Allen Cahn and Cahn Hilliard equations. To resolve this problem, a novel PINN scheme is proposed that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, the Cahn Hilliard and Allen Cahn equations are solved. These equations are widely used to describe phase separation and reaction–diffusion systems. Additionally, two new techniques have been introduced to improve the proposed bc-PINN scheme. The first technique uses the initial condition of a time-segment to guide the neural network map closer to the true map over that segment. The second technique is a transfer learning approach where the features learned from the previous training are preserved. We have demonstrated that these two techniques improve the accuracy and efficiency of the bc-PINN scheme significantly. It has also been demonstrated that the convergence is improved by using a phase space representation for higher-order PDEs. It is shown that the proposed bc-PINN technique is significantly more accurate and efficient than PINN.
We develop backstepping state feedback control to stabilize a moving shockwave in a freeway segment under bilateral boundary actuations of traffic flow. A moving shockwave, consisting of light ...traffic upstream of the shockwave and heavy traffic downstream, is usually caused by changes of local road situations. The density discontinuity travels upstream and drivers caught in the shockwave experience transitions from free to congested traffic. Boundary control design in this article brings the shockwave front to a static setpoint position, hindering the upstream propagation of traffic congestion. The traffic dynamics are described with Lighthil-Whitham-Richard model, leading to a system of two first-order hyperbolic partial differential equations (PDEs). Each represents the traffic density of a spatial domain segregated by the moving interface. By Rankine-Hugoniot condition, the interface position is driven by flux discontinuity and thus governed by an ordinary differential equation (ODE) dependent on the PDE states. The control objective is to stabilize both the PDE states of traffic density and the ODE state of moving shock position to setpoint values. Using delay representation and backstepping method, we design predictor feedback controllers to cooperatively compensate state-dependent input delays to the ODE. From Lyapunov stability analysis, we show local stability of the closed-loop system in <inline-formula><tex-math notation="LaTeX">H^1</tex-math></inline-formula> norm with an arbitrarily fast convergence rate. The stabilization result is demonstrated by a numerical simulation and the total travel time of the open-loop system is reduced by <inline-formula><tex-math notation="LaTeX">12 \%</tex-math></inline-formula> in the closed loop.
An implicit unconditional stable difference scheme is presented for a kind of linear space–time fractional convection–diffusion equation. The equation is obtained from the classical integer order ...convection–diffusion equations with fractional order derivatives for both space and time. First-order consistency, unconditional stability, and first-order convergence of the method are proven using a novel shifted version of the classical Grünwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence.
•Novel connections between neural network architectures and HJ PDE viscosity solutions.•Exact solutions of certain high dimensional HJ PDEs using neural networks.•Grid-free methods for certain ...Hamilton-Jacobi Partial Differential Equations.
We propose novel connections between several neural network architectures and viscosity solutions of some Hamilton–Jacobi (HJ) partial differential equations (PDEs) whose Hamiltonian is convex and only depends on the spatial gradient of the solution. To be specific, we prove that under certain assumptions, the two neural network architectures we proposed represent viscosity solutions to two sets of HJ PDEs with zero error. We also implement our proposed neural network architectures using Tensorflow and provide several examples and illustrations. Note that these neural network representations can avoid curve of dimensionality for certain HJ PDEs, since they do not involve neither grids nor discretization. Our results suggest that efficient dedicated hardware implementation for neural networks can be leveraged to evaluate viscosity solutions of certain HJ PDEs.
In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the ...least-squares sense as the loss function and a deep neural network (DNN) to approximate the PDE solution. In this work, we propose a deep mixed residual method (MIM) to solve PDEs with high-order derivatives. Notable examples include Poisson equation, Monge-Ampére equation, biharmonic equation, and Korteweg-de Vries equation. In MIM, we first rewrite a high-order PDE into a first-order system, very much in the same spirit as local discontinuous Galerkin method and mixed finite element method in classical numerical methods for PDEs. We then use the residual of the first-order system in the least-squares sense as the loss function, which is in close connection with least-squares finite element method. For aforementioned classical numerical methods, the choice of trial and test functions is important for stability and accuracy issues in many cases. MIM shares this property when DNNs are employed to approximate unknowns functions in the first-order system. In one case, we use nearly the same DNN to approximate all unknown functions and in the other case, we use totally different DNNs for different unknown functions. Numerous results of MIM with different loss functions and different choices of DNNs are given for four types of PDEs. In most cases, MIM provides better approximations (not only for high-order derivatives of the PDE solution but also for the PDE solution itself) than DGM with nearly the same DNN and the same execution time, sometimes by more than one order of magnitude. MIM with multiple DNNs often provides better approximations than MIM with only one DNN, sometimes by more than one order of magnitude. Numerical results also indicate interesting connections between MIM and classical numerical methods. Therefore, we expect MIM to open up a possibly systematic way to understand and improve deep learning for solving PDEs from the perspective of classical numerical analysis.
We propose a new reduced basis element-cum-component mode synthesis approach for parametrized elliptic coercive partial differential equations. In the Offline stage we construct a Library of ...interoperable parametrized reference components relevant to some family of problems; in the Online stage we instantiate and connect reference components (at ports) to rapidly form and query parametric systems. The method is based on static condensation at the interdomain level, a conforming eigenfunction “port” representation at the interface level, and finally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at the intradomain level. We show under suitable hypotheses that the RB Schur complement is close to the FE Schur complement: we can thus demonstrate the stability of the discrete equations; furthermore, we can develop inexpensive and rigorous (system-level) a posteriori error bounds. We present numerical results for model many-parameter heat transfer and elasticity problems with particular emphasis on the Online stage; we discuss flexibility, accuracy, computational performance, and also the effectivity of the a posteriori error bounds.
In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of ...an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as
k
-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.
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