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# 数学代写|常微分方程代考Ordinary Differential Equations代写|The neural network solution of ODE problems

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|The neural network solution of ODE problems

In this section, we illustrate the use of the MLP network to solve ODE problems. For this purpose, a common approach is to define a trial function that includes the given initial/boundary conditions and the output of the MLP network. Specifically, consider the following Cauchy problem:
\left{\begin{aligned} y^{\prime}(x) & =f(x, y(x)) \ y(a) & =y_a \end{aligned}\right.
Then a suitable trial function is given by
$$\bar{y}(x)=y_a+(x-a) O(x ; w)$$
where the $\mathrm{NN}$ function $O(x ; w)$ represents the output of the network. With this setting, the initial condition is automatically satisfied.
Further, consider the following boundary-value problem:
\left{\begin{aligned} y^{\prime \prime}(x) & =f\left(x, y(x), y^{\prime}(x)\right) \ y(a) & =y_a, \quad y(b)=y_b \end{aligned}\right.
In this case, an appropriate choice of the trial function on the interval $[a, b]$ is as follows:
$$\bar{y}(x)=y_a \frac{b-x}{b-a}+y_b \frac{x-a}{b-a}+\frac{(b-x)(x-a)}{(b-a)^2} O(x ; w) .$$
In the following, we refer to the MLP network of Figure 14.5 with input nodes for the value of $x$ and the bias, $K$ hidden nodes, and one output node.
For clarity, we explicitly write the bias weight, thus $W^{(1)}$ is a one-column matrix, and $W^{(2)}$ is a one-row matrix. The output is linear (no activation function), and its value is denoted with $O(x ; w, \theta)$. We have
$$O(x ; w, \theta)=\sum_{j=1}^K w_j^{(2)} \sigma\left(v_j^{(1)}\right)=\sum_{j=1}^K w_j^{(2)} \sigma\left(w_j^{(1)} x+\theta_j\right) .$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Parameter identification with neural networks

A NN represents a versatile computation framework that can be applied to large classes of ODE problems. In particular, it can be used to solve control and inverse problems governed by ODEs; see [110] for a detailed discussion and further references.

In the case of control problems and system identification problems where part of the dynamics function is unknown, one can use the setting of the trial function introduced in the previous section, together with the data representing part of the evolution of the real system to identify the missing part of the dynamics. On the other hand, we may have an adequate model of the system up to some parameters that need to be determined. This latter case is discussed in the following considering the Lotka-Volterra model (6.20). As discussed in Section 6.8, this is a model of evolution of a prey-predator system, which is written as follows:
\begin{aligned} & z_1^{\prime}=z_1\left(a-b z_2\right), \ & z_2^{\prime}=z_2\left(c z_1-d\right), \end{aligned}
where $z_1(x)$ and $z_2(x)$ represent the sizes of the prey and predator population, respectively, at the time instant $x$.

In this section, we discuss a NN approach to the identification of the parameters $(a, b, c, d)$ of (14.26). As in a real experiment, we assume that the population sizes have been measured and recorded with a time label for a representative period of time (e.g., one year), and this is all the data available. Thus, we have the following
$$\left(z_1^k, z_2^k, x^k\right), \quad k=1, \ldots, N$$
where $x^k, k=1, \ldots, N$, are the times when the measurement were performed.
Now, let us pursue an approach similar to the one of the previous section, and consider a MLP network as in Figure 14.5, but with two input nodes for the values of $z_1$ and $z_2$ (and no bias), $K$ hidden nodes, and four (linear) output nodes, corresponding to the four parameters sought. Specifically, we consider the following NN output:
$$O_i(w)=\sum_{k=1}^K w_{i k}^{(2)} \sigma\left(\sum_{j=1}^2 w_{k j}^{(1)} z_j\right), \quad i=1, \ldots, 4$$

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The neural network solution of ODE problems

\left{\begin{aligned} y^{\prime}(x) & =f(x, y(x)) \ y(a) & =y_a \end{aligned}\right.

$$\bar{y}(x)=y_a+(x-a) O(x ; w)$$

\left{\begin{aligned} y^{\prime \prime}(x) & =f\left(x, y(x), y^{\prime}(x)\right) \ y(a) & =y_a, \quad y(b)=y_b \end{aligned}\right.

$$\bar{y}(x)=y_a \frac{b-x}{b-a}+y_b \frac{x-a}{b-a}+\frac{(b-x)(x-a)}{(b-a)^2} O(x ; w) .$$

$$O(x ; w, \theta)=\sum_{j=1}^K w_j^{(2)} \sigma\left(v_j^{(1)}\right)=\sum_{j=1}^K w_j^{(2)} \sigma\left(w_j^{(1)} x+\theta_j\right) .$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Parameter identification with neural networks

\begin{aligned} & z_1^{\prime}=z_1\left(a-b z_2\right), \ & z_2^{\prime}=z_2\left(c z_1-d\right), \end{aligned}

$$\left(z_1^k, z_2^k, x^k\right), \quad k=1, \ldots, N$$

$$O_i(w)=\sum_{k=1}^K w_{i k}^{(2)} \sigma\left(\sum_{j=1}^2 w_{k j}^{(1)} z_j\right), \quad i=1, \ldots, 4$$

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