Artificial Neural Networks

Artificial neural networks works pretty well for solving some differential equations.

Universal Approximators

Maxwell Stinchcombe and Halber White proved that no theoretical constraints for the feedforward networks to approximate any measurable function. In principle, one can use feedforward networks to approximate measurable functions to any accuracy.

However, the convergence slows down if we have a lot of hidden units. There is a balance between accuracy and convergence rate. More hidden units lead to slow convergence but more accuracy.

Here is a quick review of the history of this topic.

1. Cybenko 1989

   Cybenko proved that

   $$ \sum_k v_k \sigma(w_k x + u_k) $$

   is a good approximation of continuous functions because it is dense in continuous function space. In this result, $\sigma$ is a continuous sigmoidal function and the parameters are real.

2. Hornik 1989

   “Single hidden layer feedforward networks can approximate any measurable functions arbitrarily well regardless of the activation function, the dimension of the input and the input space environment.”

   Reference: http://deeplearning.cs.cmu.edu/notes/Sonia_Hornik.pdf

Activation Functions

Neural networks usually consists of some affine transformations and some activation functions¹

$$ \mathbf Y = H(\mathbf X, \mathbf W) $$

For example, $H$ can be a combination of a linear transformation $\hat L = \mathbf W \cdot $ and an nonlinear activation function $\sigma(\cdot)$, i.e., $H(\mathbf X, \mathbf W) = \sigma( \mathbf W \mathbf X)$. This is a super simple example and this transformation can be much more complicated.

There are many activation functions.

• [[Uni-Polar Sigmoid Function]] Uni-Polar Sigmoid Uni-polar sigmoid function and its properties
• [[Bipolar Sigmoid Function]] BiPolar Sigmoid BiPolar sigmoid function and its properties
• [[Hyperbolic Tangent]] Hyperbolic Tanh Tanh function and its properties
• [[Radial Basis Function]] Radial Basis Function Radial Basis Function function and its properties
• [[Conic Section Function]] Conic Section Function Conic Section Function and its properties
• [[ReLu]] ReLu Rectified Linear Unit, aka ReLu, and its properties
• [[Leaky ReLu]] Leaky ReLu Leaky ReLu and its properties
• [[ELU]] ELU ELU and its properties
• [[Swish]] Swish Swish and its properties

Solving Differential Equations

The problem here to solve is

$$ \frac{d}{dt}y(t)= - y(t), $$

with initial condition $y(0)=1$.

To construct a single layered neural network, the function is decomposed using

$$ \begin{align} y(t_i) & = y(t_0) + t_i v_k f(t_i w_k+u_k) \\ &= 1+t_i v_k f(t_i w_k+u_k) , \end{align} $$

where $y(t_0)$ is the initial condition and $k$ is summed over.

Presumably this should be the gate controlling trigering of the neuron or not. Therefore the following expit function serves this purpose well,

$$ f(x) = \frac{1}{1+\exp(-x)}. $$

One important reason for choosing this is that a lot of expressions can be calculated analytically and easily.

With the form of the function to be solved, we can define a cost

$$ I=\sum_i\left( \frac{dy}{dt}(t_i)+y(t_i) \right)^2, $$

which should be minimized to 0 if our structure of networks is optimized for this problem.

Now the task becomes clear:

1. Write down the cost analytically;
2. Minimized cost to find structure;
3. Substitute back to the function and we are done.

Overfitting

It is possible that we could over fit a network so that it works only for the training data. To avoid that, people use several strategies.

1. Split data into two parts, one for training and one for testing. A youtube video
2. Throw more data in. At least 10 times as many as examples as the DoFs of the model. A youtube video
3. Regularization by plugin a artificial term to the cost function, as an example we could add the . A youtube video

Neural Network and Finite Element Method

We consider the solution to a differential equation

$$ \mathcal L \psi - f = 0. $$

Neural network is quite similar to finite element method. In terms of finite element method, we can write down a neural network structured form of a function ²

$$ \psi(x_i) = A(x_i) + F(x_i, \mathcal N_i), $$

where $\mathcal N$ is the neural network structure. Specifically,

$$ \mathcal N_i = \sigma( w_{ij} x_j + u_i ). $$

The function is parameterized using the network. Such parameterization is similar to collocation method in finite element method, where multiple basis is used for each location.

One of the choices of the function $F$ is a linear combination,

$$ F(x_i, \mathcal N_i) = x_i \mathcal N_i, $$

and $A(x_i)$ should take care of the boundary condition.

With such parameterization, the differential equation itself is parameterized such that

$$ \mathcal L \psi - f = 0, $$

such that the minimization should be

$$ \lvert \mathcal L \psi - f \rvert^2 \to 0 $$

at each point.