Introduction to Neural Networks for Dynamic Systems

Neural networks have revolutionized how we approach complex dynamical systems. In this post, we'll explore how these powerful tools can be applied to engineering problems.

Mathematical Foundation

When modeling dynamical systems, we often start with a differential equation of the form:

$\frac{dx}{dt} = f(x, t, \theta)$

where $x$ represents the state variables, $t$ is time, and $\theta$ are the system parameters.

Simple Implementation Example

Here's a basic implementation of a neural network for modeling a dynamical system:

import torch
import torch.nn as nn
import numpy as np

class DynamicSystemNN(nn.Module):
    def __init__(self, input_size, hidden_size, output_size):
        super(DynamicSystemNN, self).__init__()
        self.layer1 = nn.Linear(input_size, hidden_size)
        self.layer2 = nn.Linear(hidden_size, hidden_size)
        self.layer3 = nn.Linear(hidden_size, output_size)
        self.relu = nn.ReLU()
        
    def forward(self, x):
        x = self.relu(self.layer1(x))
        x = self.relu(self.layer2(x))
        x = self.layer3(x)
        return x

# Example usage
model = DynamicSystemNN(input_size=2, hidden_size=20, output_size=2)