def test_solve_spherical():
pde = laplacian_spherical
generator = ExampleGeneratorSpherical(512)
# 0-boundary condition; solution should be u(r, theta, phi) = 0 identically
f = lambda th, ph: 0.
g = lambda th, ph: 0.
condition = DirichletBVPSpherical(r_0=0., f=f, r_1=1., g=g)
solution, loss_history = solve_spherical(pde,
condition,
0.0,
1.0,
max_epochs=500,
return_best=True)
rs, thetas, phis = generator.get_examples()
us = solution(rs, thetas, phis, as_type='np')
assert np.isclose(us, np.zeros_like(us),
atol=0.005).all(), f"Solution is not straight 0s: {us}"
# 1-boundary condition; solution should be u(r, theta, phi) = 1 identically
f = lambda th, ph: 1.
g = lambda th, ph: 1.
condition = DirichletBVPSpherical(r_0=0., f=f, r_1=1., g=g)
solution, loss_history = solve_spherical(pde,
condition,
0.0,
1.0,
max_epochs=500,
return_best=True)
rs, thetas, phis = generator.get_examples()
us = solution(rs, thetas, phis, as_type='np')
assert np.isclose(us, np.ones_like(us),
atol=0.005).all(), f"Solution is not straight 1s: {us}"
print("solve_spherical tests passed")
def test_monitor_spherical():
pde = laplacian_spherical
f = lambda th, ph: 0.
g = lambda th, ph: 0.
condition = DirichletBVPSpherical(r_0=0., f=f, r_1=1., g=g)
monitor = MonitorSpherical(0.0, 1.0, check_every=1)
solve_spherical(pde, condition, 0.0, 1.0, max_epochs=50, monitor=monitor)
print("MonitorSpherical test passed")
def test_train_generator_spherical():
pde = laplacian_spherical
condition = NoConditionSpherical()
train_generator = ExampleGeneratorSpherical(size=64,
r_min=0.,
r_max=1.,
method='equally-spaced-noisy')
r, th, ph = train_generator.get_examples()
assert (0. < r.min()) and (r.max() < 1.)
assert (0. <= th.min()) and (th.max() <= np.pi)
assert (0. <= ph.min()) and (ph.max() <= 2 * np.pi)
valid_generator = ExampleGeneratorSpherical(size=64,
r_min=1.,
r_max=1.,
method='equally-radius-noisy')
r, th, ph = valid_generator.get_examples()
assert (r == 1).all()
assert (0. <= th.min()) and (th.max() <= np.pi)
assert (0. <= ph.min()) and (ph.max() <= 2 * np.pi)
solve_spherical(pde,
condition,
0.0,
1.0,
train_generator=train_generator,
valid_generator=valid_generator,
max_epochs=5)
with raises(ValueError):
_ = ExampleGeneratorSpherical(64, method='bad_generator')
with raises(ValueError):
_ = ExampleGeneratorSpherical(64, r_min=-1.0)
with raises(ValueError):
_ = ExampleGeneratorSpherical(64, r_min=1.0, r_max=0.0)
print("GeneratorSpherical tests passed")
def test_electric_potential_uniformly_charged_ball():
"""
electric potential on uniformly charged solid sphere
refer to http://www.phys.uri.edu/~gerhard/PHY204/tsl94.pdf
"""
# free charge volume density
rho = 1.
# medium permittivity
epsilon = 1.
# Coulomb constant
k = 1. / (4 * np.pi * epsilon)
# radius of the ball
R = 1.
# total electric charge on solid sphere
Q = (4 / 3) * np.pi * (R**3) * rho
# electric potential at sphere center
v_0 = 1.5 * k * Q / R
# electric potential on sphere boundary
v_R = k * Q / R
# analytic solution of electric potential
analytic_solution = lambda r, th, ph: k * Q / (2 * R) * (3 - (r / R)**2)
pde = lambda u, r, theta, phi: laplacian_spherical(u, r, theta, phi
) + rho / epsilon
condition = DirichletBVPSpherical(r_0=0.,
f=lambda th, ph: v_0,
r_1=R,
g=lambda th, ph: v_R)
monitor = MonitorSpherical(0.0, R, check_every=50)
solution, loss_history, analytic_mse = solve_spherical(
pde,
condition,
0.,
R,
max_epochs=500,
return_best=True,
analytic_solution=analytic_solution,
monitor=monitor)
generator = ExampleGeneratorSpherical(512)
rs, thetas, phis = generator.get_examples()
us = solution(rs, thetas, phis, as_type="np")
vs = analytic_solution(rs, thetas, phis).detach().cpu().numpy()
abs_diff = abs(us - vs)
assert np.isclose(us, vs, atol=0.008).all(), \
f"Solution doesn't match analytic expectation {us} != {vs}, abs_diff={abs_diff}"
print("electric-potential-on-uniformly-charged-solid-sphere passed")
def test_solve_spherical():
pde = laplacian_spherical
generator = GeneratorSpherical(512)
# 0-boundary condition; solution should be u(r, theta, phi) = 0 identically
f = lambda th, ph: 0.
g = lambda th, ph: 0.
condition = DirichletBVPSpherical(r_0=0., f=f, r_1=1., g=g)
with pytest.warns(FutureWarning):
solution, loss_history = solve_spherical(pde,
condition,
0.0,
1.0,
max_epochs=2,
return_best=True)
assert isinstance(solution, SolutionSpherical)
def test_solve_spherical():
pde = laplacian_spherical
generator = GeneratorSpherical(512)
# 0-boundary condition; solution should be u(r, theta, phi) = 0 identically
f = lambda th, ph: 0.
g = lambda th, ph: 0.
condition = DirichletBVPSpherical(r_0=0., f=f, r_1=1., g=g)
solution, loss_history = solve_spherical(pde,
condition,
0.0,
1.0,
max_epochs=2,
return_best=True)
rs, thetas, phis = generator.get_examples()
us = solution(rs, thetas, phis, as_type='np')
def test_electric_potential_gaussian_charged_density():
# total charge
Q = 1.
# standard deviation of gaussian
sigma = 1.
# medium permittivity
epsilon = 1.
# Coulomb constant
k = 1 / (4 * np.pi * epsilon)
# coefficient of gaussian term
gaussian_coeff = Q / (sigma**3) / np.power(2 * np.pi, 1.5)
# distribution of charge
rho_f = lambda r: gaussian_coeff * torch.exp(-r.pow(2) / (2 * sigma**2))
# analytic solution, refer to https://en.wikipedia.org/wiki/Poisson%27s_equation
analytic_solution = lambda r, th, ph: (k * Q / r) * torch.erf(r / (np.sqrt(
2) * sigma))
pde = lambda u, r, th, ph: laplacian_spherical(u, r, th, ph) + rho_f(
r) / epsilon
r_0, r_1 = 0.1, 3.
v_0 = (k * Q / r_0) * erf(r_0 / (np.sqrt(2) * sigma))
v_1 = (k * Q / r_1) * erf(r_1 / (np.sqrt(2) * sigma))
condition = DirichletBVPSpherical(r_0, lambda th, ph: v_0, r_1,
lambda th, ph: v_1)
monitor = MonitorSpherical(r_0, r_1, check_every=50)
solution, loss_history, analytic_mse = solve_spherical(
pde,
condition,
r_0,
r_1,
max_epochs=500,
return_best=True,
analytic_solution=analytic_solution,
monitor=monitor)
generator = ExampleGeneratorSpherical(512, r_min=r_0, r_max=r_1)
rs, thetas, phis = generator.get_examples()
us = solution(rs, thetas, phis, as_type="np")
vs = analytic_solution(rs, thetas, phis).detach().cpu().numpy()
rdiff = abs(us - vs) / vs
assert np.isclose(us, vs, rtol=0.05).all(), \
f"Solution doesn't match analytic expectattion {us} != {vs}, relative-diff={rdiff}"
print("electric-potential-on-gaussian-charged-density passed")
def test_electric_potential_gaussian_charged_density():
# total charge
Q = 1.
# standard deviation of gaussian
sigma = 1.
# medium permittivity
epsilon = 1.
# Coulomb constant
k = 1 / (4 * np.pi * epsilon)
# coefficient of gaussian term
gaussian_coeff = Q / (sigma**3) / np.power(2 * np.pi, 1.5)
# distribution of charge
rho_f = lambda r: gaussian_coeff * torch.exp(-r.pow(2) / (2 * sigma**2))
# analytic solution, refer to https://en.wikipedia.org/wiki/Poisson%27s_equation
analytic_solution = lambda r, th, ph: (k * Q / r) * torch.erf(r / (np.sqrt(
2) * sigma))
# interior and exterior radius
r_0, r_1 = 0.1, 3.
# values at interior and exterior boundary
v_0 = (k * Q / r_0) * erf(r_0 / (np.sqrt(2) * sigma))
v_1 = (k * Q / r_1) * erf(r_1 / (np.sqrt(2) * sigma))
def validate(solution):
generator = GeneratorSpherical(512, r_min=r_0, r_max=r_1)
rs, thetas, phis = generator.get_examples()
us = solution(rs, thetas, phis, to_numpy=True)
vs = analytic_solution(rs, thetas, phis).detach().cpu().numpy()
assert us.shape == vs.shape
# solving the problem using normal network (subject to the influence of polar singularity of laplacian operator)
pde1 = lambda u, r, th, ph: laplacian_spherical(u, r, th, ph) + rho_f(
r) / epsilon
condition1 = DirichletBVPSpherical(r_0, lambda th, ph: v_0, r_1,
lambda th, ph: v_1)
monitor1 = MonitorSpherical(r_0, r_1, check_every=50)
with pytest.warns(FutureWarning):
solution1, metrics_history = solve_spherical(
pde1,
condition1,
r_0,
r_1,
max_epochs=2,
return_best=True,
analytic_solution=analytic_solution,
monitor=monitor1,
)
validate(solution1)
# solving the problem using spherical harmonics (laplcian computation is optimized)
max_degree = 2
harmonics_fn = RealSphericalHarmonics(max_degree=max_degree)
harmonic_laplacian = HarmonicsLaplacian(max_degree=max_degree)
pde2 = lambda R, r, th, ph: harmonic_laplacian(R, r, th, ph) + rho_f(
r) / epsilon
R_0 = torch.tensor([v_0 * 2] + [0 for _ in range((max_degree + 1)**2 - 1)])
R_1 = torch.tensor([v_1 * 2] + [0 for _ in range((max_degree + 1)**2 - 1)])
def analytic_solution2(r, th, ph):
sol = torch.zeros(r.shape[0], (max_degree + 1)**2)
sol[:, 0:1] = 2 * analytic_solution(r, th, ph)
return sol
condition2 = DirichletBVPSphericalBasis(r_0=r_0,
R_0=R_0,
r_1=r_1,
R_1=R_1,
max_degree=max_degree)
monitor2 = MonitorSphericalHarmonics(r_0,
r_1,
check_every=50,
harmonics_fn=harmonics_fn)
net2 = FCNN(n_input_units=1, n_output_units=(max_degree + 1)**2)
with pytest.warns(FutureWarning):
solution2, metrics_history = solve_spherical(
pde2,
condition2,
r_0,
r_1,
net=net2,
max_epochs=2,
return_best=True,
analytic_solution=analytic_solution2,
monitor=monitor2,
harmonics_fn=harmonics_fn,
)
validate(solution2)