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_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_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_solution_ode():
diff_eq = LotkaVolterraEquation()
cp = ConstrainedProblem(diff_eq)
ic = ContinuousInitialCondition(cp, lambda _: np.array([5., 10.]))
ivp = InitialValueProblem(cp, (0., 10.), ic)
t_coordinates = np.array([5., 10.])
discrete_y = np.arange(4).reshape((2, 2))
solution = Solution(ivp, t_coordinates, discrete_y)
assert solution.initial_value_problem == ivp
assert np.array_equal(solution.t_coordinates, t_coordinates)
assert np.isclose(solution.d_t, 5.)
assert solution.vertex_oriented is None
assert np.allclose(solution.y(), [[0., 1.], [2., 3.]])
assert np.allclose(solution.discrete_y(), [[0., 1.], [2., 3.]])
other_solutions = [
Solution(ivp, np.linspace(2.5, 10., 4),
np.arange(8).reshape((4, 2))),
Solution(ivp, np.linspace(1.25, 10., 8),
np.arange(16).reshape((8, 2)))
]
expected_differences = [
[[2., 2.], [4., 4.]],
[[6., 6.], [12., 12.]],
]
diff = solution.diff(other_solutions)
assert np.allclose(diff.matching_time_points, [5., 10.])
assert np.allclose(diff.differences, expected_differences)
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_discrete_initial_condition_ode():
diff_eq = LotkaVolterraEquation()
cp = ConstrainedProblem(diff_eq)
initial_condition = DiscreteInitialCondition(cp, np.array([10., 100.]))
assert np.all(initial_condition.y_0(None) == [10., 100.])
assert np.all(initial_condition.discrete_y_0() == [10., 100.])
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_pidon_operator_in_ar_mode_training_with_invalid_t_interval():
diff_eq = PopulationGrowthEquation()
cp = ConstrainedProblem(diff_eq)
t_interval = (0., 1.)
ic = ContinuousInitialCondition(cp, lambda _: np.array([1.]))
sampler = UniformRandomCollocationPointSampler()
pidon = PIDONOperator(sampler, .25, True, auto_regression_mode=True)
with pytest.raises(ValueError):
pidon.train(
cp,
t_interval,
training_data_args=DataArgs(
y_0_functions=[ic.y_0],
n_domain_points=50,
n_batches=1
),
model_args=ModelArgs(
latent_output_size=1,
trunk_hidden_layer_sizes=[50, 50, 50],
),
optimization_args=OptimizationArgs(
optimizer={'class_name': 'Adam'},
epochs=100,
verbose=False
)
)
def __init__(
self,
cp: ConstrainedProblem,
y_0: np.ndarray,
vertex_oriented: Optional[bool] = None,
interpolation_method: str = 'linear'):
"""
:param cp: the constrained problem to turn into an initial value
problem by providing the initial conditions for it
:param y_0: the array containing the initial values of y over a spatial
mesh (which may be 0 dimensional in case of an ODE)
:param vertex_oriented: whether the initial conditions are evaluated at
the vertices or cell centers of the spatial mesh; it the
constrained problem is an ODE, it can be None
:param interpolation_method: the interpolation method to use to
calculate values that do not exactly fall on points of the y_0
grid; if the constrained problem is based on an ODE, it can be None
"""
if cp.differential_equation.x_dimension and vertex_oriented is None:
raise ValueError('vertex orientation must be defined for PDEs')
if y_0.shape != cp.y_shape(vertex_oriented):
raise ValueError(
f'discrete initial value shape {y_0.shape} must match '
'constrained problem solution shape '
f'{cp.y_shape(vertex_oriented)}')
self._cp = cp
self._y_0 = np.copy(y_0)
self._vertex_oriented = vertex_oriented
self._interpolation_method = interpolation_method
if vertex_oriented:
apply_constraints_along_last_axis(
cp.static_y_vertex_constraints, self._y_0)
def test_initial_value_problem_with_invalid_time_interval():
diff_eq = PopulationGrowthEquation()
cp = ConstrainedProblem(diff_eq)
ic = ContinuousInitialCondition(cp, lambda _: np.array([0.]))
with pytest.raises(ValueError):
InitialValueProblem(cp, (3., 2.), ic)
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_auto_regression_operator_on_ode_with_isolated_perturbations():
set_random_seed(0)
diff_eq = LotkaVolterraEquation(2., 1., .8, 1.)
cp = ConstrainedProblem(diff_eq)
ic = ContinuousInitialCondition(cp, lambda _: np.array([1., 2.]))
ivp = InitialValueProblem(cp, (0., 10.), ic)
oracle = ODEOperator('DOP853', .001)
ref_solution = oracle.solve(ivp)
ml_op = AutoRegressionOperator(2.5, True)
ml_op.train(ivp,
oracle,
RandomForestRegressor(),
25,
lambda t, y: y + np.random.normal(0., .01, size=y.shape),
isolate_perturbations=True)
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, 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])) < .01
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_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_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_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_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_solution_with_invalid_t_coordinate_dimensions():
diff_eq = LotkaVolterraEquation()
cp = ConstrainedProblem(diff_eq)
ic = ContinuousInitialCondition(cp, lambda _: np.array([5., 10.]))
ivp = InitialValueProblem(cp, (0., 10.), ic)
discrete_y = np.zeros((2, 2))
with pytest.raises(ValueError):
Solution(ivp, np.array([[5., 10.]]), discrete_y)
def test_solution_with_mismatched_discrete_y_shape():
diff_eq = LotkaVolterraEquation()
cp = ConstrainedProblem(diff_eq)
ic = ContinuousInitialCondition(cp, lambda _: np.array([5., 10.]))
ivp = InitialValueProblem(cp, (0., 10.), ic)
discrete_y = np.zeros((2, 3))
with pytest.raises(ValueError):
Solution(ivp, np.array([5., 10.]), discrete_y)
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_ode_with_analytic_solution():
set_random_seed(0)
r = 4.
y_0 = 1.
diff_eq = PopulationGrowthEquation(r)
cp = ConstrainedProblem(diff_eq)
t_interval = (0., .25)
ic = ContinuousInitialCondition(cp, lambda _: np.array([y_0]))
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=25,
n_batches=5,
n_ic_repeats=5
),
model_args=ModelArgs(
latent_output_size=1,
trunk_hidden_layer_sizes=[50, 50, 50],
),
optimization_args=OptimizationArgs(
optimizer=optimizers.SGD(),
epochs=100,
verbose=False
),
secondary_optimization_args=SecondaryOptimizationArgs(
max_iterations=100,
verbose=False
)
)
assert len(training_loss_history) == 101
assert len(test_loss_history) == 0
assert training_loss_history[-1].weighted_total_loss.numpy() < 5e-5
ivp = InitialValueProblem(
cp,
t_interval,
ic,
lambda _ivp, t, x: np.array([y_0 * np.e ** (r * t)])
)
solution = pidon.solve(ivp)
assert solution.d_t == .001
assert solution.discrete_y().shape == (250, 1)
analytic_y = np.array([ivp.exact_y(t) for t in solution.t_coordinates])
assert np.mean(np.abs(analytic_y - solution.discrete_y())) < 1e-3
assert np.max(np.abs(analytic_y - solution.discrete_y())) < 2.5e-3
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_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_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_initial_value_problem_without_exact_solution():
diff_eq = PopulationGrowthEquation()
cp = ConstrainedProblem(diff_eq)
ic = ContinuousInitialCondition(cp, lambda _: np.array([0.]))
ivp = InitialValueProblem(cp, (0., 2.), ic)
assert not ivp.has_exact_solution
with pytest.raises(RuntimeError):
ivp.exact_y(2.)
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_ode_operator_on_ode():
diff_eq = LotkaVolterraEquation()
cp = ConstrainedProblem(diff_eq)
ic = ContinuousInitialCondition(cp, lambda _: np.array([100., 15.]))
ivp = InitialValueProblem(cp, (0., 10.), ic)
op = ODEOperator('DOP853', 1e-3)
solution = op.solve(ivp)
assert solution.vertex_oriented is None
assert solution.d_t == 1e-3
assert solution.discrete_y().shape == (1e4, 2)
def test_auto_regression_operator_with_wrong_perturbed_initial_value_shape():
diff_eq = LorenzEquation()
cp = ConstrainedProblem(diff_eq)
ic = ContinuousInitialCondition(cp, lambda _: np.ones(3))
ivp = InitialValueProblem(cp, (0., 10.), ic)
oracle = ODEOperator('DOP853', .001)
ml_op = AutoRegressionOperator(2.5, True)
with pytest.raises(ValueError):
ml_op.train(ivp, oracle, LinearRegression(), 25,
lambda t, y: np.array([1.]))
