MOEA
Contents
MOEA#
!wget https://raw.githubusercontent.com/cfteach/modules/master/detector2.py
!pip install pymoo
!pip install plotly
!pip install ipyvolume
!pip install altair
--2024-04-18 17:56:25-- https://raw.githubusercontent.com/cfteach/modules/master/detector2.py
Resolving raw.githubusercontent.com (raw.githubusercontent.com)... 185.199.108.133, 185.199.109.133, 185.199.110.133, ...
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Length: 7976 (7.8K) [text/plain]
Saving to: ‘detector2.py’
detector2.py 100%[===================>] 7.79K --.-KB/s in 0s
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Collecting pymoo
Downloading pymoo-0.6.1.1-cp310-cp310-manylinux_2_17_x86_64.manylinux2014_x86_64.whl (4.1 MB)
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Collecting cma==3.2.2 (from pymoo)
Downloading cma-3.2.2-py2.py3-none-any.whl (249 kB)
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Downloading about_time-4.2.1-py3-none-any.whl (13 kB)
Collecting grapheme==0.6.0 (from alive-progress->pymoo)
Downloading grapheme-0.6.0.tar.gz (207 kB)
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Building wheels for collected packages: grapheme
Building wheel for grapheme (setup.py) ... ?25l?25hdone
Created wheel for grapheme: filename=grapheme-0.6.0-py3-none-any.whl size=210078 sha256=d82ed9986cf25fff8c28a851ae8cc2a25a3e07f88d7a986eb128717c5037066c
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Successfully built grapheme
Installing collected packages: grapheme, dill, Deprecated, cma, about-time, alive-progress, pymoo
Successfully installed Deprecated-1.2.14 about-time-4.2.1 alive-progress-3.1.5 cma-3.2.2 dill-0.3.8 grapheme-0.6.0 pymoo-0.6.1.1
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%load_ext autoreload
%autoreload 2
import ipyvolume as ipv
import ipywidgets as widgets
from IPython.display import display, Math, Latex
import os
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
#import AI4NP_detector_opt.sol2.detector2 as detector2
import detector2
import re
import pickle
import dill
from pymoo.algorithms.moo.nsga2 import NSGA2
from pymoo.core.problem import Problem
from pymoo.optimize import minimize
from pymoo.visualization.scatter import Scatter
from pymoo.util.display.column import Column
from pymoo.util.display.output import Output
from pymoo.util.nds.non_dominated_sorting import NonDominatedSorting
Create detector geometry and simulate tracks#
The module detector creates a simple 2D geometry of a wire based tracker made by 4 planes.
The adjustable parameters are the radius of each wire, the pitch (along the y axis), and the shift along y and z of a plane with respect to the previous one.
A total of 8 parameters can be tuned.
The goal of this toy model, is to tune the detector design so to optimize the efficiency (fraction of tracks which are detected) as well as the cost for its realization. As a proxy for the cost, we use the material/volume (the surface in 2D) of the detector. For a track to be detetected, in the efficiency definition we require at least two wires hit by the track.
So we want to maximize the efficiency (defined in detector.py) and minimize the cost.
LIST OF PARAMETERS#
(baseline values)
R = .5 [cm]
pitch = 4.0 [cm]
y1 = 0.0, y2 = 0.0, y3 = 0.0, z1 = 2.0, z2 = 4.0, z3 = 6.0 [cm]
# CONSTANT PARAMETERS
#------ define mother region ------#
y_min=-10.1
y_max=10.1
N_tracks = 1000
print("::::: BASELINE PARAMETERS :::::")
R = 0.5
pitch = 4.0
y1 = 2.0
y2 = 1.0
y3 = 0.0
z1 = 2.0
z2 = 4.0
z3 = 6.0
print("R, pitch, y1, y2, y3, z1, z2, z3: ", R, pitch, y1, y2, y3, z1, z2, z3,"\n")
#------------- GEOMETRY ---------------#
print(":::: INITIAL GEOMETRY ::::")
tr = detector2.Tracker(R, pitch, y1, y2, y3, z1, z2, z3)
Z, Y = tr.create_geometry()
num_wires = detector2.calculate_wires(Y, y_min, y_max)
volume = detector2.wires_volume(Y, y_min, y_max,R)
detector2.geometry_display(Z, Y, R, y_min=y_min, y_max=y_max,block=False,pause=5) #5
print("# of wires: ", num_wires, ", volume: ", volume)
#------------- TRACK GENERATION -----------#
print(":::: TRACK GENERATION ::::")
t = detector2.Tracks(b_min=y_min, b_max=y_max, alpha_mean=0, alpha_std=0.2)
tracks = t.generate(N_tracks)
detector2.geometry_display(Z, Y, R, y_min=y_min, y_max=y_max,block=False, pause=-1)
detector2.tracks_display(tracks, Z,block=False,pause=-1)
#a track is detected if at least two wires have been hit
score = detector2.get_score(Z, Y, tracks, R)
frac_detected = score[0]
resolution = score[1]
print("fraction of tracks detected: ",frac_detected)
print("resolution: ",resolution)
::::: BASELINE PARAMETERS :::::
R, pitch, y1, y2, y3, z1, z2, z3: 0.5 4.0 2.0 1.0 0.0 2.0 4.0 6.0
:::: INITIAL GEOMETRY ::::
# of wires: 21 , volume: 65.94
:::: TRACK GENERATION ::::
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fraction of tracks detected: 0.243
resolution: 0.23594329069824552
Define Objectives#
Defines a class for the objectives of the problem that can be used in the MOO.
class objectives():
def __init__(self,tracks,y_min,y_max):
self.tracks = tracks
self.y_min = y_min
self.y_max = y_max
def wrapper_geometry(fun):
def inner(self):
R, pitch, y1, y2, y3, z1, z2, z3 = self.X
self.geometry(R, pitch, y1, y2, y3, z1, z2, z3)
return fun(self)
return inner
def update_tracks(self, new_tracks):
self.tracks = new_tracks
def update_design_point(self,X):
self.X = X
def geometry(self,R, pitch, y1, y2, y3, z1, z2, z3):
tr = detector2.Tracker(R, pitch, y1, y2, y3, z1, z2, z3)
self.R = R
self.Z, self.Y = tr.create_geometry()
@wrapper_geometry
def calc_score(self):
res = detector2.get_score(self.Z, self.Y, self.tracks, self.R)
assert res[0] >= 0 and res[1] >= 0,"Fraction or Resolution negative."
return res
def get_score(self,X):
R, pitch, y1, y2, y3, z1, z2, z3 = X
self.geometry(R, pitch, y1, y2, y3, z1, z2, z3)
res = detector2.get_score(self.Z, self.Y, self.tracks, self.R)
return res
def get_volume(self):
volume = detector2.wires_volume(self.Y, self.y_min, self.y_max,self.R)
return volume
res = objectives(tracks,y_min,y_max)
#res.geometry(R, pitch, y1, y2, y3, z1, z2, z3)
X = R, pitch, y1, y2, y3, z1, z2, z3
#fscore = res.get_score(X)
res.update_design_point(X)
fscore = res.calc_score()[0]
fvolume = res.get_volume()
print("...check: ", fvolume, fscore)
...check: 65.94 0.243
Multi-Objective Optimization#
We will be using pymoo (https://pymoo.org/getting_started.html).
In the constructor method we specify number of variables N, objectives M, and constraint functions, as well as the lower and upper boundaries of each variable. In our toy model, these boundaries are taken in such a way that all solutions are feasible and no constraint function is needed. You can try to change this and introduce some constraint.
The _evaluate method takes a one-dimensional NumPy array x number of entries equal to n_var. This behavior is enabled by setting elementwise_evaluation=True while calling the super() method.
Notice that every function is minimized. Our efficiency is defined as an tracking inefficiency = 1 - efficiency
We add the resolution as a third objective. The average residual of the track hit from the wire centre is used as a proxy for the resolution for this toy-model
from pymoo.core.problem import ElementwiseProblem
class MyProblem(ElementwiseProblem):
#--------- vectorized ---------#
def __init__(self):
super().__init__(n_var=8,
n_obj=3, #<------------
n_constr=0,
xl=np.array([0.5,2.5,0.,0.,0.,2.,2.,2.]),
xu=np.array([1.0,5.0,4.,4.,4.,10.,10.,10.]))
def _evaluate(self, x, out, *args, **kwargs):
f1 = 1.-res.get_score(x)[0] # efficiency
f2 = res.get_volume() # volume
f3 = res.get_score(x)[1] #resolution
out["F"] = [f1, f2, f3]
Creation of Problem and choice of optimization algorithm.#
We will use NSGA-II, as explained in the lectures. You can decide the population size and the number of offsprings, based on what we discussed.
Pymoo offers different algorithms that can be used which are highly customizable and can be easily extended. https://pymoo.org/algorithms/index.html
Before dealing with a problem, it’s useful to compare with a list of test problems reported in https://pymoo.org/problems/index.html, where different scenarios in terms of Variables, Objectives, Constraints are described.
problem = MyProblem()
algorithm = NSGA2(pop_size=100,n_offsprings=20) #n_offsprings=10
res = minimize(problem,
algorithm,
("n_gen", 50),
verbose=True,
seed=1,
save_history=True)
==========================================================
n_gen | n_eval | n_nds | eps | indicator
==========================================================
1 | 100 | 18 | - | -
2 | 120 | 18 | 0.0317658940 | f
3 | 140 | 19 | 0.0086681196 | f
4 | 160 | 25 | 0.0060704585 | ideal
5 | 180 | 26 | 0.0073991461 | f
6 | 200 | 31 | 0.0098656270 | f
7 | 220 | 36 | 0.0029239766 | ideal
8 | 240 | 42 | 0.1000000000 | ideal
9 | 260 | 45 | 0.0068599632 | f
10 | 280 | 48 | 0.0037308676 | f
11 | 300 | 49 | 0.0061795697 | f
12 | 320 | 55 | 0.0054861726 | f
13 | 340 | 50 | 0.0124104215 | ideal
14 | 360 | 54 | 0.0093864031 | f
15 | 380 | 59 | 0.0281329923 | nadir
16 | 400 | 62 | 0.0024959459 | f
17 | 420 | 64 | 0.0156984746 | ideal
18 | 440 | 69 | 0.0952689460 | nadir
19 | 460 | 72 | 0.0027375378 | f
20 | 480 | 72 | 0.0113780025 | nadir
21 | 500 | 74 | 0.0025348542 | nadir
22 | 520 | 75 | 0.0171987532 | ideal
23 | 540 | 77 | 0.0019749087 | f
24 | 560 | 79 | 0.0029671449 | f
25 | 580 | 73 | 0.0027486134 | f
26 | 600 | 73 | 0.0198757764 | ideal
27 | 620 | 74 | 0.0413971539 | nadir
28 | 640 | 77 | 0.0383106751 | nadir
29 | 660 | 80 | 0.0021919528 | f
30 | 680 | 79 | 0.0041187368 | f
31 | 700 | 80 | 0.0026270497 | f
32 | 720 | 84 | 0.0025096075 | f
33 | 740 | 83 | 0.0021244723 | f
34 | 760 | 82 | 0.0295086495 | ideal
35 | 780 | 85 | 0.0010025574 | f
36 | 800 | 89 | 0.0023646349 | f
37 | 820 | 91 | 0.0040466056 | f
38 | 840 | 92 | 0.0013260417 | f
39 | 860 | 90 | 0.0027962201 | f
40 | 880 | 90 | 0.0238410596 | nadir
41 | 900 | 92 | 0.0147031664 | nadir
42 | 920 | 93 | 0.0017278338 | f
43 | 940 | 94 | 0.0029719466 | f
44 | 960 | 89 | 0.0718459407 | nadir
45 | 980 | 91 | 0.0065789474 | ideal
46 | 1000 | 92 | 0.0009939480 | f
47 | 1020 | 89 | 0.0029172000 | f
48 | 1040 | 93 | 0.0120082842 | ideal
49 | 1060 | 95 | 0.0218790219 | nadir
50 | 1080 | 99 | 0.0164556962 | ideal
Analysis of Results#
import plotly.express as px
fig = px.scatter_3d(x = res.F[:, 0], y = res.F[:, 1], z = res.F[:, 2], labels={
"x": "InEfficiency",
"y": "Volume",
"z": "Resolution"
},width = 800, height = 800, title = "Final Call feasible solutions")
fig.update_traces(marker=dict(size=8,
line=dict(width=2,
color='DarkSlateGrey')),
selector=dict(mode='markers'))
fig.show()
# Making a animation of evolution
import pandas as pd
obj1 = []
obj2 = []
obj3 = []
calls = []
for r in res.history:
objs = r.pop.get("F")
obj1.extend(objs[:, 0])
obj2.extend(objs[:, 1])
obj3.extend(objs[:, 2])
calls.extend([r.n_gen]*len(objs))
df = pd.DataFrame(data = {"InEfficiency": obj1, "Volume": obj2,
"Resolution": obj3, "n_gen": calls})
obj_fig = px.scatter_3d(df, x="InEfficiency", y="Volume", z = "Resolution",
animation_frame="n_gen", color="n_gen",
range_x=[0., 0.6], range_y=[0. , 400.], range_z=[0., 0.6],
hover_data = df.columns,
width = 800, height = 800)
obj_fig.update(layout_coloraxis_showscale=False)
obj_fig.layout.updatemenus[0].buttons[0].args[1]["frame"]["duration"] = 10
obj_fig.update_layout(transition = {'duration': 0.001})
obj_fig.show()
import matplotlib.pyplot as plt
fig = plt.figure(figsize = (12, 10))
ax = fig.add_subplot(projection='3d')
ax.scatter(res.F[:, 0], res.F[:, 1], res.F[:, 2], marker = "o", s = 55)
ax.set_xlabel('Ineff', fontsize = 15)
ax.set_ylabel('Vol', fontsize = 15)
ax.set_zlabel('Res', fontsize = 15)
plt.show()
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Exercise 3#
Determine the Pareto set from the 3D front and choose an optimal point
Plot the optimal configuration of the tracker corresponding to that point
Do analysis of convergence
Visualize the point with a radar or petal diagram, following https://pymoo.org/visualization/index.html