<commit_before>{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Automatically identifying governing partial differential equations with reverse finite differencing\n",
    "\n",
    "So far, most \"higher level\" physical knowledge has been derived from simpler first principles. However, these techniques have still not given a full understanding of turbulent flow. As such, it is of interes to find a different quantity, for which we can find some conservation or transport equation, that will be simpler to solve, e.g., does not have the scale resolution requirements of a direct numerical simulation.\n",
    "\n",
    "As a first step towards this goal, we want to see if the process of deriving physical laws or theories can be automated, i.e., can we posit a generic homogeneous PDE, such as\n",
    "$$\n",
    "A \\frac{\\partial u}{\\partial t} = B \\frac{\\partial u}{\\partial x} + C \\frac{\\partial^2 u}{\\partial x^2} ...\n",
    "$$\n",
    "then solve for the coefficients, eliminating the terms for which coefficients are small?\n",
    "\n",
    "For a first example, we analyze the heat or diffusion equation, for which an analytical solution can be obtained.\n",
    "We will use two terms we know apply, and one that doesn't\n",
    "\n",
    "$$\n",
    "A u_t = B u_x + C u_{xx} + D uu_x + E\n",
    "$$\n",
    "\n",
    "with the initial condition\n",
    "$$\n",
    "u(x, 0) = \\sin(4\\pi x)\n",
    "$$\n",
    "\n",
    "There exists an analytical solution\n",
    "$$\n",
    "u(x,t) = \\sin(4\\pi x) e^{-16\\pi^2 \\nu t}\n",
    "$$\n",<commit_msg>Fix typo<commit_after>{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Automatically identifying governing partial differential equations with reverse finite differencing\n",
    "\n",
    "So far, most \"higher level\" physical knowledge has been derived from simpler first principles. However, these techniques have still not given a full understanding of turbulent flow. As such, it is of interest to find a different quantity, for which we can find some conservation or transport equation, that will be simpler to solve, e.g., does not have the scale resolution requirements of a direct numerical simulation.\n",
    "\n",
    "As a first step towards this goal, we want to see if the process of deriving physical laws or theories can be automated, i.e., can we posit a generic homogeneous PDE, such as\n",
    "$$\n",
    "A \\frac{\\partial u}{\\partial t} = B \\frac{\\partial u}{\\partial x} + C \\frac{\\partial^2 u}{\\partial x^2} ...\n",
    "$$\n",
    "then solve for the coefficients, eliminating the terms for which coefficients are small?\n",
    "\n",
    "For a first example, we analyze the heat or diffusion equation, for which an analytical solution can be obtained.\n",
    "We will use two terms we know apply, and one that doesn't\n",
    "\n",
    "$$\n",
    "A u_t = B u_x + C u_{xx} + D uu_x + E\n",
    "$$\n",
    "\n",
    "with the initial condition\n",
    "$$\n",
    "u(x, 0) = \\sin(4\\pi x)\n",
    "$$\n",
    "\n",
    "There exists an analytical solution\n",
    "$$\n",
    "u(x,t) = \\sin(4\\pi x) e^{-16\\pi^2 \\nu t}\n",
    "$$\n",
{'fraction_non_alphanumeric': 0.15284653465346534, 'fraction_numerical': 0.0049504950495049506, 'mean_word_length': 2.904589371980676, 'pattern_counts': {'":': 8, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'dir': 'git-commits-cleaned', 'id': '1402990', 'n_tokens_mistral': 998, 'n_tokens_neox': 973, 'n_words': 430}