def test_polar_stream_plot():
file_path = 'polar_stream_plot'
mesh = Mesh([(.1, 1.1), (0., 2 * np.pi)], [.2, .4 * np.pi],
CoordinateSystem.POLAR)
plot = StreamPlot(np.random.rand(2, 5, 5, 2), mesh, False)
plot.save(file_path).close()
os.remove(f'{file_path}.gif')
def test_fdm_operator_on_2d_pde():
diff_eq = NavierStokesEquation(5000.)
mesh = Mesh([(0., 10.), (0., 10.)], [1., 1.])
bcs = [(DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (1., .1, None, None)),
is_static=True),
DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (0., 0., None, None)),
is_static=True)),
(DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (0., 0., None, None)),
is_static=True),
DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (0., 0., None, None)),
is_static=True))]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = ContinuousInitialCondition(cp, lambda x: np.zeros((len(x), 4)))
ivp = InitialValueProblem(cp, (0., 10.), ic)
op = FDMOperator(RK4(), ThreePointCentralDifferenceMethod(), .25)
solution = op.solve(ivp)
assert solution.vertex_oriented
assert solution.d_t == .25
assert solution.discrete_y().shape == (40, 11, 11, 4)
assert solution.discrete_y(False).shape == (40, 10, 10, 4)
def test_spherical_quiver_plot():
file_path = 'spherical_quiver_plot'
mesh = Mesh([(.1, 1.1), (0., 2 * np.pi), (.05 * np.pi, .95 * np.pi)],
[.2, .4 * np.pi, .1 * np.pi], CoordinateSystem.SPHERICAL)
plot = QuiverPlot(np.random.rand(2, 6, 6, 10, 3), mesh, True)
plot.save(file_path).close()
os.remove(f'{file_path}.gif')
def test_discrete_initial_condition_2d_pde():
diff_eq = WaveEquation(2)
mesh = Mesh([(0., 2.), (0., 2.)], [1., 1.])
bcs = [(DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (0., 2.)), is_static=True),
DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (1., 2.)), is_static=True)),
(DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (3., 2.)), is_static=True),
DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (4., 2.)), is_static=True))]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
initial_condition = DiscreteInitialCondition(cp, np.zeros((3, 3, 2)), True)
y = initial_condition.y_0(np.array([1.5, .5]).reshape((1, 2)))
assert np.allclose(y, [1.75, 1.5])
y_0_vertices = initial_condition.discrete_y_0(True)
assert y_0_vertices.shape == (3, 3, 2)
assert np.all(y_0_vertices[0, 1:-1, 0] == 0.)
assert np.all(y_0_vertices[0, 1:-1, 1] == 2.)
assert np.all(y_0_vertices[-1, 1:-1, 0] == 1.)
assert np.all(y_0_vertices[-1, 1:-1, 1] == 2.)
assert np.all(y_0_vertices[:, 0, 0] == 3.)
assert np.all(y_0_vertices[:, 0, 1] == 2.)
assert np.all(y_0_vertices[:, -1, 0] == 4.)
assert np.all(y_0_vertices[:, -1, 1] == 2.)
assert np.all(y_0_vertices[1:-1, 1:-1, :] == 0.)
y_0_cell_centers = initial_condition.discrete_y_0(False)
assert y_0_cell_centers.shape == (2, 2, 2)
def test_continuous_initial_condition_pde_with_wrong_shape():
diff_eq = WaveEquation(1)
mesh = Mesh([(0., 10.)], [1.])
bcs = [(DirichletBoundaryCondition(lambda x: np.zeros((len(x), 2))), ) * 2]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
with pytest.raises(ValueError):
ContinuousInitialCondition(cp, lambda x: np.zeros((3, 2)))
def test_beta_initial_condition_with_wrong_number_of_alpha_and_betas():
diff_eq = DiffusionEquation(1)
mesh = Mesh([(0., 1.)], [.1])
bcs = [(NeumannBoundaryCondition(lambda x: np.zeros((len(x), 1))), ) * 2]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
with pytest.raises(ValueError):
BetaInitialCondition(cp, [(1., 1.), (1., 1)])
def test_discrete_initial_condition_pde_with_wrong_shape():
diff_eq = WaveEquation(1)
mesh = Mesh([(0., 10.)], [1.])
bcs = [(DirichletBoundaryCondition(lambda x: np.zeros((len(x), 2))), ) * 2]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
with pytest.raises(ValueError):
DiscreteInitialCondition(cp, np.zeros((10, 2)), vertex_oriented=True)
def test_gaussian_initial_condition_2d_pde():
diff_eq = WaveEquation(2)
mesh = Mesh([(0., 2.), (0., 2.)], [1., 1.])
bcs = [(DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (0., 2.)), is_static=True),
DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (1., 2.)), is_static=True)),
(DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (3., 2.)), is_static=True),
DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (4., 2.)), is_static=True))]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
initial_condition = GaussianInitialCondition(cp, [
(np.array([1., 1.]), np.array([[1., 0.], [0., 1.]])),
(np.array([1., 1.]), np.array([[.75, .25], [.25, .75]])),
], [1., 2.])
x_coordinates = np.array([[1., 1.], [.5, 1.5]])
expected_y_0 = [[.15915494, .45015816], [.12394999, .27303472]]
actual_y_0 = initial_condition.y_0(x_coordinates)
assert np.allclose(actual_y_0, expected_y_0)
expected_vertex_discrete_y_0 = [[[3., 2.], [0., 2.], [4., 2.]],
[[3., 2.], [.15915494, .45015816],
[4., 2.]], [[3., 2.], [1., 2.], [4., 2.]]]
actual_vertex_discrete_y_0 = initial_condition.discrete_y_0(True)
assert np.allclose(actual_vertex_discrete_y_0,
expected_vertex_discrete_y_0)
expected_cell_discrete_y_0 = [[[.12394999, .35058353],
[.12394999, .27303472]],
[[.12394999, .27303472],
[.12394999, .35058353]]]
actual_cell_discrete_y_0 = initial_condition.discrete_y_0(False)
assert np.allclose(actual_cell_discrete_y_0, expected_cell_discrete_y_0)
def test_gaussian_initial_condition_pde_with_wrong_means_and_cov_length():
diff_eq = WaveEquation(1)
mesh = Mesh([(0., 10.)], [1.])
bcs = [(DirichletBoundaryCondition(lambda x: np.zeros((len(x), 2))), ) * 2]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
with pytest.raises(ValueError):
GaussianInitialCondition(cp, [(np.array([1.]), np.array([[1.]]))] * 1)
def test_continuous_initial_condition_1d_pde():
diff_eq = DiffusionEquation(1)
mesh = Mesh([(0., 20.)], [.1])
bcs = [(DirichletBoundaryCondition(lambda x, t: np.zeros((len(x), 1)),
is_static=True),
DirichletBoundaryCondition(lambda x, t: np.full((len(x), 1), 1.5),
is_static=True))]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
initial_condition = ContinuousInitialCondition(
cp, lambda x: np.exp(-np.square(np.array(x) - 10.) / (2 * 5**2)))
assert np.isclose(initial_condition.y_0(np.full((1, 1), 10.)), 1.)
assert np.isclose(
initial_condition.y_0(np.full((1, 1),
np.sqrt(50) + 10.)), np.e**-1)
assert np.allclose(initial_condition.y_0(np.full((5, 1), 10.)),
np.ones((5, 1)))
y_0_vertices = initial_condition.discrete_y_0(True)
assert y_0_vertices.shape == (201, 1)
assert y_0_vertices[0, 0] == 0.
assert y_0_vertices[-1, 0] == 1.5
assert y_0_vertices[100, 0] == 1.
assert np.all(0. < y_0_vertices[1:100, 0]) \
and np.all(y_0_vertices[1:100, 0] < 1.)
assert np.all(0. < y_0_vertices[101:-1, 0]) \
and np.all(y_0_vertices[101:-1, 0] < 1.)
y_0_cell_centers = initial_condition.discrete_y_0(False)
assert y_0_cell_centers.shape == (200, 1)
assert np.all(0. < y_0_cell_centers) and np.all(y_0_cell_centers < 1.)
def test_fdm_operator_on_pde_with_t_and_x_dependent_rhs():
class TestDiffEq(DifferentialEquation):
def __init__(self):
super(TestDiffEq, self).__init__(2, 1)
@property
def symbolic_equation_system(self) -> SymbolicEquationSystem:
return SymbolicEquationSystem([
self.symbols.t / 100. *
(self.symbols.x[0] + self.symbols.x[1])**2
])
diff_eq = TestDiffEq()
mesh = Mesh([(-5., 5.), (0., 3.)], [2., 1.])
bcs = [(NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 1))),
NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 1))))] * 2
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = ContinuousInitialCondition(cp, lambda x: np.zeros((len(x), 1)))
ivp = InitialValueProblem(cp, (0., 5.), ic)
op = FDMOperator(RK4(), ThreePointCentralDifferenceMethod(), .25)
solution = op.solve(ivp)
y = solution.discrete_y()
assert solution.vertex_oriented
assert solution.d_t == .25
assert y.shape == (20, 6, 4, 1)
def test_pidon_operator_on_spherical_pde():
set_random_seed(0)
diff_eq = DiffusionEquation(3)
mesh = Mesh(
[(1., 11.), (0., 2 * np.pi), (.25 * np.pi, .75 * np.pi)],
[2., np.pi / 5., np.pi / 4],
CoordinateSystem.SPHERICAL)
bcs = [
(DirichletBoundaryCondition(
lambda x, t: np.ones((len(x), 1)), is_static=True),
DirichletBoundaryCondition(
lambda x, t: np.full((len(x), 1), 1. / 11.), is_static=True)),
(NeumannBoundaryCondition(
lambda x, t: np.zeros((len(x), 1)), is_static=True),
NeumannBoundaryCondition(
lambda x, t: np.zeros((len(x), 1)), is_static=True)),
(NeumannBoundaryCondition(
lambda x, t: np.zeros((len(x), 1)), is_static=True),
NeumannBoundaryCondition(
lambda x, t: np.zeros((len(x), 1)), is_static=True))
]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = ContinuousInitialCondition(cp, lambda x: 1. / x[:, :1])
t_interval = (0., .5)
ivp = InitialValueProblem(cp, t_interval, ic)
sampler = UniformRandomCollocationPointSampler()
pidon = PIDONOperator(sampler, .001, True)
training_loss_history, test_loss_history = pidon.train(
cp,
t_interval,
training_data_args=DataArgs(
y_0_functions=[ic.y_0],
n_domain_points=20,
n_boundary_points=10,
n_batches=1
),
model_args=ModelArgs(
latent_output_size=20,
branch_hidden_layer_sizes=[30, 30],
trunk_hidden_layer_sizes=[30, 30],
),
optimization_args=OptimizationArgs(
optimizer=optimizers.Adam(learning_rate=2e-5),
epochs=3,
verbose=False
)
)
assert len(training_loss_history) == 3
for i in range(2):
assert np.all(
training_loss_history[i + 1].weighted_total_loss.numpy() <
training_loss_history[i].weighted_total_loss.numpy())
solution = pidon.solve(ivp)
assert solution.d_t == .001
assert solution.discrete_y().shape == (500, 6, 11, 3, 1)
def test_solution_generate_plots_for_2d_pde_with_scalar_and_vector_fields():
diff_eq = ShallowWaterEquation(.5)
mesh = Mesh([(0., 5.), (0., 5.)], [1., 1.])
bcs = [(NeumannBoundaryCondition(
vectorize_bc_function(lambda x, t: (.0, None, None)), is_static=True),
NeumannBoundaryCondition(
vectorize_bc_function(lambda x, t: (.0, None, None)),
is_static=True))] * 2
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = GaussianInitialCondition(
cp, [(np.array([2.5, 1.25]), np.array([[1., 0.], [0., 1.]]))] * 3,
[1., .0, .0])
ivp = InitialValueProblem(cp, (0., 20.), ic)
t_coordinates = np.array([10., 20.])
discrete_y = np.arange(216).reshape((2, 6, 6, 3))
solution = Solution(ivp, t_coordinates, discrete_y, vertex_oriented=True)
plots = list(solution.generate_plots())
try:
assert len(plots) == 4
assert isinstance(plots[0], QuiverPlot)
assert isinstance(plots[1], StreamPlot)
assert isinstance(plots[2], ContourPlot)
assert isinstance(plots[3], SurfacePlot)
finally:
for plot in plots:
plot.close()
def test_beta_initial_condition_with_more_than_1d_pde():
diff_eq = DiffusionEquation(2)
mesh = Mesh([(0., 1.), (0., 1.)], [.1, .1])
bcs = [(NeumannBoundaryCondition(lambda x: np.zeros((len(x), 1))), ) * 2
] * 2
cp = ConstrainedProblem(diff_eq, mesh, bcs)
with pytest.raises(ValueError):
BetaInitialCondition(cp, [(1., 1.), (1., 1)])
def test_pidon_operator_on_pde_system():
set_random_seed(0)
diff_eq = NavierStokesEquation()
mesh = Mesh([(-2.5, 2.5), (0., 4.)], [1., 1.])
ic_function = vectorize_ic_function(lambda x: [
2. * x[0] - 4.,
2. * x[0] ** 2 + 3. * x[1] - x[0] * x[1] ** 2,
4. * x[0] - x[1] ** 2,
2. * x[0] * x[1] - 3.
])
bcs = [
(DirichletBoundaryCondition(
lambda x, t: ic_function(x),
is_static=True),
DirichletBoundaryCondition(
lambda x, t: ic_function(x),
is_static=True))
] * 2
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = ContinuousInitialCondition(cp, ic_function)
t_interval = (0., .5)
ivp = InitialValueProblem(cp, t_interval, ic)
sampler = UniformRandomCollocationPointSampler()
pidon = PIDONOperator(sampler, .001, True)
training_loss_history, test_loss_history = pidon.train(
cp,
t_interval,
training_data_args=DataArgs(
y_0_functions=[ic.y_0],
n_domain_points=20,
n_boundary_points=10,
n_batches=1
),
model_args=ModelArgs(
latent_output_size=20,
branch_hidden_layer_sizes=[20, 20],
trunk_hidden_layer_sizes=[20, 20],
),
optimization_args=OptimizationArgs(
optimizer=optimizers.Adam(learning_rate=1e-5),
epochs=3,
verbose=False
)
)
assert len(training_loss_history) == 3
for i in range(2):
assert np.all(
training_loss_history[i + 1].weighted_total_loss.numpy() <
training_loss_history[i].weighted_total_loss.numpy())
solution = pidon.solve(ivp)
assert solution.d_t == .001
assert solution.discrete_y().shape == (500, 6, 5, 4)
def test_fdm_operator_on_pde_with_t_and_x_dependent_rhs():
class TestDiffEq(DifferentialEquation):
def __init__(self):
super(TestDiffEq, self).__init__(2, 1)
@property
def symbolic_equation_system(self) -> SymbolicEquationSystem:
return SymbolicEquationSystem([
self.symbols.t / 100. *
(self.symbols.x[0] + self.symbols.x[1]) ** 2
])
diff_eq = TestDiffEq()
mesh = Mesh([(-1., 1.), (0., 2.)], [2., 1.])
bcs = [
(NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 1))),
NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 1))))
] * 2
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = ContinuousInitialCondition(cp, lambda x: np.zeros((len(x), 1)))
t_interval = (0., 1.)
ivp = InitialValueProblem(cp, t_interval, ic)
sampler = UniformRandomCollocationPointSampler()
pidon = PIDONOperator(sampler, .05, True)
training_loss_history, test_loss_history = pidon.train(
cp,
t_interval,
training_data_args=DataArgs(
y_0_functions=[ic.y_0],
n_domain_points=20,
n_boundary_points=10,
n_batches=1
),
model_args=ModelArgs(
latent_output_size=20,
branch_hidden_layer_sizes=[30, 30],
trunk_hidden_layer_sizes=[30, 30],
),
optimization_args=OptimizationArgs(
optimizer=optimizers.Adam(learning_rate=2e-5),
epochs=3,
verbose=False
)
)
assert len(training_loss_history) == 3
for i in range(2):
assert np.all(
training_loss_history[i + 1].weighted_total_loss.numpy() <
training_loss_history[i].weighted_total_loss.numpy())
solution = pidon.solve(ivp)
assert solution.d_t == .05
assert solution.discrete_y().shape == (20, 2, 3, 1)
def test_cp_pde_with_wrong_boundary_constraint_width():
diff_eq = WaveEquation(2)
mesh = Mesh([(0., 5.), (-5., 5.)], [.1, .2])
bcs = [(DirichletBoundaryCondition(lambda x, t: np.zeros((len(x), 1)),
is_static=True), ) * 2] * 2
with pytest.raises(ValueError):
ConstrainedProblem(diff_eq, mesh, bcs)
bcs = [(DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: [0.]), is_static=True), ) * 2] * 2
with pytest.raises(ValueError):
ConstrainedProblem(diff_eq, mesh, bcs)
def test_mesh():
x_intervals = [(-10., 10.), (0., 50.)]
d_x = [.1, .2]
mesh = Mesh(x_intervals, d_x)
assert mesh.vertices_shape == (201, 251)
assert mesh.cells_shape == (200, 250)
assert mesh.shape(True) == mesh.vertices_shape
assert mesh.shape(False) == mesh.cells_shape
for axis in range(2):
assert np.array_equal(
mesh.vertex_axis_coordinates[axis],
np.linspace(*x_intervals[axis], mesh.vertices_shape[axis]))
assert np.array_equal(
mesh.cell_center_axis_coordinates[axis],
np.linspace(x_intervals[axis][0] + d_x[axis] / 2.,
x_intervals[axis][1] - d_x[axis] / 2.,
mesh.cells_shape[axis]))
assert np.array_equal(
mesh.axis_coordinates(True)[axis],
mesh.vertex_axis_coordinates[axis])
assert np.array_equal(
mesh.axis_coordinates(False)[axis],
mesh.cell_center_axis_coordinates[axis])
all_vertex_x = mesh.all_index_coordinates(True)
assert np.allclose(all_vertex_x[2, 3], (-9.8, .6))
all_vertex_x_flattened = mesh.all_index_coordinates(True, True)
assert np.array_equal(all_vertex_x[2, 3],
all_vertex_x_flattened[2 * 251 + 3])
all_cell_x = mesh.all_index_coordinates(False)
assert np.allclose(all_cell_x[2, 3], (-9.75, .7))
all_cell_x_flattened = mesh.all_index_coordinates(False, True)
assert np.array_equal(all_cell_x[2, 3], all_cell_x_flattened[2 * 250 + 3])
def test_ode_operator_on_pde():
diff_eq = DiffusionEquation(1, 1.5)
mesh = Mesh([(0., 10.)], [.1])
bcs = [
(NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 1))),
DirichletBoundaryCondition(lambda x, t: np.zeros((len(x), 1)))),
]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = GaussianInitialCondition(cp, [(np.array([5.]), np.array([[2.5]]))],
[20.])
ivp = InitialValueProblem(cp, (0., 10.), ic)
op = ODEOperator('RK23', 2.5e-3)
with pytest.raises(ValueError):
op.solve(ivp)
def test_cp_pde_with_wrong_boundary_constraint_length():
diff_eq = DiffusionEquation(2)
mesh = Mesh([(0., 5.), (-5., 5.)], [.1, .2])
static_bcs = [(DirichletBoundaryCondition(lambda x, t: np.zeros((13, 1)),
is_static=True), ) * 2] * 2
with pytest.raises(ValueError):
ConstrainedProblem(diff_eq, mesh, static_bcs)
dynamic_bcs = [
(DirichletBoundaryCondition(lambda x, t: np.zeros((13, 1))), ) * 2
] * 2
cp = ConstrainedProblem(diff_eq, mesh, dynamic_bcs)
with pytest.raises(ValueError):
cp.create_boundary_constraints(True, 0.)
def test_pidon_operator_in_ar_mode_on_pde():
set_random_seed(0)
diff_eq = WaveEquation(1)
mesh = Mesh([(0., 1.)], (.2,))
bcs = [
(NeumannBoundaryCondition(
lambda x, t: np.zeros((len(x), 2)), is_static=True),
NeumannBoundaryCondition(
lambda x, t: np.zeros((len(x), 2)), is_static=True)),
]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
t_interval = (0., 1.)
ic = BetaInitialCondition(cp, [(3.5, 3.5), (3.5, 3.5)])
ivp = InitialValueProblem(cp, t_interval, ic)
training_y_0_functions = [
BetaInitialCondition(cp, [(p, p), (p, p)]).y_0
for p in [2., 3., 4., 5.]
]
sampler = UniformRandomCollocationPointSampler()
pidon = PIDONOperator(sampler, .25, False, auto_regression_mode=True)
assert pidon.auto_regression_mode
pidon.train(
cp,
(0., .25),
training_data_args=DataArgs(
y_0_functions=training_y_0_functions,
n_domain_points=50,
n_boundary_points=20,
n_batches=2
),
model_args=ModelArgs(
latent_output_size=50,
branch_hidden_layer_sizes=[50, 50],
trunk_hidden_layer_sizes=[50, 50],
),
optimization_args=OptimizationArgs(
optimizer=optimizers.Adam(learning_rate=1e-4),
epochs=2,
ic_loss_weight=10.,
verbose=False
)
)
sol = pidon.solve(ivp)
assert np.allclose(sol.t_coordinates, [.25, .5, .75, 1.])
assert sol.discrete_y().shape == (4, 5, 2)
def test_solution_pde_with_no_vertex_orientation_defined():
diff_eq = WaveEquation(1)
mesh = Mesh([(0., 2.)], [1.])
bcs = [(NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 2))),
NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 2))))]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = GaussianInitialCondition(cp, [(np.array([1.]), np.array([[1.]]))] * 2)
ivp = InitialValueProblem(cp, (0., 2.), ic)
t_coordinates = np.array([1., 2.])
discrete_y = np.arange(12).reshape((2, 3, 2))
with pytest.raises(ValueError):
Solution(ivp, t_coordinates, discrete_y)
def test_fdm_operator_on_1d_pde():
diff_eq = BurgerEquation(1, 1000.)
mesh = Mesh([(0., 10.)], [.1])
bcs = [(NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 1)),
is_static=True),
NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 1)),
is_static=True))]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = GaussianInitialCondition(cp, [(np.array([2.5]), np.array([[1.]]))])
ivp = InitialValueProblem(cp, (0., 50.), ic)
op = FDMOperator(RK4(), ThreePointCentralDifferenceMethod(), .25)
solution = op.solve(ivp)
assert solution.vertex_oriented
assert solution.d_t == .25
assert solution.discrete_y().shape == (200, 101, 1)
assert solution.discrete_y(False).shape == (200, 100, 1)
def _verify_pde_solution_shape_matches_problem(
y: np.ndarray,
mesh: Mesh,
vertex_oriented: bool,
expected_x_dims: Union[int, Tuple[int, int]],
is_vector_field: bool):
"""
Verifies that the shape of the input array representing the solution
of a partial differential equation over the provided mesh and with the
specified vertex orientation matches expectations.
:param y: an array representing the solution of the partial
differential equation
:param mesh: the spatial mesh over which the solution is evaluated
:param vertex_oriented: whether the solution is evaluated over the
vertices or the cell centers of the mesh
:param expected_x_dims: the expected number of spatial dimensions
:param is_vector_field: whether the solution is supposed to be a vector
field or a scalar field
"""
if isinstance(expected_x_dims, int):
if mesh.dimensions != expected_x_dims:
raise ValueError(f'mesh must be {expected_x_dims} dimensional')
elif not (expected_x_dims[0] <= mesh.dimensions <= expected_x_dims[1]):
raise ValueError(
f'mesh must be between {expected_x_dims[0]} and '
f'{expected_x_dims[1]} dimensional')
if y.ndim != mesh.dimensions + 2:
raise ValueError(
f'number of y axes ({y.ndim}) must be two larger than mesh '
f'dimensions ({mesh.dimensions})')
if y.shape[1:-1] != mesh.shape(vertex_oriented):
raise ValueError(
f'y shape {y.shape} must be compatible with mesh shape '
f'{mesh.shape(vertex_oriented)}')
if is_vector_field:
if y.shape[-1] != mesh.dimensions:
raise ValueError(
f'number of y components ({y.shape[-1]}) must match '
f'x dimensions {mesh.dimensions}')
elif y.shape[-1] != 1:
raise ValueError(
f'number of y components ({y.shape[-1]}) must be one')
def test_fdm_operator_on_spherical_pde():
diff_eq = DiffusionEquation(3)
mesh = Mesh([(1., 11.), (0., 2. * np.pi), (.1 * np.pi, .9 * np.pi)],
[2., np.pi / 5., np.pi / 5], CoordinateSystem.SPHERICAL)
bcs = [(NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 1)),
is_static=True),
NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 1)),
is_static=True))] * 3
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = ContinuousInitialCondition(cp, lambda x: 1. / x[:, :1])
ivp = InitialValueProblem(cp, (0., 5.), ic)
op = FDMOperator(RK4(), ThreePointCentralDifferenceMethod(), .1)
solution = op.solve(ivp)
assert solution.vertex_oriented
assert solution.d_t == .1
assert solution.discrete_y().shape == (50, 6, 11, 5, 1)
assert solution.discrete_y(False).shape == (50, 5, 10, 4, 1)
def test_fdm_operator_on_3d_pde():
diff_eq = CahnHilliardEquation(3)
mesh = Mesh([(0., 5.), (0., 5.), (0., 10.)], [.5, 1., 2.])
bcs = [(NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 2)),
is_static=True),
NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 2)),
is_static=True))] * 3
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = DiscreteInitialCondition(
cp, .05 * np.random.uniform(-1., 1., cp.y_shape(True)), True)
ivp = InitialValueProblem(cp, (0., 5.), ic)
op = FDMOperator(RK4(), ThreePointCentralDifferenceMethod(), .05)
solution = op.solve(ivp)
assert solution.vertex_oriented
assert solution.d_t == .05
assert solution.discrete_y().shape == (100, 11, 6, 6, 2)
assert solution.discrete_y(False).shape == (100, 10, 5, 5, 2)
def test_cp_1d_pde():
diff_eq = DiffusionEquation(1)
mesh = Mesh([(0., 1.)], [.1])
bcs = [(NeumannBoundaryCondition(lambda x, t: np.zeros((1, 1)),
is_static=True), ) * 2]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
assert cp.are_all_boundary_conditions_static
assert not cp.are_there_boundary_conditions_on_y
assert cp.y_shape(True) == (11, 1)
assert cp.y_shape(False) == (10, 1)
assert cp.differential_equation == diff_eq
assert cp.mesh == mesh
assert np.array_equal(cp.boundary_conditions, bcs)
y_vertex_constraints = cp.static_y_vertex_constraints
assert y_vertex_constraints.shape == (1, )
assert np.all(y_vertex_constraints[0].mask == [False])
assert np.all(y_vertex_constraints[0].values == [])
vertex_boundary_constraints = cp.static_boundary_constraints(True)
y_vertex_boundary_constraints = vertex_boundary_constraints[0]
assert y_vertex_boundary_constraints.shape == (1, 1)
assert y_vertex_boundary_constraints[0, 0][0] is None
assert y_vertex_boundary_constraints[0, 0][1] is None
d_y_vertex_boundary_constraints = vertex_boundary_constraints[1]
assert d_y_vertex_boundary_constraints.shape == (1, 1)
assert np.all(d_y_vertex_boundary_constraints[0, 0][0].mask == [True])
assert np.all(d_y_vertex_boundary_constraints[0, 0][0].values == [0.])
assert np.all(d_y_vertex_boundary_constraints[0, 0][1].mask == [True])
assert np.all(d_y_vertex_boundary_constraints[0, 0][1].values == [0.])
cell_boundary_constraints = cp.static_boundary_constraints(False)
y_cell_boundary_constraints = cell_boundary_constraints[0]
assert y_cell_boundary_constraints.shape == (1, 1)
assert y_cell_boundary_constraints[0, 0][0] is None
assert y_cell_boundary_constraints[0, 0][1] is None
d_y_cell_boundary_constraints = cell_boundary_constraints[1]
assert d_y_cell_boundary_constraints.shape == (1, 1)
assert np.all(d_y_cell_boundary_constraints[0, 0][0].mask == [True])
assert np.all(d_y_cell_boundary_constraints[0, 0][0].values == [0.])
assert np.all(d_y_cell_boundary_constraints[0, 0][1].mask == [True])
assert np.all(d_y_cell_boundary_constraints[0, 0][1].values == [0.])
def test_auto_regression_operator_on_pde():
set_random_seed(0)
diff_eq = WaveEquation(2)
mesh = Mesh([(-5., 5.), (-5., 5.)], [1., 1.])
bcs = [(DirichletBoundaryCondition(lambda x, t: np.zeros((len(x), 2)),
is_static=True),
DirichletBoundaryCondition(lambda x, t: np.zeros((len(x), 2)),
is_static=True))] * 2
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = GaussianInitialCondition(
cp, [(np.array([0., 2.5]), np.array([[.1, 0.], [0., .1]]))] * 2,
[3., .0])
ivp = InitialValueProblem(cp, (0., 10.), ic)
oracle = FDMOperator(RK4(), ThreePointCentralDifferenceMethod(), .1)
ref_solution = oracle.solve(ivp)
ml_op = AutoRegressionOperator(2.5, True)
ml_op.train(
ivp, oracle,
SKLearnKerasRegressor(
DeepONet([
np.prod(cp.y_shape(True)).item(), 100, 50,
diff_eq.y_dimension * 10
], [1 + diff_eq.x_dimension, 50, 50, diff_eq.y_dimension * 10],
diff_eq.y_dimension),
optimizer=optimizers.Adam(
learning_rate=optimizers.schedules.ExponentialDecay(
1e-2, decay_steps=500, decay_rate=.95)),
batch_size=968,
epochs=500,
), 20, lambda t, y: y + np.random.normal(0., t / 75., size=y.shape))
ml_solution = ml_op.solve(ivp)
assert ml_solution.vertex_oriented
assert ml_solution.d_t == 2.5
assert ml_solution.discrete_y().shape == (4, 11, 11, 2)
diff = ref_solution.diff([ml_solution])
assert np.all(diff.matching_time_points == np.linspace(2.5, 10., 4))
assert np.max(np.abs(diff.differences[0])) < .5
def test_fdm_operator_on_polar_pde():
diff_eq = ShallowWaterEquation(.5)
mesh = Mesh([(1., 11.), (0., 2 * np.pi)], [2., np.pi / 5.],
CoordinateSystem.POLAR)
bcs = [(NeumannBoundaryCondition(
vectorize_bc_function(lambda x, t: (.0, None, None)), is_static=True),
NeumannBoundaryCondition(
vectorize_bc_function(lambda x, t: (.0, None, None)),
is_static=True))] * 2
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = GaussianInitialCondition(
cp, [(np.array([-6., 0.]), np.array([[.25, 0.], [0., .25]]))] * 3,
[1., .0, .0])
ivp = InitialValueProblem(cp, (0., 5.), ic)
op = FDMOperator(RK4(), ThreePointCentralDifferenceMethod(), .1)
solution = op.solve(ivp)
assert solution.vertex_oriented
assert solution.d_t == .1
assert solution.discrete_y().shape == (50, 6, 11, 3)
assert solution.discrete_y(False).shape == (50, 5, 10, 3)
def test_solution_pde():
diff_eq = WaveEquation(1)
mesh = Mesh([(0., 2.)], [1.])
bcs = [(NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 2))),
NeumannBoundaryCondition(lambda x, t: np.zeros((len(x), 2))))]
cp = ConstrainedProblem(diff_eq, mesh, bcs)
ic = GaussianInitialCondition(cp, [(np.array([1.]), np.array([[1.]]))] * 2)
ivp = InitialValueProblem(cp, (0., 2.), ic)
t_coordinates = np.array([1., 2.])
discrete_y = np.arange(12).reshape((2, 3, 2))
solution = Solution(ivp, t_coordinates, discrete_y, vertex_oriented=True)
assert solution.initial_value_problem == ivp
assert np.array_equal(solution.t_coordinates, t_coordinates)
assert np.isclose(solution.d_t, 1.)
assert solution.vertex_oriented
x_coordinates = np.array([[.5], [1.]])
expected_y = [[[1., 2.], [2., 3.]], [[7., 8.], [8., 9.]]]
assert np.allclose(solution.y(x_coordinates), expected_y)
expected_cell_y = [[[1., 2.], [3., 4.]], [[7., 8.], [9., 10.]]]
assert np.allclose(solution.discrete_y(False), expected_cell_y)
assert np.allclose(solution.discrete_y(), discrete_y)
other_solutions = [
Solution(ivp,
np.linspace(.5, 2., 4),
np.arange(16).reshape((4, 2, 2)),
vertex_oriented=False)
]
expected_differences = [
[[3., 3.], [3., 3.], [3., 3.]],
[[5., 5.], [5., 5.], [5., 5.]],
]
diff = solution.diff(other_solutions)
assert np.allclose(diff.matching_time_points, [1., 2.])
assert np.allclose(diff.differences, expected_differences)
