Momentum term

The momentum terms can be rewritten using the mass balance to yield ... 

In Example 10.9(b), if the target values for the outputs are d 2 = 0 and 1 22 = 1, ealeulate new values for the weights and biases using the baek-propagation algorithm. Assume a learning rate of 0.5 with no momentum term. 

Add Some Inertia Another way is to add a so-called momentum term to our previous expression for calculating the weight-changes (see equations 10.59 and... 

The first of these assumptions drops the momentum terms from the equations of motion, giving a situation known as creeping flow. This leaves Vr and coupled through a pair of simultaneous, partial differential equations. The pair can be solved when circumstances warrant, but the second assumption allows much greater simplification. It allows to be given by a single, ordinary differential equation ... 

To reduce the chance that the network will be trapped by a set of endlessly oscillating connection weights, a momentum term can be added to the update of the weights. This term adds a proportion of the update of the weight in the previous epoch to the weight update in the current epoch ... 

Note that the vector (directional) character of the convected momentum terms (i.e., mV) is that of the velocity, because m is a scalar (i.e., tit = pV A is a scalar product). 

The second term represents the conserved current and the third is a momentum term. Noting that dL/dq = P, the field momentum bscomes... 

Similarly, by equating the unsteady momentum term with buoyancy gives a timescale... 

Equation 7.10 shows that the total pressure gradient comprises three components that are due to fluid friction, the rate of change of momentum and the static head. The momentum term is usually called the accelerative component. Thus... 

Using Ui = pl/2mi + 4>i(x) in the classical limit [the number of energetically degenerate states = (dpdq)/h and integrating the momentum terms for Z yields... 

The four experiments done previously with Rnp (= 0.5, 1, 3, 4) were used to train the neural network and the experiment with / exp = 2 was used to validate the system. Dynamic models of process-model mismatches for three state variables (i.e. X) of the system are considered here. They are the instant distillate composition (xD), accumulated distillate composition (xa) and the amount of distillate (Ha). The inputs and outputs of the network are as in Figure 12.2. A multilayered feed forward network, which is trained with the back propagation method using a momentum term as well as an adaptive learning rate to speed up the rate of convergence, is used in this work. The error between the actual mismatch (obtained from simulation and experiments) and that predicted by the network is used as the error signal to train the network as described earlier. 

Next consider the momentum terms in Eq. (2.160), To find the rate at which momentum enters through ab it is noted that the rate at which mass crosses the strip on ab indicated in Fig. 2.20 is equal to (pudy), the control surface having a unit width. The rate at which momentum enters through this strip is, therefore, equal to pu2 dy). The rate at which momentum enters through the face ab, which will here be termed Mab, is, therefore, given by... 

To optimize the neural network design, important choices must be made for the selection of numerous parameters. Many of these are internal parameters that need to be tuned with the help of experimental results and experience with the specific application under study. The following discussion focuses on back-propagation design choices for the learning rate, momentum term, activation function, error function, initial weights, and termination condition. 

Now, the original two-dimensional one-particle Hamiltonian H y which mixes the x- and -directions via the angular momentum term has been transformed into the sum of two Hamiltonians of independent particles with masses Ml and M2 in one-dimensional harmonic potentials with frequencies u>i and (O2, respectively. The new masses and frequencies are given in terms of the original frequencies co, oiy, and the cyclotron frequency oi = Via oj there is an implicit dependence of the Hamiltonian H3 on the magnetic field strength B. 

Material, energy, and momentum balances are essential to fully describe the performance of reactors, and often simplifying assumptions and phenomenological assumptions are needed especially for energy and momentum terms, as indicated in greater detail in Sec. 19... 

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