I am the first author of this work and was primarily responsible for the research described therein. This is the culmination of a research project with Andrew Nagel and supervised by Prof.
Computational biophysics
Cryogenic electron microscopy, for example, requires samples to be frozen at temperatures that are not hospitable to most life forms. 1The Abbe direction limit prevents classical optical instruments from resolving features less than a few times smaller than the wavelength of light used by the device.
Microuidic and nanouidic devices
Nanopores, and the translocation of polymers through nanopores, are discussed in Chapter 3 as part of a new MNFD design. In particular, the work in Chapter 6 aims to connect the results of this thesis to all MNFDs that have periodic geometric designs.
Numerical methods meet deep learning
Introduction to the neural network method
This section provides a brief introduction to the basic elements of the neural network method (NNM) for solving differential equations. The basic idea of NNM is to train a deep neural network to approximately satisfy the target PDE.
Implicit solvent models: Langevin dynamics
Another force with which the solvent acts on the particles is the stochastic forcing term, which represents the total effect of the thermal motion of the solvent. Mathematical models 20 a way to show the real dynamics of a system more like an idealized one.
Electric eld models: Laplace's equation
Drift velocity, eective charge, and mobility
The overdamped Langevin equation in the presence of a constant and uniform electric field is of the form As a result, it is often stated that the Smoluchowski model predicts that the mobility µ is independent of the size and shape of the charged object under consideration.
Modelling ensembles: The Smoluchowski equation
Then the first moment of FPT, i.e. the mean first transit time (MFPT), is written as. The intramolecular variability of the translocation time τ can be conveniently quantified by the coefficient of variation.
Results
The different sort orders are produced by varying the size and shape of the channels, which controls the relative importance of the two trends.
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Another intrinsic difference between the two devices is in the shape of the electric field. As mentioned above, the starting radius of the microscopic model is included in the mesoscopic model as an absorbing boundary. Results of the microscopic simulations. a) The average of the microscopic time, tmicro, as a function of the chain length.
Thus, the origin of the mesoscopic coordinate system was in the center of the channel. The polymers were assumed to have a negligible effect on the shape of the electric field.
Results
- Irregularity is a bottleneck
- Convergence of error with loss
- No pathologically unphysical solutions
- Convergence of loss with capacity
With this relaxation in place, Magill, Nagel and Haan [26] found that the standard formulation of the NNM performed reliably. Neural network solutions for differential equations in non-convex domains: Solving the electric field in the slit-well microuidic device 68 invariances of the system under study. Neural Network Solutions for Differential Equations in Nonconvex Domains: Solving the Electric Field in the Slit-Well Microuidic Device 69 alleviating this concern about the NNM.
First, it is shown that the NNM solutions recover the left-right (anti) symmetries of the true solution. Neural network solutions for differential equations in nonconvex domains: Solving the electrical part in the 72-well microfluidic device memory-efficient representations of PDE solutions with NNM.
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Specifically, the NNM is used to solve a model of the electric field in the slit-well microfluidic device, which is an application. 4(c) shows a pronounced relative error in the electric field near the corners at the bottom of the well. Summary of the NNM solutions selected for flux retention and particle simulation tests.
The behavior of E( ˜u;) for the FEM solution differs from that of the NNM solutions in some important ways. In particular, some of the NNM solutions roughly conserve flux globally as well as the FEM solution.
Results
Intrinsic dimensionality
Basically, a large ensemble of neural networks was trained to solve a family of PDEs associated with a problem parameter x0, and the resulting internal representations were analyzed as x0 varied. The main result is that, for a given choice of x0, neural networks always learn essentially the same representations as long as they have sufficient capacity. The intrinsic dimensionality, as more precisely defined in the manuscript, is approximately equal to the number of important principal components in the hidden layers of different neural networks.
The most prominent result is that the intrinsic dimensionality is much smaller than the total width: very wide neural networks do not use all their degrees of freedom. Furthermore, it appears that the effective number of degrees of freedom converges to a maximum at large latitudes.
Reproducibility and specicity
The fact that many of these functions are approximately independent of x0 has implications for the use of NNM on parameterized PDEs. In particular, this suggests that transfer learning protocols can be practically useful if a PDE needs to be solved repeatedly for many values of its parameters in succession. The solution for one choice of the parameters can be a very good initial guess for the solution for another choice of the parameters.
Interpreting the features
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We visualize the main components of the first layers, which turned out to be general. The remaining layers of the receiver are initialized randomly and it is trained in problem B. 19] successfully used their method to confirm the generality of the first layers of image-based CNNs.
14], we will use the SVCCA to define a scalar measure of the similarity of two layers. The last row of Figure 3 contains the same components shown in Figure 5 of the main text.
Results
Direct and indirect mobilities are equivalent
Indirect mobility better exploits parallel computing
However, the numerical analyzes of the algorithms were found to break down in the low Péclet number regime. The above discussion neglects a crucial component of the computational cost of the indirect mobility: the calculation of the stationary distribution. In the slot well example, the stationary distribution was found to be essentially uniform when the inter-period delimiter of the device was placed in the center of the slots.
This was assumed to occur due to the geometrical barrier in the system near these points. In any case, for the example of nanoparticles traversing the crack well, proper sampling of the immobile scattering increases the total computational cost of indirect mobility by no more than a factor of 2 or 3.
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2(b) shows the relative error of the indirect mobility values compared to the direct mobility measurements. Ratio of the predicted run times of the direct and indirect mobility estimators, for different choices of Npara (indicated by line color) and CV =σ0/hτ1i (indicated by line style). Equation C2 states that the mean relative error of the direct mobility estimator is proportional to .
Figure 7(a) shows the predicted ratio of transit times for the direct and indirect mobility estimators. This will be exponentially faster than the convergence of the direct small business mobility estimator.
Basic algorithms
The first two terms on the RHS resemble the standard Euler discretization of a deterministic equation of overdamped motion under the influence of the force qE. This discussion of the Euler-Maruyama method is sufficient to illustrate the qualitative differences that arise in the numerical treatment of SDEs versus ODEs. When solving more challenging biophysical SDE models, the Euler-Maruyama discretization of the overdamped Langevin equation is not always adequate.
When simulating the dynamics of generic polymer models, motion occurs over a wide range of time scales, effectively approximating the dynamics with the Euler-Maruyama of the. Numerical methods 159 involve discretizing the full (i.e. not overdamped) Langevin equation with a stochastic variant of the velocity Verlet method [41].
Rates of convergence
- Timestep
- Geometry
- Ensemble size
- Many-body systems
Numerical methods It is worth noting that the strong convergence rate of the Euler-Maruyama method is only 0.5: the error in a given trajectory generated by the solver decreases only in proportion to the square root of ∆t. The implementation of boundary conditions is the main path through which the complexity of the problem geometry affects the computational cost of particle simulations. The ease with which particle simulations can be implemented in domains with complicated geometries has historically been one of the main motivations for using SDE models over PDE models.
As mentioned above, biophysical research is mainly concerned with the statistical properties of SDE models. Finally, it is important to consider the behavior of particle systems in many-body systems as a function of the number of interacting particles n.
The nite element method
Basic algorithms
- Mesh generation
- The method of weighted residuals
- Mixed formulation
- Model order reduction
Now assume that the PDE problem domain has been decomposed into a fine mesh and that a basis for the trial functions has been determined on this mesh. The coefficients should only be chosen using an implicit description of u that is classified by the data of the PDE problem (ie, the PDE and its boundary conditions). Each of the trial functions E˜ and u˜ is expressed in a separate base of piecewise polynomial functions with a compact support.
Depending on the details of the model, some of these steps may need to be performed only once. That is, the solution at a new value of p would be approximated by a sample function of the form.
Rates of convergence
The numerical method used to solve the system of equations (linear or non-linear) that negates the coordinates of the trial function. Remember that you are not necessarily the best approximation of you in the scope of the trial function base. Specifically, dene w∗ as the projection of the true solutionu into the space defined by the trial function basis φi, given by.
We can now appeal to results on the rate of convergence of the interpolation of a function on the FEM basis. More appropriate rejections of regularity for the solutions of PDEs are rejected in terms of the number of times a function is weakly differentiable (see Sobolev spaces).
The neural network method
Background
Basic algorithms
- Neural network architectures
- Training neural networks
- Loss functionals
- Regarding generalization
- Model order reduction with the NNM
Rates of convergence
- Expressivity
- Trainability