Reaction-diffusion master equation, diffusion-limited reactions, and singular potentials

Experimental results have shown that anti-cancer therapies, such as radiotherapy and chemotherapy, can modulate the cell cycle and generate cell cycle phase-dependent responses. As a result, obtaining a detailed understanding of the cell cycle is one possible path towards improving the efficacy of many of these therapies. Here, we consider a basic structured partial differential equation (PDE) model for cell progression through the cell cycle, and derive expressions for key quantities, such as the population growth rate and cell phase proportions. These quantities are shown to be periodic and, as such, we compare the PDE model to a corresponding ordinary differential equation (ODE) model in which the parameters are linked by ensuring that the long-term ODE behaviour agrees with the average PDE behaviour. By design, we find that the ODE model does an excellent job of representing the mean dynamics of the PDE model within just a few cell cycles. However, by probing the parameter space we find cases in which this mean behaviour is not a good measure of the PDE population growth. Our analytical comparison of two caricature models (one PDE and one ODE system) provides insight into cases in which the simple ODE model is an appropriate approximation to the PDE model.

In this study, problems of the partial differential equation (PDE) modelling and vibration control design are first resolved for an overhead crane bridge system, which is regarded as a Timoshenko beam with an attached rigid body. With the established infinite‐dimensional PDE model, a Nussbaum function‐based adaptive control approach is developed in order to settle out parametric uncertainties of the system with unknown control directions. Under the proposed control method, states of the overhead crane bridge are globally bounded and finally converge to an adjustable region of zero. The stability of the closed‐loop overhead crane bridge system is analysed by employing Lyapunov's direct method. The effectiveness of the developed control strategy is verified through the simulation results.

A closed loop model of pulmonary gas exchange

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat "patchy" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

Loss landscape is a useful tool for characterizing and comparing neural network models. The main challenge for analysis of loss landscape for the deep neural networks is that they are generally highly nonconvex in very high-dimensional space. In this paper, we develop the “roughness” concept for understanding such landscapes in high dimensions and apply this technique to study two neural network models arising from solving differential equations. Our main innovation is the proposal of a well-defined and easy-to-compute roughness index (RI) which is based on the mean and variance of the (normalized) total variation for one-dimensional functions projected on randomly sampled directions. A large RI at the local minimizer indicates an oscillatory landscape profile and indicates a severe challenge for the first-order optimization method. Particularly, we observe the increasing-then-decreasing pattern for RI along the gradient descent path in most models. We apply our method to two types of loss functions used to solve partial differential equations (PDEs) when the solution of PDE is parametrized by neural networks. Our empirical results on these PDE problems reveal important and consistent observations that the landscapes from the deep Galerkin method around its local minimizers are less rough than the deep Ritz method.

Accurate reporting and forecasting of PM2.5 concentration are important for improving public health. In this paper, we propose a partial differential equation (PDE) model, specially, a linear diffusive equation, to describe the spatial-temporal characteristics of PM2.5 in order to make short-term prediction. We analyze the temporal and spatial patterns of a real dataset from China’s National Environmental Monitoring and validate the PDE-based model in terms of predicting the PM2.5 concentration of the next day by the former days’ history data. Our experiment results show that the PDE model is able to characterize and predict the process of PM2.5 transport. For example, for 300 continuous days of 2016, the average prediction accuracy of the PDE model over all city-regions is 93% or 83% based on different accuracy definitions. To our knowledge, this is the first attempt to use PDE-based model to study PM2.5 prediction in both temporal and spatial dimensions.

A partial differential equation (PDE) model for a flexible inverted pendulum system (FIPS) is derived by the use of the Hamilton principle. To solve the coupling system model, a singular perturbation method was adopted. The PDE model was divided into a fast subsystem and a slow subsystem using the singular perturbation method. To stabilize the fast subsystem, a boundary control force was applied at the free end of the beam. It then was proven that the closed‐loop subsystem is appropriate and exponentially stable. For the slow subsystem, a sliding mode control method was employed to design a controller and the Linear Matrix Inequality (LMI) method was used to design the sliding surface. It then was shown that the slow subsystem is exponentially stable.

Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

Populations interact in a wide variety of ways: through cooperation,competition, or predation. In this work we consider two interacting populations, individuals and a stimulus. Individuals move in response to the stimulus population while the stimulus only diffuses. Both populations grow while the stimulus population is being depleted by the individuals. In order to account for the random nature of the system, an individual-based model (IBM) is first developed and then upscaled into a continuous partial differential equation (PDE) model by considering transition probabilities of the individuals at each site. Finally, a set of numerical experiments is presented showing very good agreement between the IBM and the PDE model.

Summary There is no standard framework for solving optimization problems for systems described by agent-based models (ABMs). We present a method for constructing individual-level controls that steer the population-level dynamics of an ABM towards a desired state. Our method uses a system of partial differential equations (PDEs) with control functions to approximate the dynamics of the ABM with control. An optimal control problem is formulated in terms of the PDE model to mimic the optimization goal of the ABM. Mathematical theory is used to derive optimal controls for the PDE model, which are numerically approximated and transformed for use in the ABM. We use the Sugarscape ABM, a prototype ABM that includes agent and environmental heterogeneity and accumulation of agent resources over time. We present a PDE model that approximates well the spatial, temporal, and resource dynamics of the Sugarscape ABM. In both models, control represents taxation of agent wealth with the goal to maximize total taxes collected while minimizing the impact of taxation on the population over a finite time. Solutions to the optimal control problem yield taxation rates specific to an agent's location and current wealth. The use of optimal controls (generated by the PDE model) within the ABM performed better than other controls we evaluated, even though some error was introduced between the ABM and PDE models. Our results demonstrate the feasibility of using a PDE to approximate an ABM for control purposes and illustrate challenges that can arise in applying this technique to sophisticated ABMs. Copyright © 2016 John Wiley & Sons, Ltd.

Analytical solutions of partial differential equation (PDE) models describing reactive transport phenomena in saturated porous media are often used as screening tools to provide insight into contaminant fate and transport processes. While many practical modelling scenarios involve spatially variable coefficients, such as spatially variable flow velocity, v(x), or spatially variable decay rate, k(x), most analytical models deal with constant coefficients. Here we present a framework for constructing exact solutions of PDE models of reactive transport. Our approach is relevant for advection-dominant problems, and is based on a regular perturbation technique. We present a description of the solution technique for a range of one-dimensional scenarios involving constant and variable coefficients, and we show that the solutions compare well with numerical approximations. Our general approach applies to a range of initial conditions and various forms of v(x) and k(x). Instead of simply documenting specific solutions for particular cases, we present a symbolic worksheet, as supplementary material, which enables the solution to be evaluated for different choices of the initial condition, v(x) and k(x). We also discuss how the technique generalizes to apply to models of coupled multispecies reactive transport as well as higher dimensional problems.

The aim of the present study was to investigate a new type of Bayesian reconstruction method which utilizes partial differential equations (PDE) image models as a prior. PDE image models are very popular in image restoration and segmentation. Our method introduces such models to emission tomography reconstruction by using Bayesian one step late (OSL) algorithm and an OS acceleration one. In a PDE based model, the image can be viewed as the solution of an evolutionary differential equation. It can be thought as a descent of an energy function which entitled us to use PDE models in Bayesian reconstruction. In this paper, three different PDE models are studied, two of them are based on anisotropic diffusion model, and another is based on a complex diffusion model. All of them have the properties of edge-preserving and denoising as the latest median root prior (MRP). We validated the effectiveness of the method using a Zubal phantom in numerical experiments and compared it to the classical MLEM and MRP reconstruction. The results show that the proposed PDE model method is better than the MLEM and the MRP reconstruction methods in visualization, bias and variance, and are more suitable for OS acceleration than MRP.

Modelica is a new object-oriented language for modeling of large, complex, and heterogeneous physical systems. It is designed to allow for convenient and efficient modeling and simulation of the differential-algebraic equations (DAEs) systems. However, in many engineering fields, partial differential equations (PDEs) are applied. So it is necessary to be able to handle PDE models in Modelica. In this paper, we present how the Modelica language can be extended to support object-oriented modeling with partial differential equations. The description of PDE models in Modelica is introduced, which uses a natural way to describe the PDE models so that the extending language is easier to be understood and more convenient to be used. The prototype implementation included external translator and processor is described. An example modeled with PDE in Modelica and solved using the prototype is presented.

The radio frequency (RF) ablation is a promising minimally invasive form of treatment for hepatic metastases and primary tumors. Thereby a needle like applicator, which is interstitially placed into the lesion, induces an electric current that causes heating and consequent destruction of the tissue due to its Ohmic resistance. In order to be a true alternative to the standard surgical resection, RF ablation must lead to a result similar to R0 resections. Here, patient specific mathematical modeling and numerical simulation of the bio-physical processes lead to a valuable support of the therapy planning, because they allow for an a priori estimation of the success as well as an optimization of the therapy parameters. In this article we discuss a mathematical model of partial differential equations (PDEs) for the patient specific numerical simulation of RF ablation. A particular focus lies on the consideration of uncertainties in the material properties, the underlying image data as well as the computational results. We discuss a stochastic PDE model, which allows for a sensitivity analysis of the computational results with respect to perturbations in the material properties. Furthermore a method for the fast estimation of the thermal necrosis is shown, which bases on the separation of non patient specific pre-calculations and patient specific computations, leading to an interactive real-time simulation tool.