def test_cp_ode():
diff_eq = LotkaVolterraEquation()
cp = ConstrainedProblem(diff_eq)
assert cp.mesh is None
assert cp.static_y_vertex_constraints is None
assert cp.static_boundary_vertex_constraints is None
assert cp.static_boundary_cell_constraints is None
assert cp.static_boundary_constraints(True) is None
assert cp.static_boundary_constraints(False) is None
assert cp.boundary_conditions is None
assert cp.y_shape(True) == cp.y_shape(False) == (diff_eq.y_dimension, )
assert not cp.are_all_boundary_conditions_static
assert not cp.are_there_boundary_conditions_on_y
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_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_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_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_cp_2d_pde():
diff_eq = WaveEquation(2)
mesh = Mesh([(2., 6.), (-3., 3.)], [.1, .2])
bcs = ((DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (999., None)), is_static=True),
NeumannBoundaryCondition(
vectorize_bc_function(lambda x, t: (100., -100.)),
is_static=True)),
(NeumannBoundaryCondition(
vectorize_bc_function(lambda x, t: (-x[0], None)),
is_static=True),
DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (x[0], x[1])),
is_static=True)))
cp = ConstrainedProblem(diff_eq, mesh, bcs)
assert cp.are_all_boundary_conditions_static
assert cp.are_there_boundary_conditions_on_y
y_vertices = np.full(cp.y_shape(True), 13.)
apply_constraints_along_last_axis(cp.static_y_vertex_constraints,
y_vertices)
assert np.all(y_vertices[0, :-1, 0] == 999.)
assert np.all(y_vertices[0, :-1, 1] == 13.)
assert np.all(y_vertices[-1, :-1, :] == 13.)
assert np.all(y_vertices[1:, 0, :] == 13.)
assert np.allclose(y_vertices[:, -1, 0],
np.linspace(2., 6., y_vertices.shape[0]))
assert np.all(y_vertices[:, -1, 1] == 3.)
y_vertices = np.zeros(cp.y_shape(True))
diff = ThreePointCentralDifferenceMethod()
d_y_boundary_constraints = cp.static_boundary_vertex_constraints[1]
d_y_0_over_d_x_0 = diff.gradient(y_vertices[..., :1], mesh, 0,
d_y_boundary_constraints[:, :1])
assert np.all(d_y_0_over_d_x_0[-1, :, :] == 100.)
assert np.all(d_y_0_over_d_x_0[:-1, :, :] == 0.)
d_y_0_over_d_x_1 = diff.gradient(y_vertices[..., :1], mesh, 1,
d_y_boundary_constraints[:, :1])
assert np.allclose(d_y_0_over_d_x_1[:, 0, 0],
np.linspace(-2., -6., y_vertices.shape[0]))
assert np.all(d_y_0_over_d_x_1[:, 1:, :] == 0.)
d_y_1_over_d_x_0 = diff.gradient(y_vertices[..., 1:], mesh, 0,
d_y_boundary_constraints[:, 1:])
assert np.all(d_y_1_over_d_x_0[-1, :, :] == -100.)
assert np.all(d_y_1_over_d_x_0[:-1, :, :] == 0.)
d_y_1_over_d_x_1 = diff.gradient(y_vertices[..., 1:], mesh, 1,
d_y_boundary_constraints[:, 1:])
assert np.all(d_y_1_over_d_x_1 == 0.)
y_boundary_cell_constraints = cp.static_boundary_cell_constraints[0]
assert np.all(y_boundary_cell_constraints[0, 0][0].mask == [True] *
cp.y_cells_shape[1])
assert np.all(y_boundary_cell_constraints[0, 0][0].values == 999.)
assert np.all(y_boundary_cell_constraints[0, 1][0].mask == [False] *
cp.y_cells_shape[1])
assert y_boundary_cell_constraints[0, 1][0].values.size == 0
assert np.all(y_boundary_cell_constraints[1, 0][1].mask == [True] *
cp.y_cells_shape[0])
assert np.allclose(y_boundary_cell_constraints[1, 0][1].values,
np.linspace(2.05, 5.95, cp.y_cells_shape[0]))
assert np.all(y_boundary_cell_constraints[1, 1][1].mask == [True] *
cp.y_cells_shape[0])
assert np.all(y_boundary_cell_constraints[1, 1][1].values == 3.)
assert y_boundary_cell_constraints[1, 0][0] is None
def test_cp_3d_pde():
mesh = Mesh([(2., 6.), (-3., 3.), (10., 12.)], [.1, .2, .5])
assert mesh.shape(True) == (41, 31, 5)
assert mesh.shape(False) == (40, 30, 4)
diff_eq = WaveEquation(3)
cp = ConstrainedProblem(
diff_eq, mesh,
((DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (999., None)), is_static=True),
NeumannBoundaryCondition(
vectorize_bc_function(lambda x, t: (None, None)),
is_static=True)),
(DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (0., 0.)), is_static=True),
NeumannBoundaryCondition(lambda x, t: np.full((len(x), 2), t))),
(NeumannBoundaryCondition(lambda x, t: -x[:, :2] * x[:, 1:3],
is_static=True),
DirichletBoundaryCondition(
vectorize_bc_function(lambda x, t: (-999., None))))))
assert cp.y_shape(True) == (41, 31, 5, 2)
assert cp.y_shape(False) == (40, 30, 4, 2)
assert not cp.are_all_boundary_conditions_static
assert cp.are_there_boundary_conditions_on_y
assert cp.static_y_vertex_constraints.shape == (2, )
y = np.full(cp._y_vertices_shape, -1)
cp.static_y_vertex_constraints[0].apply(y[..., :1])
cp.static_y_vertex_constraints[1].apply(y[..., 1:])
assert np.all(y[0, 1:, :, 0] == 999.)
assert np.all(y[:, 0, :, 0] == 0.)
assert np.all(y[1:, 1:, :, 0] == -1.)
assert np.all(y[:, 0, :, 1] == 0.)
assert np.all(y[:, 1:, :, 1] == -1.)
vertex_boundary_constraints = cp.static_boundary_vertex_constraints
cell_boundary_constraints = cp.static_boundary_cell_constraints
for y_boundary_constraints in \
[vertex_boundary_constraints[0], cell_boundary_constraints[0]]:
assert y_boundary_constraints.shape == (3, 2)
assert y_boundary_constraints[0, 0][0] is not None
assert y_boundary_constraints[0, 1][0] is not None
assert y_boundary_constraints[0, 0][1] is None
assert y_boundary_constraints[0, 1][1] is None
assert y_boundary_constraints[1, 0][0] is not None
assert y_boundary_constraints[1, 1][0] is not None
assert y_boundary_constraints[1, 0][1] is None
assert y_boundary_constraints[1, 1][1] is None
assert y_boundary_constraints[2, 0][0] is None
assert y_boundary_constraints[2, 1][0] is None
assert y_boundary_constraints[2, 0][1] is None
assert y_boundary_constraints[2, 1][1] is None
for d_y_boundary_constraints in \
[vertex_boundary_constraints[1], cell_boundary_constraints[1]]:
assert d_y_boundary_constraints.shape == (3, 2)
assert d_y_boundary_constraints[0, 0][0] is None
assert d_y_boundary_constraints[0, 1][0] is None
assert d_y_boundary_constraints[0, 0][1] is not None
assert d_y_boundary_constraints[0, 1][1] is not None
assert d_y_boundary_constraints[1, 0][0] is None
assert d_y_boundary_constraints[1, 1][0] is None
assert d_y_boundary_constraints[1, 0][1] is None
assert d_y_boundary_constraints[1, 1][1] is None
assert d_y_boundary_constraints[2, 0][0] is not None
assert d_y_boundary_constraints[2, 1][0] is not None
assert d_y_boundary_constraints[2, 0][1] is None
assert d_y_boundary_constraints[2, 1][1] is None
new_vertex_boundary_constraints = cp.create_boundary_constraints(True, 1.)
new_y_boundary_constraints = new_vertex_boundary_constraints[0]
new_d_y_boundary_constraints = new_vertex_boundary_constraints[1]
assert new_y_boundary_constraints[2, 0][1] is not None
assert new_y_boundary_constraints[2, 1][1] is not None
assert new_d_y_boundary_constraints[1, 0][1] is not None
assert new_d_y_boundary_constraints[1, 1][1] is not None
d_y_boundary = np.full((41, 1, 5, 2), np.nan)
new_d_y_boundary_constraints[1, 0][1].apply(d_y_boundary[..., :1])
new_d_y_boundary_constraints[1, 1][1].apply(d_y_boundary[..., 1:])
assert np.all(d_y_boundary == 1.)
new_y_vertex_constraints = \
cp.create_y_vertex_constraints(new_y_boundary_constraints)
assert new_y_vertex_constraints.shape == (2, )
y = np.full(cp._y_vertices_shape, -1)
new_y_vertex_constraints[0].apply(y[..., :1])
new_y_vertex_constraints[1].apply(y[..., 1:])
assert np.all(y[0, 1:, :-1, 0] == 999.)
assert np.all(y[:, 0, :-1, 0] == 0.)
assert np.all(y[:, :, -1, 0] == -999.)
assert np.all(y[1:, 1:, :-1, 0] == -1.)
assert np.all(y[:, 0, :, 1] == 0.)
assert np.all(y[:, 1:, :, 1] == -1.)
