got transition recursion working, and created a notebook for the next step: connecting transition to optimality and creating a loss function
Will King
5 years ago
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3e8b8f5284
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "closed-glenn",
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"import torch\n",
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"from torch.autograd.functional import jacobian\n",
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"import itertools"
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]
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},
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{
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"cell_type": "markdown",
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"id": "naval-ivory",
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"metadata": {},
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"source": [
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"# Setup Functions\n",
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"## General CompositionFunctions"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "italian-enforcement",
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"metadata": {},
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"outputs": [],
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"source": [
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"### Set up functions to compose functions \n",
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"# These functions will \n",
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"# - compose two functions together\n",
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"# - compose a function to itself n times.\n",
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"\n",
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"def compose(f,g):\n",
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" \"\"\"\n",
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" this function composes two functions f and g in\n",
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" the order f(g(*args))\n",
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" \n",
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" it returns a function\n",
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" \"\"\"\n",
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" return lambda *args: f(g(*args))\n",
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"\n",
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"def compose_recursive_functions(fn,n):\n",
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" \"\"\"\n",
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" This function takes a function fn and composes it with itself n times\n",
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" \n",
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" Returns an array/list that contains each of the n composition levels, ordered\n",
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" from fn() to fn^n()\n",
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" \"\"\"\n",
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"\n",
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" \n",
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" #Set base conditions\n",
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" out_func = None\n",
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" out_func_list =[]\n",
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"\n",
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" #build the compositions of functions\n",
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" for f in itertools.repeat(fn, n):\n",
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" #if first iteration\n",
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" if out_func == None:\n",
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" out_func = f\n",
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" else:\n",
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" out_func = compose(f,out_func)\n",
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"\n",
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" #append the ou\n",
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" out_func_list.append(out_func)\n",
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" \n",
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" return out_func_list"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fancy-tucson",
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"metadata": {},
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"source": [
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"## Setup functions related to the problem"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "outside-arrangement",
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"metadata": {},
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"outputs": [],
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"source": [
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"### Classes\n",
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"class States():\n",
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" \"\"\"\n",
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" This class represents the state variables associated with the problems.\n",
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" \n",
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" In this problem, the two types of states are constellation stocks and debris.\n",
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" \n",
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" I'm not sure how useful it will be. We'll see.\n",
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" \"\"\"\n",
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" def __init__(self, satellite_stock, debris):\n",
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" self.stock = satellite_stock\n",
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" self.debris = debris\n",
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" \n",
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" \n",
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"\n",
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"### functions\n",
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"\n",
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"def survival(stock, debris):\n",
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" \"\"\"\n",
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" This is a basic, deterministic survival function. \n",
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" It is based on the CDF of an exponential distribution.\n",
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" \"\"\"\n",
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" #TODO: ACTUALLY DERIVE A SURVIVAL FUNCTION. THIS IS JUST A PLACEHOLDER. PROBABLY SHOULD BE AN EXPONENTIAL DISTRIBUTION\n",
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" return 1 - torch.exp(-SCALING * stock - debris)\n",
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"\n",
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"def test_launch(stock, debris):\n",
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" \"\"\"\n",
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" Temporary launch function \n",
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" \"\"\"\n",
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" return torch.ones(5, requires_grad=True)\n",
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"\n",
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"def laws_of_motion(stock, debris, launches):\n",
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" \"\"\"\n",
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" This function updates state variables (stock and debris), according \n",
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" to the laws of motion.\n",
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" \n",
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" It returns the state variables as \n",
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" \"\"\"\n",
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" \n",
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" new_stock = stock*survival(stock,debris) + launches#(stock,debris) #TODO: Launches will become a function (neural network)\n",
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" \n",
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" #TODO: Currently Ignoring autocatalysis\n",
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" new_debris = (1-DELTA)*debris + LAUNCH_DEBRIS_RATE * launches.sum() + COLLISION_DEBRIS_RATE*(1-survival(stock,debris)) @ stock\n",
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" \n",
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" return (new_stock, new_debris)\n",
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"\n",
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"#This is not a good specification of the profit function, but it will work for now.\n",
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"def profit(stock, debris):\n",
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" return UTIL_WEIGHTS @ stock"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "romance-generation",
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"metadata": {},
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"outputs": [],
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"source": [
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"def single_transition(item_to_iterate, laws_motion, profit, launch, stocks, debris ):\n",
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" \"\"\"\n",
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" This function represents the inverted envelope conditions.\n",
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" It allows us to describe the derivatives of the value function evaluated at time $t+1$ in terms based in time period $t$.\n",
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" \n",
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" It takes a few derivatives (jacobians), and due to the nature of the return as tuples\n",
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" they must be concatenated.\n",
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" \n",
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" It returns the transitioned values\n",
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" \"\"\"\n",
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" \n",
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" #Find the matrix to invert. \n",
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" #it consists of the derivative of the laws of motion with respect to stocks and debris\n",
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" \n",
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" #Get the jacobian\n",
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" a = jacobian(laws_motion, (stocks,debris,launch))\n",
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" \n",
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" #Reassemble the Jacobian nested tuples into the appropriate tensor\n",
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" A = BETA * torch.cat((torch.cat((a[0][0],a[0][1]),dim=1),torch.cat((a[1][0],a[1][1]),dim=1)), dim=0)\n",
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" \n",
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" #TODO: figure out some diagnostics for this section\n",
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" #Possibilities include:\n",
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" # - Det(A) ~= 0\n",
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" # - EigVal(A) ~= 0\n",
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" # - A.inverse() with a try catch system to record types of returns\n",
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" #Alternatively, \n",
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" \n",
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" #Calculate the item to transition\n",
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" T = item_to_iterate - torch.cat(jacobian(profit,(stocks, debris))) \n",
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" #Includes rearranging the jacobian of profit.\n",
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"\n",
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" #Return the transitioned values\n",
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" return ( A.inverse()/BETA ) @ T\n",
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"\n",
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"# This function wraps the single transition and handles updating dates etc.\n",
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"def transition_wrapper(data_in):\n",
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" \"\"\"\n",
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" \"\"\"\n",
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" #unpack states and functions\n",
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" stocks, debris,profit, launch, laws_motion, item_to_transition = data_in\n",
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" \n",
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" #Calculate new states\n",
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" new_stocks, new_debris = laws_motion(stocks,debris,launch)\n",
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" \n",
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" #WARNING: RECURSION: You may break your head...\n",
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" #This gets the transition of the value function derivatives over time.\n",
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" transitioned = single_transition(\n",
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" item_to_transition, #item to iterate, i.e. the derivatives of the value function\n",
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" #functions\n",
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" laws_motion, \n",
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" profit, \n",
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" launch, #TODO: reimplement with launch as a function\n",
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" stocks, debris #states\n",
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" )\n",
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" \n",
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" #collects the data back together for return, including the updated state variables\n",
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" data_out = new_stocks, new_debris, profit, launch, laws_motion, transitioned\n",
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" \n",
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" return data_out"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fluid-parks",
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"metadata": {},
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"source": [
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"# Actual calculations"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "changing-january",
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"metadata": {},
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"outputs": [],
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"source": [
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"#set states\n",
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"stocks = torch.ones(5)\n",
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"#Last one is different\n",
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"stocks[-1] = 0.5\n",
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"#now add the tracking requirement in place\n",
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"stocks.requires_grad_()\n",
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"\n",
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"#Setup Debris\n",
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"debris = torch.tensor([2.2],requires_grad=True)\n",
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"\n",
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"#CHANGE LATER: Launch is currently a value, should be a function (i.e. neural network)\n",
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"launch = torch.ones(5, requires_grad=True)\n",
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"\n",
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"#compose the functions together.\n",
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"base_data = (stocks,debris, profit, launch, laws_of_motion, torch.ones(6, requires_grad=True))\n",
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"\n",
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"#Parameters\n",
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"SCALING = torch.ones(5)\n",
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"DELTA = 0.9 \n",
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"LAUNCH_DEBRIS_RATE = 0.005\n",
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"COLLISION_DEBRIS_RATE = 0.0007\n",
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"UTIL_WEIGHTS = torch.tensor([1,-0.2,0,0,0])\n",
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"BETA = 0.95"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "dominant-boost",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"tensor([-0.4523, 0.8774, 0.6558, 0.6558, 0.7608, 11.0954],\n",
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" grad_fn=<MvBackward>) \n",
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"\n",
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"\n",
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"\n",
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"tensor([-25.6591, -23.1244, -23.5468, -23.5468, -29.4675, 123.8316],\n",
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" grad_fn=<MvBackward>) \n",
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"\n",
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"\n",
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"\n",
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"tensor([-236.0462, -232.3070, -232.9302, -232.9302, -312.7054, 1379.8452],\n",
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" grad_fn=<MvBackward>) \n",
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"\n",
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"\n",
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"\n",
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"tensor([-1638.0408, -1632.9258, -1633.7783, -1633.7783, -2380.2834, 15341.8955],\n",
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" grad_fn=<MvBackward>) \n",
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"\n",
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"\n",
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"\n",
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"tensor([ -9855.1680, -9848.4297, -9849.5527, -9849.5527, -15547.5283,\n",
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" 170308.7656], grad_fn=<MvBackward>) \n",
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"\n",
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"\n",
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"\n"
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]
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}
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],
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"source": [
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"#Get the values from 5 transitions\n",
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"for f in compose_recursive_functions(transition_wrapper,5):\n",
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" result = f(base_data)\n",
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" print(result[5], \"\\n\"*3)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "unnecessary-architect",
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"metadata": {},
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"source": [
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"Also, maybe I can create a `Model` class that upon construction will capture the necesary constants, functions, etc.\n",
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"\n",
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"## Next steps\n",
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"- implement 'launch function as a function' portion\n",
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"- substitute the transition functions into the optimality conditions.\n",
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"- use these optimality conditions to create a loss function\n",
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"- add boundary conditions to loss function\n",
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"- get a basic gradient descent/optimization of launch function working.\n",
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"- add satellite deorbit to model.\n",
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"- turn this into a framework in a module, not just a single notebook (long term goal)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "varying-organization",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.9.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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