def test_4d_approx(n):
t, x, y, z = sp.symbols('t x y z')
t_domain = np.linspace(-1, 4, n) # Physical
x_domain = np.linspace(0, 2, n)
y_domain = np.linspace(0, 1, n)
z_domain = np.linspace(-2, 0, n)
T, X, Y, Z = np.meshgrid(t_domain,
x_domain,
y_domain,
z_domain,
indexing='ij')
# I will look at 3 points aroung the center
pts_indices = [(n / 2 - 1, ) * 4, (n / 2, ) * 4, (n / 2 + 1, ) * 4]
# Their physical representations
pts = map(
lambda
(i, j, k, l): [t_domain[i], x_domain[j], y_domain[k], z_domain[k]],
pts_indices)
symbols = (t, x, y, z)
functions = [
sp.sin(sp.pi * t) * sp.exp(-(x**2 + y**3 - z**2)),
sp.cos(sp.pi * t) * sp.exp(-(x**2 + 3 * y**2 + x * y * z))
]
# Some deriveatives
derivs = [(0, 0, 3, 0), (1, 0, 1, 1), (0, 1, 0, 2), (2, 1, 0, 3),
(1, 1, 2, 1)]
numeric = np.zeros(len(pts))
errors = defaultdict(list)
for foo in functions:
foo_values = sp.lambdify((t, x, y, z), foo, 'numpy')(T, X, Y, Z)
grid_foo = GridFunction([t_domain, x_domain, y_domain, z_domain],
foo_values)
for d in derivs:
if d != (0, 0, 0, 0):
dfoo = foo
for var, deg in zip(symbols, d):
dfoo = dfoo.diff(var, deg)
else:
dfoo = foo
# True
df = sp.lambdify((t, x, y, z), dfoo)
exact = np.array([df(*pt) for pt in pts])
dgrid_foo = diff(grid_foo, d)
[
dgrid_foo(pi, numeric[i:i + 1])
for i, pi in enumerate(pts_indices)
]
error = exact - numeric
e = np.linalg.norm(error)
errors[foo].append(e)
return errors
def test_3d_poly(n):
t, x, y = sp.symbols('t x y')
t_domain = np.linspace(-1, 4, n) # Physical
x_domain = np.linspace(0, 2, n)
y_domain = np.linspace(0, 1, n)
T, X, Y = np.meshgrid(t_domain, x_domain, y_domain, indexing='ij')
# I will look at 3 points aroung the center
pts_indices = [(n / 2 - 1, ) * 3, (n / 2, ) * 3, (n / 2 + 1, ) * 3]
# Their physical representations
pts = map(lambda (i, j, k): [t_domain[i], x_domain[j], y_domain[k]],
pts_indices)
symbols = (t, x, y)
functions = [
2 * t + 1 * x - 2 * y,
3 * (t**2 - 2 * t + 1) + x**2 + t * x - 4 * t * y + 3 * t * x,
y * (t**4 - 2 * t**2 + t - 1) + 2 * t**2 * 4 * x**3 - y * x * 4 * t**4
]
# Some deriveatives
derivs = [(0, 0, 0), (1, 0, 1), (0, 1, 0), (2, 1, 0), (1, 2, 0), (3, 1, 1),
(1, 3, 1)]
status = True
numeric = np.zeros(len(pts))
for foo in functions:
print foo
foo_values = sp.lambdify((t, x, y), foo, 'numpy')(T, X, Y)
grid_foo = GridFunction([t_domain, x_domain, y_domain], foo_values)
for d in derivs:
if d != (0, 0, 0):
dfoo = foo
for var, deg in zip(symbols, d):
dfoo = dfoo.diff(var, deg)
else:
dfoo = foo
print '\t', d, dfoo,
# True
df = sp.lambdify((t, x, y), dfoo)
exact = np.array([df(*pt) for pt in pts])
dgrid_foo = diff(grid_foo, d)
[
dgrid_foo(pi, numeric[i:i + 1])
for i, pi in enumerate(pts_indices)
]
error = exact - numeric
e = np.linalg.norm(error)
print 'OK' if e < TOL else (exact, numeric)
status = status and e < TOL
return status
def test_2d_poly(n):
'''Error for some approx at [-1, 4]x[0, 2]'''
t, x = sp.symbols('t x')
t_domain = np.linspace(-1, 4, n) # Physical
x_domain = np.linspace(0, 2, n) # Physical
T, X = np.meshgrid(t_domain, x_domain, indexing='ij')
# I will look at 3 points aroung the center
pts_indices = [(n / 2 - 1, ) * 2, (n / 2, ) * 2, (n / 2 + 1, ) * 2]
# Their physical representations
pts = map(lambda (i, j): [t_domain[i], x_domain[j]], pts_indices)
symbols = (t, x)
functions = [
2 * t + x, 3 * (t**2 - 2 * t + 1) + x**2,
(t**4 - 2 * t**2 + t - 1) + 2 * t**2 * 4 * x**3 - x * 3 * t**2
]
# Some deriveatives
derivs = [(0, 0), (1, 0), (0, 1), (1, 1), (2, 1), (3, 1), (1, 3), (2, 2)]
numeric = np.zeros(len(pts))
status = True
for foo in functions:
print foo
foo_values = sp.lambdify((t, x), foo, 'numpy')(T, X)
grid_foo = GridFunction([t_domain, x_domain], foo_values)
for d in derivs:
if d != (0, 0):
dfoo = foo
for var, deg in zip(symbols, d):
dfoo = dfoo.diff(var, deg)
else:
dfoo = foo
print '\t', d, dfoo,
# True
df = sp.lambdify((t, x), dfoo)
exact = np.array([df(*pt) for pt in pts])
dgrid_foo = diff(grid_foo, d)
[
dgrid_foo(pi, numeric[i:i + 1])
for i, pi in enumerate(pts_indices)
]
error = exact - numeric
e = np.linalg.norm(error)
print 'OK' if e < TOL else (exact, numeric)
status = status and e < TOL
return status
def test_2d_approx(n):
'''Error for some approx at [-1, 4]x[0, 2].Just convergence'''
t, x = sp.symbols('t x')
t_domain = np.linspace(-1, 4, n) # Physical
x_domain = np.linspace(0, 2, n) # Physical
T, X = np.meshgrid(t_domain, x_domain, indexing='ij')
# I will look at 3 points aroung the center
pts_indices = [(n / 2 - 1, ) * 2, (n / 2, ) * 2, (n / 2 + 1, ) * 2]
# Their physical representations
pts = map(lambda (i, j): [t_domain[i], x_domain[j]], pts_indices)
symbols = (t, x)
functions = [
sp.sin(sp.pi * (t + x)), t * sp.sin(2 * sp.pi * (x - 3 * t * x)),
sp.cos(2 * sp.pi * x * t)
]
# Some deriveatives
derivs = [(1, 0), (0, 1), (1, 1), (2, 1), (3, 1), (1, 3), (2, 2)]
errors = {foo: list() for foo in functions}
numeric = np.zeros(len(pts))
for foo in functions:
foo_values = sp.lambdify((t, x), foo, 'numpy')(T, X)
grid_foo = GridFunction([t_domain, x_domain], foo_values)
for d in derivs:
if d != (0, 0):
dfoo = foo
for var, deg in zip(symbols, d):
dfoo = dfoo.diff(var, deg)
else:
dfoo = foo
# True
df = sp.lambdify((t, x), dfoo)
exact = np.array([df(*pt) for pt in pts])
dgrid_foo = diff(grid_foo, d)
[
dgrid_foo(pi, numeric[i:i + 1])
for i, pi in enumerate(pts_indices)
]
error = exact - numeric
e = np.linalg.norm(error)
errors[foo].append(e)
return errors
def test_1d_poly_vec(n):
'''Error for some approx at [-1, 4]'''
t = sp.Symbol('t')
t_domain = np.linspace(-1, 4, n) # Physical
t_domain_index = range(len(t_domain))
# I will look at 3 points aroung the center
pts_index = [
t_domain_index[len(t_domain) / 2 - 1],
t_domain_index[len(t_domain) / 2],
t_domain_index[len(t_domain) / 2 + 1]
]
pts = t_domain[pts_index]
symbols = (t, )
functions = [2 * t, 3 * (t**2 - 2 * t + 1), (t**4 - 2 * t**2 + t - 1)]
# We look at up to 3 derivatives
derivs = [(0, ), (1, ), (2, ), (3, )]
status = True
numeric = np.zeros((len(pts), 2))
for foo in functions:
print foo
foo_values = np.array(map(sp.lambdify(t, foo), t_domain))
grid_foo = GridFunction([t_domain], [foo_values, 2 * foo_values])
for d in derivs:
if d != (0, ):
dfoo = foo
for var, deg in zip(symbols, d):
dfoo = dfoo.diff(var, deg)
else:
dfoo = foo
print '\t', d, dfoo,
# True
df = sp.lambdify(t, dfoo)
exact = df(pts)
exact = np.c_[exact, 2 * exact]
dgrid_foo = diff(grid_foo, d)
for row, pi in zip(numeric, pts_index):
np.put(row, range(2), dgrid_foo((pi, )))
error = exact - numeric
e = np.linalg.norm(error)
print 'OK' if e < TOL else (exact, numeric)
status = status and e < TOL
return status
def test_1d_approx(n, width=7):
'''Error for some approx at [-1, 4]. Just convergence'''
t = sp.Symbol('t')
t_domain = np.linspace(-1, 4, n) # Physical
t_domain_index = range(len(t_domain))
# I will look at 3 points aroung the center
pts_index = [
t_domain_index[len(t_domain) / 2 - 1],
t_domain_index[len(t_domain) / 2],
t_domain_index[len(t_domain) / 2 + 1]
]
pts = t_domain[pts_index]
symbols = (t, )
functions = [2 * sp.sin(sp.pi * t), sp.cos(2 * sp.pi * t**2)]
# We look at up to 3 derivatives
derivs = [(1, ), (2, ), (3, ), (4, )]
numeric = np.zeros(len(pts))
status = {f: list() for f in functions}
for foo in functions:
foo_values = np.array(map(sp.lambdify(t, foo), t_domain))
grid_foo = GridFunction([t_domain], foo_values)
for d in derivs:
if d != (0, ):
dfoo = foo
for var, deg in zip(symbols, d):
dfoo = dfoo.diff(var, deg)
else:
dfoo = foo
# True
df = sp.lambdify(t, dfoo)
exact = np.array(map(df, pts))
dgrid_foo = diff(grid_foo, d, width=width)
numeric.ravel()[:] = [dgrid_foo((pi, )) for pi in pts_index]
error = exact - numeric
e = np.linalg.norm(error)
status[foo].append(e)
return status
if __name__ == '__main__':
from grid_function import GridFunction
import numpy as np
f, g = sp.symbols('f g')
n = 24
grid = [
np.linspace(-1, 4, n),
np.linspace(0, 2, n),
np.linspace(0, 1, n),
np.linspace(-2, 0, n)
]
T, X, Y, Z = np.meshgrid(*grid, indexing='ij')
f, g, t, x, y, z = sp.symbols('f g t x y z')
# The one that gives us data
expr = t**2 + sp.sin(4 * x * y * z) + z**2
expr_numpy = sp.lambdify((t, x, y, z), expr, 'numpy')
data = expr_numpy(T, X, Y, Z)
grid_f = GridFunction(grid, [[data, 2 * data], [3 * data, 4 * data]])
#y = np.zeros(grid_f.value_shape)
print grid_f((8, 7, 6, 7))
#grid_g = GridFunction(grid, [data, -2*data])
#print expr_value_shape(f * g, {f: grid_f, g: grid_g})
u, wind, x, t = sp.symbols('u w x t')
# Use exact solution to generate the data
u_exact = sp.sin(sp.sin(x) + t)
u_exact = sp.lambdify((t, x), u_exact, 'numpy')
# All data
t_domain = np.linspace(0, T_end, n_samples)
x_domain = np.linspace(-0.5, 0.5, n_samples)
T, X = np.meshgrid(t_domain, x_domain, indexing='ij')
u_data = u_exact(T, X)
# To build the model we sample the data (in interior indices) to get
# the derivative right. For the rhs we keep it simple
u_grid = GridFunction([t_domain, x_domain], u_data)
x_grid = Coordinate([t_domain, x_domain], 1)
lhs_stream = compile_stream((sp.Derivative(u, t), ), {u: u_grid})
foos = polynomials([u], 3)
derivatives = Dx(u, 1, [3])[1:] # Don't include 0;th
# Combine these guys
columns = [(1./sp.cos(x))*sp.Derivative(u, x)] + \
list(foos) + \
list(derivatives) + \
[reduce(operator.mul, cs) for cs in itertools.product(foos, derivatives)]
rhs_stream = compile_stream(columns, {u: u_grid})
grid = [
np.linspace(-1, 4, n),
np.linspace(0, 2, n),
np.linspace(0, 1, n),
np.linspace(-2, 0, n)
]
T, X, Y, Z = np.meshgrid(*grid, indexing='ij')
f, g, t, x, y, z = sp.symbols('f g t x y z')
# The one that gives us data
expr = t**2 + sp.sin(4 * x * y * z) + z**2
expr_numpy = sp.lambdify((t, x, y, z), expr, 'numpy')
data = expr_numpy(T, X, Y, Z)
grid_expr = GridFunction(grid, data)
grid_expr1 = GridFunction(grid, data + data**2)
# Check the values at
i_point = (8, 7, 6, 7)
# print expand_derivative(sp.Derivative(f**2 + g**2, y, x))
# exit()
expression = (1, f, f**2, sp.sin(f) * f, sp.Derivative(f, x),
f * sp.Derivative(f, x),
f**2 + sp.sqrt(f) * sp.Derivative(f, y),
sp.Derivative(f, x, y) + sp.Derivative(g, y, y),
sp.Derivative(f**2, y, x) + sp.Derivative(g, y, x))
subs = {f: grid_expr, g: grid_expr1}
print 'A', A
print 'eigenvalues', eigw
u0 = np.ones(n_eqs)
# This is now a vector valued function
u_exact = lambda t, U=U, eigw=eigw, u0=u0: (
(U.dot(np.diag([np.exp(ei * t) for ei in eigw]))).dot(U.T)).dot(u0)
# # All data
t_domain = np.linspace(0, T_end, n_samples)
u_data = np.array(map(u_exact, t_domain))
# Let's make one grid function for evvery component of that vector
u_grids = []
for col in range(n_eqs):
u_grids.append(GridFunction([t_domain], u_data[:, col]))
# And a symbols for each
us = sp.symbols(' '.join(['u%d' % i for i in range(n_eqs)]))
symbol_foo_map = dict(zip(us, u_grids))
# We are going to build a block structured system d u0/dt @ all_pts, next component
lhs_streams = [
compile_stream((sp.Derivative(usi, t), ), symbol_foo_map) for usi in us
]
# NOTE: this is reused for each block
columns = polynomials(us, 2)
print columns
rhs_stream = compile_stream(columns, symbol_foo_map)
grid = [
np.linspace(-1, 4, n),
np.linspace(0, 2, n),
np.linspace(0, 1, n),
np.linspace(-2, 0, n)
]
T, X, Y, Z = np.meshgrid(*grid, indexing='ij')
f, t, x, y, z = sp.symbols('f t x y z')
# The one that gives us data
expr = t**2 + sp.sin(4 * x * y * z) + z**2
expr_numpy = sp.lambdify((t, x, y, z), expr, 'numpy')
data = expr_numpy(T, X, Y, Z)
grid_expr = GridFunction(grid, data)
# Check the values at
i_point = (8, 7, 6, 7)
point = [grid[i][p] for i, p in enumerate(i_point)]
compiled_expr = compile_expr(f**3 + f * sp.Derivative(f**2, x, y)**2 +
f / 3 + sp.sin(f), {f: grid_expr},
grid=grid)
values[:] = compiled_expr(i_point)
# Now take some derivative
df_expr = expr**3 + expr * (sp.Derivative(expr, x,
y))**2 + expr / 3 + sp.sin(expr)
u, x, t = sp.symbols('u x t')
# Use exact solution to generate the data
u_exact_expr = x**2*t
u_exact = sp.lambdify((t, x), u_exact_expr, 'numpy')
# All data
t_domain = np.linspace(0, T_end, n_samples)
x_domain = np.linspace(0, X_end, n_samples)
T, X = np.meshgrid(t_domain, x_domain, indexing='ij')
u_data = u_exact(T, X)
# To build the model we sample the data (in interior indices) to get
# the derivative right. For the rhs we keep it simple
grid = [t_domain, x_domain]
u_grid = GridFunction(grid, u_data)
equation = sp.Derivative(u, t) - u - sp.Derivative(u, x) - sp.Derivative(u, x, x)
lhs_stream = compile_stream((equation, ), {u: u_grid})
foos = (-t*x**2, -2*t*x, -2*t, x**2)# + (sp.sin(x), sp.cos(x), sp.sin(t), sp.cos(t))
#polynomials([x, t], 3)
# Combine these guys
columns = list(foos)
rhs_stream = compile_stream(columns, {u:u_grid})
# A subset of this data is now used for training; where we can eval deriv
# safely
np.linspace(0, 1, n),
np.linspace(-2, 0, n)
]
T, X, Y, Z = np.meshgrid(*grid, indexing='ij')
f, g, t, x, y, z = sp.symbols('f g t x y z')
expressions = (
x, #t**2 + sp.sin(4*x*y*z) + z**2,
y) #t + sp.exp(-(4*(x**2 + y**2))) + z**3)
grid_fs = []
for f_ in expressions:
expr_numpy = sp.lambdify((t, x, y, z), f_, 'numpy')
data = expr_numpy(T, X, Y, Z)
grid_fs.append(GridFunction(grid, data))
# Not the streams
expression = (1, f, g, f**2 + f * g + g**2, sp.sin(f + g),
sp.exp(-(f**2 + g**2)), sp.Derivative(f, x),
sp.Derivative(f**2, y), sp.Derivative(f + g, x),
2 * f * sp.Derivative(g, y, x), 2 * g * sp.Derivative(f, y, x),
sp.Derivative(2 * f * g, y, x, y), sp.Derivative(f**2, x, x))
from streams import expand_derivative
gg = expand_derivative(expression[0])
print
subs = {f: grid_fs[0], g: grid_fs[1]}
# Sympy version