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# | ||
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``` bash | ||
python3 -m venv venv | ||
source venv/bin/active | ||
python3 -m pip install numpy matplotlib jupyterlab ipykernel 'jax[cpu]' | ||
``` |
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{ | ||
"cells": [], | ||
"metadata": {}, | ||
"nbformat": 4, | ||
"nbformat_minor": 5 | ||
} |
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import numpy as np | ||
import matplotlib.pyplot as plt | ||
import sys | ||
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def pendulum_ideal(x): | ||
g = 1.0 # gravity | ||
l = 1.0 # pendulum arm length | ||
θ, ω = x | ||
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dθdt = ω | ||
dωdt = -(g / l) * np.sin(θ) | ||
return np.array((dθdt, dωdt)) | ||
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def pendulum_friction(x): | ||
g = 1.0 # gravity | ||
l = 1.0 # pendulum arm length | ||
γ = 1.0 # friction co-efficient | ||
θ, ω = x | ||
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dθdt = ω | ||
dωdt = -(g / l) * np.sin(θ) - γ * ω | ||
return dθdt, dωdt | ||
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def lotka_volterra(x): | ||
x, y = x | ||
a = 1.0 | ||
b = 1.0 | ||
c = 1.0 | ||
d = 1.0 | ||
dxdt = a * x - b * x * y | ||
dydt = c * x * y - d * y | ||
return dxdt, dydt | ||
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def run_lotka_volterra(): | ||
mesh_step = 0.3 | ||
x, y = np.meshgrid( | ||
np.arange(0.0, 5.0 + mesh_step, mesh_step), | ||
np.arange(0.0, 5.0 + mesh_step, mesh_step), | ||
) | ||
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dxdt, dydt = lotka_volterra((x, y)) | ||
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fig, ax = plt.subplots() | ||
# ax.quiver(x, y, dxdt, dydt, angles="xy", scale_units="xy", scale=8) | ||
ax.streamplot(x, y, dxdt, dydt) | ||
ax.set_xlabel("x") | ||
ax.set_ylabel("y") | ||
ax.set_title("Vector Field: Lotka-Volterra") | ||
plt.savefig("figures/simulation/lotka_volterra_vector_field.pdf") | ||
plt.show() | ||
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def run_pendulum_solver(): | ||
step_size_reference = 0.00001 | ||
step_size = 0.1 | ||
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t_start = 0.0 | ||
t_end = 4 * np.pi | ||
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# simulate reference | ||
t_reference = np.arange(t_start, t_end + step_size_reference, step_size_reference) | ||
x_cur = np.array((np.pi / 4, 00)) # initial state | ||
xs_reference = [x_cur] | ||
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for _ in t_reference[1:]: | ||
x_new = x_cur + step_size_reference * pendulum_ideal(x_cur) | ||
xs_reference.append(x_new) | ||
x_cur = x_new | ||
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xs_reference = np.stack(xs_reference, axis=1) # (n_states, n_steps) | ||
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# simulate Euler | ||
t = np.arange(t_start, t_end + step_size, step_size) | ||
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x_cur = np.array((np.pi / 4, 00)) # initial state | ||
xs_euler = [x_cur] | ||
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for _ in t[1:]: | ||
x_new = x_cur + step_size * pendulum_ideal(x_cur) | ||
xs_euler.append(x_new) | ||
x_cur = x_new | ||
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xs_euler = np.stack(xs_euler, axis=1) # (n_states, n_steps) | ||
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# simulate midpoint | ||
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x_cur = np.array((np.pi / 4, 00)) # initial state | ||
xs_midpoint = [x_cur] | ||
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for _ in t[1:]: | ||
x_new = x_cur + step_size * pendulum_ideal( | ||
x_cur + step_size / 2 * pendulum_ideal(x_cur) | ||
) | ||
xs_midpoint.append(x_new) | ||
x_cur = x_new | ||
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xs_midpoint = np.stack(xs_midpoint, axis=1) # (n_states, n_steps) | ||
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# plotting | ||
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fig, (ax1, ax2) = plt.subplots(2, sharex=True) | ||
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ax1.plot( | ||
t_reference, | ||
xs_reference[0], | ||
label=f"FE (step-size = {step_size_reference})", | ||
c="black", | ||
linewidth=2, | ||
) | ||
ax1.plot( | ||
t, | ||
xs_euler[0], | ||
label=f"FE (step-size = {step_size})", | ||
c="blue", | ||
linestyle="dotted", | ||
linewidth=2, | ||
) | ||
ax1.plot( | ||
t, | ||
xs_midpoint[0], | ||
label=f"MID (step-size = {step_size})", | ||
c="red", | ||
linestyle="dotted", | ||
linewidth=2, | ||
) | ||
ax1.set_ylabel(rf"$\theta$ [rad]") | ||
ax2.plot(t_reference, xs_reference[1], label="reference", c="black", linewidth=2) | ||
ax2.plot( | ||
t, | ||
xs_euler[1], | ||
c="blue", | ||
linestyle="dotted", | ||
linewidth=2, | ||
) | ||
ax2.plot( | ||
t, | ||
xs_midpoint[1], | ||
c="red", | ||
linestyle="dotted", | ||
linewidth=2, | ||
) | ||
ax2.set_ylabel(rf"$\omega$ [rad/s]") | ||
ax2.set_xlabel("t [s]") | ||
ax1.legend() | ||
plt.tight_layout() | ||
plt.show() | ||
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def run_pendulum_phase(): | ||
mesh_step = 0.3 | ||
θ, ω = np.meshgrid( | ||
np.arange(-np.pi, np.pi + mesh_step, mesh_step), | ||
np.arange(-np.pi, np.pi + mesh_step, mesh_step), | ||
) | ||
θ_wrapped, ω_wrapped = np.meshgrid( | ||
np.arange(-2 * np.pi, 2 * np.pi + mesh_step, mesh_step), | ||
np.arange(-2 * np.pi, 2 * np.pi + mesh_step, mesh_step), | ||
) | ||
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dθdt_ideal, dωdt_ideal = pendulum_ideal((θ, ω)) | ||
dθdt_ideal_wrapped, dωdt_ideal_wrapped = pendulum_ideal((θ_wrapped, ω_wrapped)) | ||
dθdt_friction, dωdt_friction = pendulum_friction((θ, ω)) | ||
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fig, ax = plt.subplots() | ||
# ax.quiver(θ, ω, dθdt_ideal, dωdt_ideal, angles="xy", scale_units="xy", scale=5) | ||
ax.streamplot(θ, ω, dθdt_ideal, dωdt_ideal) | ||
ax.set_xlabel(rf"$\theta$") | ||
ax.set_ylabel(rf"$\omega$") | ||
ax.set_title("Vector Field: Ideal Pendulum") | ||
plt.savefig("figures/simulation/pendulum_ideal_vector_field.pdf") | ||
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fig, ax = plt.subplots() | ||
ax.quiver( | ||
θ_wrapped, | ||
ω_wrapped, | ||
dθdt_ideal_wrapped, | ||
dωdt_ideal_wrapped, | ||
angles="xy", | ||
scale_units="xy", | ||
scale=10, | ||
) | ||
ax.set_xlim(-2 * np.pi, 2 * np.pi) | ||
ax.set_ylim(-np.pi, np.pi) | ||
ax.set_xlabel(rf"$\theta$") | ||
ax.set_ylabel(rf"$\omega$") | ||
ax.set_title("Vector Field: Ideal Pendulum") | ||
plt.savefig("figures/simulation/pendulum_ideal_vector_field_wrapped.pdf") | ||
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fig, ax = plt.subplots() | ||
# ax.quiver( | ||
# θ, ω, dθdt_friction, dωdt_friction, angles="xy", scale_units="xy", scale=5 | ||
# ) | ||
ax.streamplot(θ, ω, dθdt_friction, dωdt_friction) | ||
ax.set_xlabel(rf"$\theta$") | ||
ax.set_ylabel(rf"$\omega$") | ||
ax.set_title("Vector Field: Pendulum with friction") | ||
plt.savefig("figures/simulation/pendulum_friction_vector_field.pdf") | ||
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plt.show() | ||
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def simulate_with_stepsize(f, x0, t_start, t_stop, step_size): | ||
t = np.arange(t_start, t_stop + step_size, step_size) | ||
x_euler = [x0] | ||
x_midpoint = [x0] | ||
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for _ in t[1:]: | ||
x_euler.append(x_euler[-1] + step_size * f(x_euler[-1])) | ||
x_midpoint.append( | ||
x_midpoint[-1] | ||
+ step_size * f(x_midpoint[-1] + step_size / 2 * x_midpoint[-1]) | ||
) | ||
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x_exact = np.array([np.e**t for t in t]) | ||
x_euler = np.stack(x_euler) | ||
x_midpoint = np.stack(x_midpoint) | ||
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return t, x_exact, x_euler, x_midpoint | ||
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def run_convergence(): | ||
def f(x): | ||
return x | ||
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step_size_min = 2**-25 | ||
step_size_max = 1.0 | ||
n_steps_sizes_in_sweep = 5 | ||
step_size_increment = (step_size_max - step_size_min) / n_steps_sizes_in_sweep | ||
t_start = 0.0 | ||
t_end = 0.1 | ||
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def f(x): | ||
return x | ||
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x0 = np.array((1.0)) | ||
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step_sizes = np.arange( | ||
step_size_min, step_size_max + step_size_increment, step_size_increment | ||
) | ||
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errors = {} | ||
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for h in step_sizes: | ||
t_cur = t_start | ||
x_cur = x0 | ||
while t_cur < t_end: | ||
x_cur = x_cur + h * f(x_cur) | ||
t_cur += h | ||
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# assert t_cur == t_end, "simulation did not stop at the desired end time" | ||
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x_true = np.e**t_cur | ||
errors[h] = abs(x_cur - x_true) | ||
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fig, ax = plt.subplots() | ||
ax.loglog(list(errors), list(errors.values()), label="euler") | ||
# ax.loglog(np.array(list(errors)), errors_midpoint, label="midpoint") | ||
ax.set_ylabel("Mean-squared error") | ||
ax.set_xlabel("step-size") | ||
plt.legend() | ||
plt.show() | ||
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def run_integrator(): | ||
step_size = 0.001 | ||
step_size_coarse = 0.3 | ||
t_start = 0.0 | ||
t_end = 10.0 | ||
t = np.arange(t_start, t_end + step_size, step_size) | ||
t_coarse = np.arange(t_start, t_end + step_size_coarse, step_size_coarse) | ||
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# fine | ||
x_cur = np.array((0.5, 0.0)) | ||
x = [x_cur] | ||
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for _ in t[1:]: | ||
x_new = x_cur + step_size * pendulum_ideal(x_cur) | ||
x.append(x_new) | ||
x_cur = x_new | ||
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x = np.stack(x, axis=1) # (n_states, n_steps) | ||
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# coarse | ||
x_cur = np.array((0.5, 0.0)) | ||
x_coarse = [x_cur] | ||
x_straight = [] | ||
n_steps_per_coarse = step_size_coarse // step_size | ||
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for _ in t_coarse[1:]: | ||
x_new = x_cur + step_size_coarse * pendulum_ideal(x_cur) | ||
x_coarse.append(x_new) | ||
x_cur = x_new | ||
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x_coarse = np.stack(x_coarse, axis=1) # (n_states, n_steps_coarse) | ||
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# plotting | ||
fig, (ax1, ax2) = plt.subplots(2, sharex=True) | ||
ax1.plot(t, x[0]) | ||
ax1.scatter(t_coarse, x_coarse[0], marker="x", c="red") | ||
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ax2.plot(t, x[1]) | ||
ax1.set_ylabel(rf"$\theta$ [rad]") | ||
ax2.set_ylabel(rf"$\omega$ [rad/s]") | ||
ax2.set_xlabel("t [s]") | ||
plt.show() | ||
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if __name__ == "__main__": | ||
# run_lotka_volterra() | ||
# run_convergence() | ||
# run_pendulum_solver() | ||
run_pendulum_phase() |
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