Change \xi_k to \xi

examples/ODEs/Part_9_SINDy.ipynb

@@ -10,7 +10,7 @@
  "This tutorial demonstrates the use of [sparse identification of nonlinear dynamics (SINDy)](https://arxiv.org/abs/1509.03580) in Neuromancer. \n",
  "\n",
  "## SINDy for System Identification\n",
- "SINDy is a machine learning model that uses sparse regression techniques to estimate the dynamics that guide time derivatives $\\dot{x}_k$ from state variables $x$ at time $k$. SINDy does this by creating a library of candidate functions $\\theta$, at which each state variable in $x_k$ is evaluated, and coefficients $\\xi_k$. SINDy then uses linear regression to fit the coefficients in the equation $\\dot{x}_k = \\theta(x_k)\\xi_k$. SINDy aims to find as few nonzero coefficients as possible while still accurately describing the relationship between $x_k$ and $\\dot{x}_k$. With this, it is possible to use the values of $\\dot{x}_k$ to predict the trajectory of $x_k$ given some initial condition.\n",
+ "SINDy is a machine learning model that uses sparse regression techniques to estimate the dynamics that guide time derivatives $\\dot{x}_k$ from state variables $x$ at time $k$. SINDy does this by creating a library of candidate functions $\\theta$, at which each state variable in $x_k$ is evaluated, and coefficients $\\xi$. SINDy then uses linear regression to fit the coefficients in the equation $\\dot{x}_k = \\theta(x_k)\\xi$. SINDy aims to find as few nonzero coefficients as possible while still accurately describing the relationship between $x_k$ and $\\dot{x}_k$. With this, it is possible to use the values of $\\dot{x}_k$ to predict the trajectory of $x_k$ given some initial condition.\n",
  "\n",
  "<img src=\"figs/sindy.jpeg\" width=600/>\n",
  "\n",
@@ -414,7 +414,7 @@
  "source": [
  "## SINDy system model\n",
  "\n",
- "Here we construct a SINDy model based on our library: $$\\dot{x}_k = \\theta(x_k^T)\\xi_k,$$ where $x_k$ is a row within the data matrix and $\\xi_k$ is the corresponding column within the matrix of coefficients. "
+ "Here we construct a SINDy model based on our library: $$\\dot{x}_k = \\theta(x_k^T)\\xi,$$ where $x_k$ is a row within the data matrix and $\\xi$ is the corresponding column within the matrix of coefficients. "
  ]
  },
  {
@@ -435,7 +435,7 @@
  "source": [
  "## Model Integration\n",
  "\n",
- "Then, we combine this SINDy model with a built-in Neuromancer integrator. This means that when we provide our model with a single data point $x_k$, we can receive an estimation of the next data point $x_{k+1}$, instead of an estimation of the derivative $\\dot{x_k}$ at the current time step. This saves us the step of having to actually find the true values of $\\dot{x}_k$. Now our model can be represented by the expression: $$x_{k+1} = \\text{ODESolve}(\\theta(x_k^T)\\xi_k)$$"
+ "Then, we combine this SINDy model with a built-in Neuromancer integrator. This means that when we provide our model with a single data point $x_k$, we can receive an estimation of the next data point $x_{k+1}$, instead of an estimation of the derivative $\\dot{x_k}$ at the current time step. This saves us the step of having to actually find the true values of $\\dot{x}_k$. Now our model can be represented by the expression: $$x_{k+1} = \\text{ODESolve}(\\theta(x_k^T)\\xi)$$"
  ]
  },
  {
@@ -521,8 +521,8 @@
  " \n",
  "$$\n",
  "\\begin{align}\n",
- "&\\underset{\\xi_k}{\\text{minimize}} && \\sum_{i=1}^m \\Big(Q_1||x^i_1 - \\hat{x}^i_1||_2^2 + \\sum_{k=1}^{N} Q_N||x^i_k - \\hat{x}^i_k||_2^2 \\Big) \\\\\n",
- "&\\text{subject to} && x_{k+1} = \\text{ODESolve}(\\theta(x_k^T)\\xi_k) \\\\\n",
+ "&\\underset{\\xi}{\\text{minimize}} && \\sum_{i=1}^m \\Big(Q_1||x^i_1 - \\hat{x}^i_1||_2^2 + \\sum_{k=1}^{N} Q_N||x^i_k - \\hat{x}^i_k||_2^2 \\Big) \\\\\n",
+ "&\\text{subject to} && x_{k+1} = \\text{ODESolve}(\\theta(x_k^T)\\xi) \\\\\n",
  "\\end{align}\n",
  "$$ "
  ]
@@ -550,7 +550,7 @@
  "source": [
  "## Solve the problem\n",
  "\n",
- "We fit each of the unknown SINDy parameters $\\xi_k$ using stochastic gradient descent. In the [original SINDy paper](https://arxiv.org/abs/1509.03580), the authors fit $\\xi_k$ using the sequentially thresholded least squares regression algorithm. This algorithm consists of solving the normal equations for [least squares](https://en.wikipedia.org/wiki/Least_squares) regression to find coefficient values in $\\xi_k$. Then for some threshold $\\lambda$, set all values of $\\xi_k$ that are less than $\\lambda$ to $0$. This process is repeated until convergence. We use standard stochastic gradient descent based methods, which given proper training, will converge to the same values of $\\xi_k$ as sequentially thresholded least squares."
+ "We fit each of the unknown SINDy parameters $\\xi$ using stochastic gradient descent. In the [original SINDy paper](https://arxiv.org/abs/1509.03580), the authors fit $\\xi$ using the sequentially thresholded least squares regression algorithm. This algorithm consists of solving the normal equations for [least squares](https://en.wikipedia.org/wiki/Least_squares) regression to find coefficient values in $\\xi$. Then for some threshold $\\lambda$, set all values of $\\xi$ that are less than $\\lambda$ to $0$. This process is repeated until convergence. We use standard stochastic gradient descent based methods, which given proper training, will converge to the same values of $\\xi$ as sequentially thresholded least squares."
  ]
  },
  {