1,397 research outputs found
Temperature control in continuous furnace by structural diagram method
The fundamentals of the structural diagram method for distributed parameter systems (DPSs) are presented and reviewed. An example is given to illustrate the application of this method for control design
PDE models of adder mechanisms in cellular proliferation
Cell division is a process that involves many biochemical steps and complex biophysical mechanisms. To simplify the understanding of what triggers cell division, three basic models that subsume more microscopic cellular processes associated with cell division have been proposed. Cells can divide based on the time elapsed since their birth, their size, and/or the volume added since their birth-the timer, sizer, and adder models, respectively. Here, we propose unified adder-sizer models and investigate some of the properties of different adder processes arising in cellular proliferation. Although the adder-sizer model provides a direct way to model cell population structure, we illustrate how it is mathematically related to the well-known model in which cell division depends on age and size. Existence and uniqueness of weak solutions to our 2+1-dimensional PDE model are proved, leading to the convergence of the discretized numerical solutions and allowing us to numerically compute the dynamics of cell population densities. We then generalize our PDE model to incorporate recent experimental findings of a system exhibiting mother-daughter correlations in cellular growth rates. Numerical experiments illustrating possible average cell volume blowup and the dynamical behavior of cell populations with mother-daughter correlated growth rates are carried out. Finally, motivated by new experimental findings, we extend our adder model cases where the controlling variable is the added size between DNA replication initiation points in the cell cycle
A frequency-dependent -adaptive technique for spectral methods
When using spectral methods, a question arises as how to determine the
expansion order, especially for time-dependent problems in which emerging
oscillations may require adjusting the expansion order. In this paper, we
propose a frequency-dependent -adaptive technique that adaptively adjusts
the expansion order based on a frequency indicator. Using this -adaptive
technique, combined with recently proposed scaling and moving techniques, we
are able to devise an adaptive spectral method in unbounded domains that can
capture and handle diffusion, advection, and oscillations. As an application,
we use this adaptive spectral method to numerically solve the Schr\"{o}dinger
equation in the whole domain and successfully capture the solution's
oscillatory behavior at infinity
A Spectral Approach for Learning Spatiotemporal Neural Differential Equations
Rapidly developing machine learning methods has stimulated research interest
in computationally reconstructing differential equations (DEs) from
observational data which may provide additional insight into underlying
causative mechanisms. In this paper, we propose a novel neural-ODE based method
that uses spectral expansions in space to learn spatiotemporal DEs. The major
advantage of our spectral neural DE learning approach is that it does not rely
on spatial discretization, thus allowing the target spatiotemporal equations to
contain long range, nonlocal spatial interactions that act on unbounded spatial
domains. Our spectral approach is shown to be as accurate as some of the latest
machine learning approaches for learning PDEs operating on bounded domains. By
developing a spectral framework for learning both PDEs and integro-differential
equations, we extend machine learning methods to apply to unbounded DEs and a
larger class of problems.Comment: 21 pages, 5 figure
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