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13 KiB
13 KiB
import numpy as np import plotly.graph_objects as go from tqdm.notebook import tqdm import plotly.express as px import matplotlib as mpl mpl.rcParams['figure.dpi'] = 300 import matplotlib.pyplot as plt import seaborn as sns import os from wand.image import Image as WImage
sns.set(palette="husl",font_scale=1)
%config InlineBackend.figure_format = 'retina'
import copy np.random.seed(0) %load_ext line_profiler
#Define constants
#L = 2*np.pi # periodic domain size L=20
define boundaries of simulation box
x0 = 0 x1 = L z0 = 0 z1 = L
define reinforcement learning problem
N_states = 12 # number of states - one for each coarse-grained degree of vorticity N_actions = 4 # number of actions - one for each coarse-grained swimming direction
numerical parameters
dt = 0.01 # timestep size
#Utility functions
def moving_average(a, n=3) : ret = np.cumsum(a, dtype=float) ret[n:] = ret[n:] - ret[:-n] return ret[n - 1:] / n
Runga-Kutta 4(5) integration for one step
# see https://stackoverflow.com/questions/54494770/how-to-set-fixed-step-size-with-scipy-integrate
def DoPri45Step(f,t,x,h):
k1 = f(t,x)
k2 = f(t + 1./5*h, x + h*(1./5*k1) )
k3 = f(t + 3./10*h, x + h*(3./40*k1 + 9./40*k2) )
k4 = f(t + 4./5*h, x + h*(44./45*k1 - 56./15*k2 + 32./9*k3) )
k5 = f(t + 8./9*h, x + h*(19372./6561*k1 - 25360./2187*k2 + 64448./6561*k3 - 212./729*k4) )
k6 = f(t + h, x + h*(9017./3168*k1 - 355./33*k2 + 46732./5247*k3 + 49./176*k4 - 5103./18656*k5) )
v5 = 35./384*k1 + 500./1113*k3 + 125./192*k4 - 2187./6784*k5 + 11./84*k6
k7 = f(t + h, x + h*v5)
v4 = 5179./57600*k1 + 7571./16695*k3 + 393./640*k4 - 92097./339200*k5 + 187./2100*k6 + 1./40*k7;
return v4,v5
#Define useful data structures #Define a dictionary of the possible states and their assigned indices
direction_states = ["right","down","left","up"] # coarse-grained directions vort_states = ["w+", "w0", "w-"] # coarse-grained levels of vorticity product_states = [(x,y) for x in direction_states for y in vort_states] # all possible states state_lookup_table = {product_states[i]:i for i in range(len(product_states))} # returns index of given state
print(product_states) # to view mapping
#Define an agent class for reinforcement learning
class Agent: def init(self, Ns): self.r = np.zeros(Ns) # reward for each stage self.t = 0 # time
# calculate reward given from entering a new state after a selected action is undertaken
def calc_reward(self):
# enforce implementation by subclass
if self.__class__ == AbstractClass:
raise NotImplementedError
def update_state(self):
# enforce implementation by subclass
if self.__class__ == AbstractClass:
raise NotImplementedError
def take_random_action(self):
# enforce implementation by subclass
if self.__class__ == AbstractClass:
raise NotImplementedError
def take_greedy_action(self, Q):
# enforce implementation by subclass
if self.__class__ == AbstractClass:
raise NotImplementedError
#Define swimmer class derived from agent
class Swimmer(Agent): def init(self, Ns): # call init for superclass super().init(Ns)
# local position within the periodic box. X = [x, z]^T with 0 <= x < 2 pi and 0 <= z < 2 pi
self.X = np.array([np.random.uniform(0, L), np.random.uniform(0, L), 0])
# absolute position. -inf. <= x_total < inf. and -inf. <= z_total < inf.
self.X_total = self.X
# particle orientation
self.theta = np.random.uniform(0, 2*np.pi) # polar angle theta in the x-z plane
self.p = np.array([np.cos(self.theta), np.sin(self.theta)]) # p = [px, pz]^T
# translational and rotational velocity
self.U = np.zeros(3)
self.W = np.array([0, 0, 1]) #Velocidad angular aleatoria
# preferred swimming direction (equal to [1,0], [0,1], [-1,0], or [0,-1])
self.ka = np.array([0,1])
# history of local and global position. Only store information for this episode.
self.history_X = [self.X]
self.history_X_total = [self.X_total]
# local vorticity at the current location
_, _, self.w = tgv(self.X[0], self.X[1])
# update coarse-grained state
self.update_state()
#obstáculos
self.obstacles= self.generate_obstacles()
def generate_obstacles(self):
obstacles=[] #el numero de obstáculos será 10*10
cell_spacing= L/30
for i in range(20):
for j in range(20):
obstacle_x= i + cell_spacing
obstacle_z= j + cell_spacing
obstacles.append(obstacle_x)
obstacles.append(obstacle_z)
return obstacles
def interaction_with_obstacles(self,obstacles, sigma,ni,kappa,alpha,beta,gamma,Pe,dt):
F= np.array([0.,0.,0.])
for i in range(len(obstacles)//2):
#F1
obstacle_position = np.array([obstacles[2*i],obstacles[2*i+1], 0])
r=self.X - obstacle_position
r_norm=np.linalg.norm(r)
Re= sigma**2*np.linalg.norm(self.W)/ni
S=1/(1+np.exp(-kappa*((Re/r_norm**3)-Re)))
F1=alpha*(Re/r**3)*np.cross(self.U,self.W)*S
#F2
F2=beta*np.cross(self.W,r)/r_norm**3
#F de atracción
F_attr= gamma*(np.exp(-r_norm/kappa)/r_norm**2)*(kappa+r_norm)*r
#Fuerza total
F+=F1+F2+F_attr
xi=np.random.normal(0,1, size=2) #vector de números aleatorios generados a partir de una distribución normal estándar con dos componentes, xi creo que es un vector de ruido estocástico (modela el ruido térmico)
dr_therm = np.sqrt(2*sigma**2*dt/Pe)*xi
dr = F[:-1]*dt + dr_therm
#actualizamos la posición del spinner
self.X[:-1] += dr
#comprobamos que el spinner siga dentro del box periódico
self.check_in_box()
def reinitialize(self):
self.X = np.array([np.random.uniform(0, L), np.random.uniform(0, L)])
self.X_total = self.X
self.theta = np.random.uniform(0, 2*np.pi) # polar angle theta in the x-z plane
self.p = np.array([np.cos(self.theta), np.sin(self.theta)]) # p = [px, pz]^T # orientación del nadador
self.U = np.zeros(2)
self.W = np.zeros(2)
self.ka = np.array([0,1])
self.history_X = [self.X]
self.history_X_total = [self.X_total]
self.t = 0
def update_kinematics(self, Φ, Ψ, D0 = 0, Dr = 0, int_method = "euler"): # Actualiza la posición y orientación del nadador según un método de integración especificado.
if int_method == "rk45":
y0 = np.concatenate((self.X,self.p))
_, v5 = DoPri45Step(self.calc_velocity_rk45,self.t,y0,dt)
y = y0 + dt*v5
self.X = y[:2]
self.p = y[2:]
dx = self.X - self.history_X[-1]
self.X_total = self.X_total + dx
# check if still in the periodic box
self.check_in_box()
# ensure the vector p has unit length
self.p /= (self.p[0]**2 + self.p[1]**2)**(1/2)
# update polar angle
x = self.p[0]
yy = self.p[1]
self.theta = np.arctan2(yy,x) if yy >= 0 else (np.arctan2(yy,x) + 2*np.pi)
# store positions
self.history_X.append(self.X)
self.history_X_total.append(self.X_total)
elif int_method == "euler":
# calculate new translational and rotational velocity
self.calc_velocity(Φ, Ψ)
self.update_position(int_method, D0)
self.update_orientation(int_method, Dr)
else:
raise Exception("Integration method must be 'Euler' or 'rk45'")
self.t = self.t + dt
def calc_velocity_rk45(self, t, y): #calcula la velocidad del nadador en un determinado tiempo 't' y estado 'y' utilizando Rk45
x = y[0]
z = y[1]
px = y[2]
pz = y[3]
ux, uz, self.w = tgv(x, z) #tgv proporciona velocidades de flujo en la posición (x,z), w es la vorticidad
#cálculo de las velocidades translacionales
U0 = ux + Φ*px #ux y uz son las velocidades del flujo
U1 = uz + Φ*pz
#cálculo de las velocidades rotacionales
ka_dot_p = self.ka[0]*px + self.ka[1]*pz #alineación del vector de nado preferido con la dirección del nadador
W0 = 1/2/Ψ*(self.ka[0] - ka_dot_p*px) + 1/2*pz*self.w
W1 = 1/2/Ψ*(self.ka[1] - ka_dot_p*pz) + 1/2*-px*self.w
return np.array([U0, U1, W0, W1])
def update_position(self, int_method, D0): #D0 representa la difusión
# use explicit euler to update
dx = dt*self.U
if D0 > 0: dx = dx + np.sqrt(2*D0*dt)*np.random.normal(size=2) #posible efecto de la difusión browniana
self.X = self.X + dx
self.X_total = self.X_total + dx
# check if still in the periodic box
self.check_in_box()
# store positions
self.history_X.append(self.X)
self.history_X_total.append(self.X_total)
def update_orientation(self, int_method, Dr):
self.p = self.p + dt*self.W #W velocidad angular
# ensure the vector p has unit length
self.p /= (self.p[0]**2 + self.p[1]**2)**(1/2)
# if rotational diffusion is present
if Dr > 0: #Dr representa difucion rotacional
px = self.p[0]
pz = self.p[1]
cross = px*pz
A = np.array([[1-px**2, -cross], [-cross, 1-pz**2]]) #A es una matriz
v = np.sqrt(2*Dr*dt)*np.random.normal(size=2) #v es un vector de valores aleatorios
self.p[0] = self.p[0] + A[0,0]*v[0] + A[0,1]*v[1] #Se calcula un cambio aleatorio en la orientación usando A y v
self.p[1] = self.p[1] + A[1,0]*v[0] + A[1,1]*v[1]
self.p /= (self.p[0]**2 + self.p[1]**2)**(1/2)
# update polar angle
x = self.p[0]
y = self.p[1]
self.theta = np.arctan2(y,x) if y >= 0 else (np.arctan2(y,x) + 2*np.pi)
def calc_velocity(self, Φ, Ψ):
ux, uz, self.w = tgv(self.X[0], self.X[1])
# careful - computing in the following way is significantly slower: self.U = np.array(ux, uz) + Φ*self.p
self.U[0] = ux + Φ*self.p[0]
self.U[1] = uz + Φ*self.p[1]
px = self.p[0]
pz = self.p[1]
ka_dot_p = self.ka[0]*px + self.ka[1]*pz
self.W[0] = 1/2/Ψ*(self.ka[0] - ka_dot_p*px) + 1/2*pz*self.w
self.W[1] = 1/2/Ψ*(self.ka[1] - ka_dot_p*pz) + 1/2*-px*self.w
def check_in_box(self): # Este método verifica si el nadador todavía está dentro del cuadro periódico
if self.X[0] < x0:
self.X[0] += L
elif self.X[0] > x1:
self.X[0] -= L
if self.X[1] < z0:
self.X[1] += L
elif self.X[1] > z1:
self.X[1] -= L
def calc_reward(self, n):
self.r[n] = self.history_X_total[-1][1]-self.history_X_total[-2][1]
def update_state(self):
if self.w < -0.33:
w_state = "w-"
elif self.w >= -0.33 and self.w <= 0.33:
w_state = "w0"
elif self.w > 0.33:
w_state = "w+"
else:
raise Exception("Invalid value of w detected: ", w)
if self.theta >= np.pi/4 and self.theta < 3*np.pi/4:
p_state = "up"
elif self.theta >= 3*np.pi/4 and self.theta < 5*np.pi/4:
p_state = "left"
elif self.theta >= 5*np.pi/4 and self.theta < 7*np.pi/4:
p_state = "down"
elif (self.theta >= 7*np.pi/4 and self.theta <= 2*np.pi) or (self.theta >= 0 and self.theta < np.pi/4):
p_state = "right"
else:
raise Exception("Invalid value of theta detected: ", theta)
self.my_state = (p_state, w_state)
def take_greedy_action(self, Q):
state_index = state_lookup_table[self.my_state]
action_index = np.argmax(Q[state_index]) # find largest entry in this row of Q (i.e. this state)
if action_index == 0: # up
self.ka = [0, 1]
elif action_index == 1: # down
self.ka = [0, -1]
elif action_index == 2: # right
self.ka = [1, 0]
else: # left
self.ka = [-1, 0]
return action_index
def take_random_action(self):
action_index = np.random.randint(4)
if action_index == 0: # up
self.ka = [0, 1]
elif action_index == 1: # down
self.ka = [0, -1]
elif action_index == 2: # right
self.ka = [1, 0]
else: # left
self.ka = [-1, 0]
return action_index
#Define Taylor-Green vortex
given position, return local velocity and vorticity
def tgv(x, z): ux = -1/2*np.cos(x)np.sin(z) uz = 1/2np.sin(x)*np.cos(z) w = -np.cos(x)*np.cos(z) return ux, uz, w
#Plot
Ns = 5000 spinner = Swimmer(Ns) traj = [] obstacles = spinner.generate_obstacles() for i in range(Ns): spinner.interaction_with_obstacles(obstacles, 1, 1e-6, 2.5, 1, 1., 2.5e-4, 10000, 0.001) traj.append(spinner.X[0]) traj.append(spinner.X[1])
fig, ax= plt.subplots(1,1) ax.plot(traj[::2], traj[1::2]) ax.plot(obstacles[::2], obstacles[1::2], '.') print(obstacles[::2]) plt.show()