def test_3d():
x, xi, xj = symbols('x xi xj')
y, yi, yj = symbols('y yi yj')
z, zi, zj = symbols('z zi zj')
X = Tuple(x, y, z)
Xi = Tuple(xi, yi, zi)
Xj = Tuple(xj, yj, zj)
u = Unknown('u')
alpha = Constant('alpha')
beta = Constant('beta')
mu = Constant('mu')
nu = Constant('nu')
theta = Constant('theta')
# expr = alpha * u
# expr = alpha * dx(u)
# expr = alpha * dy(u)
# expr = alpha * dz(u)
# expr = alpha * u + beta * dx(u)
# expr = alpha * u + beta * dy(u)
# expr = alpha * u + beta * dz(u)
# expr = mu * u + alpha * dx(u) + beta * dx(dx(u))
# expr = mu * u + alpha * dx(u) + beta * dy(dy(u))
# expr = mu * u + alpha * dx(u) + beta * dz(dz(u))
expr = mu * u + alpha * dx(u) + beta * dy(dz(u)) + nu * dx(dz(u))
# print('> generic_kernel := ', expand(generic_kernel(expr, u, Xi)))
# print('> generic_kernel := ', expand(generic_kernel(expr, u, Xj)))
print('> generic_kernel := ', expand(generic_kernel(expr, u, (Xi, Xj))))
def test_1():
u, v, a = symbols('u v a')
# ...
expr = u + v
print('> expr := {0}'.format(expr))
expr = dx(expr)
print('> gelatized := {0}'.format(expr))
print('')
# ...
# ...
expr = 2 * u * v
print('> expr := {0}'.format(expr))
expr = dx(expr)
print('> gelatized := {0}'.format(expr))
print('')
# ...
# ... dx should not operate on u^2,
# since we consider only linearized weak formulations
expr = u * u
print('> expr := {0}'.format(expr))
expr = dx(expr)
print('> gelatized := {0}'.format(expr))
print('')
def test_generic_kernel_1d():
x, xi, xj = symbols('x xi xj')
u = Unknown('u')
# ... testing u
assert (generic_kernel(u, u, xi) == Function('u')(xi))
assert (generic_kernel(u, u, xj) == Function('u')(xj))
assert (generic_kernel(u, u, (xi, xj)) == Function('u')(xi, xj))
# ...
# ... testing dx(u)
assert (generic_kernel(dx(u), u, xi) == Derivative(Function('u')(xi), xi))
assert (generic_kernel(dx(u), u, xj) == Derivative(Function('u')(xj), xj))
assert (generic_kernel(dx(u), u, (xi, xj)) == Derivative(
Function('u')(xi, xj), xi, xj))
# ...
# ... testing dx(dx(u))
assert (generic_kernel(dx(dx(u)), u,
xi) == Derivative(Function('u')(xi), xi, xi))
assert (generic_kernel(dx(dx(u)), u,
xj) == Derivative(Function('u')(xj), xj, xj))
assert (generic_kernel(dx(dx(u)), u, (xi, xj)) == Derivative(
Function('u')(xi, xj), xi, xi, xj, xj))
def test_generic_kernel_3d():
x, xi, xj = symbols('x xi xj')
y, yi, yj = symbols('y yi yj')
z, zi, zj = symbols('z zi zj')
X = Tuple(x, y, z)
Xi = Tuple(xi, yi, zi)
Xj = Tuple(xj, yj, zj)
u = Unknown('u')
# ... testing u
assert (generic_kernel(u, u, xi) == Function('u')(xi))
assert (generic_kernel(u, u, xj) == Function('u')(xj))
assert (generic_kernel(u, u, (xi, xj)) == Function('u')(xi, xj))
# ...
# ... testing dx(u)
assert (generic_kernel(dx(u), u, Xi) == Derivative(Function('u')(*Xi), xi))
assert (generic_kernel(dx(u), u, Xj) == Derivative(Function('u')(*Xj), xj))
assert (generic_kernel(dx(u), u, (Xi, Xj)) == Derivative(
Function('u')(*Xi, *Xj), xi, xj))
# ...
# ... testing dy(u)
assert (generic_kernel(dy(u), u, Xi) == Derivative(Function('u')(*Xi), yi))
assert (generic_kernel(dy(u), u, Xj) == Derivative(Function('u')(*Xj), yj))
assert (generic_kernel(dy(u), u, (Xi, Xj)) == Derivative(
Function('u')(*Xi, *Xj), yi, yj))
# ...
# ... testing dz(u)
assert (generic_kernel(dz(u), u, Xi) == Derivative(Function('u')(*Xi), zi))
assert (generic_kernel(dz(u), u, Xj) == Derivative(Function('u')(*Xj), zj))
assert (generic_kernel(dz(u), u, (Xi, Xj)) == Derivative(
Function('u')(*Xi, *Xj), zi, zj))
# ...
# ... testing dx(dx(u))
assert (generic_kernel(dx(dx(u)), u,
Xi) == Derivative(Function('u')(*Xi), xi, xi))
assert (generic_kernel(dx(dx(u)), u,
Xj) == Derivative(Function('u')(*Xj), xj, xj))
assert (generic_kernel(dx(dx(u)), u, (Xi, Xj)) == Derivative(
Function('u')(*Xi, *Xj), xi, xi, xj, xj))
def test_est_2dkernel():
"""example from Harsha."""
x, xi, xj = symbols('x xi xj')
y, yi, yj = symbols('y yi yj')
X = Tuple(x, y)
Xi = Tuple(xi, yi)
Xj = Tuple(xj, yj)
u = Unknown('u')
phi = Constant('phi')
theta = Constant('theta')
expr = phi * u + dx(u) + dy(dy(u))
print('> generic_kernel := ', expand(generic_kernel(expr, u, (Xi, Xj))))
print('')
kuu = theta * exp(-0.5 * ((xi - xj)**2 + (yi - yj)**2))
kuf = compute_kernel(expr, kuu, Xi)
kfu = compute_kernel(expr, kuu, Xj)
kff = compute_kernel(expr, kuu, (Xi, Xj))
print('> kuf := ', kuf)
print('> kfu := ', kfu)
print('> kff := ', kff)
def test_2():
u, v = symbols('u v')
F = Field('F')
# ...
expr = F * v * u
print('> expr := {0}'.format(expr))
expr = dx(expr)
print('> gelatized := {0}'.format(expr))
print('')
def test_1d():
x, xi, xj = symbols('x xi xj')
u = Unknown('u')
alpha = Constant('alpha')
beta = Constant('beta')
mu = Constant('mu')
theta = Constant('theta')
# expr = alpha * u
# expr = alpha * dx(u)
# expr = alpha * u + beta * dx(u)
# expr = mu * u + dx(u)
# expr = mu * u + dx(dx(u))
# expr = mu * u + alpha * dx(u) + beta * dx(dx(u))
expr = mu * u + dx(u) + dx(dx(u))
# print('> generic_kernel := ', expand(generic_kernel(expr, u, xi)))
# print('> generic_kernel := ', expand(generic_kernel(expr, u, xj)))
print('> generic_kernel := ', expand(generic_kernel(expr, u, (xi, xj))))
def generic_kernel(expr, func, y, args=None):
if isinstance(y, Symbol):
_derivatives = tuple([dx])
elif isinstance(y, Tuple):
_derivatives = tuple(_partial_derivatives[:len(y)])
elif not isinstance(y, (list, tuple)):
raise TypeError('expecting a Symbol or Tuple')
_args = []
if args:
for a in args:
if isinstance(a, Symbol):
_args += [a]
elif isinstance(a, Tuple):
_args += [*a]
else:
raise TypeError('expecting a Symbol or Tuple')
args = _args
if isinstance(y, (list, tuple)):
args = y
ei = func
for xi in y:
ej = generic_kernel(expr, ei, xi, args=args)
ei = ej
expr = ei
# check that there are no partial derivatives in the final expression
ops = find_partial_derivatives(expr)
if not (len(ops) == 0):
raise ValueError(
'Found unexpected partial differential operators in \n{}'.
format(expr))
return expr
else:
if isinstance(func, Unknown):
func = [func]
elif isinstance(func, (Add, Mul)):
fn = [
i for i in preorder_traversal(expr) if isinstance(i, Unknown)
]
# make it unique
fn = set(fn)
fn = list(fn)
if not (len(fn) == 1):
raise ValueError('expecting only one unknown')
expr = expr.subs({fn[0]: func})
func = fn
elif isinstance(func, Derivative):
fn = [i for i in expr.free_symbols if isinstance(i, Unknown)]
if not (len(fn) == 1):
raise ValueError('expecting only one unknown')
u = fn[0]
expr = expr.subs({u: func})
func = [u]
# TODO improve
elif isinstance(func, Function):
fname = type(func).__name__
func = [Unknown(fname)]
else:
raise NotImplementedError('type = ', type(func), func)
if not args:
args = [y]
if isinstance(y, Tuple):
args = [*y]
# ... TODO is there a way to remove this part?
for f in func:
fnew = Function(f.name)
if isinstance(y, Tuple):
for d in _derivatives:
for D in _derivatives:
i_d = d.grad_index
i_D = D.grad_index
dD_f = fnew(*args).diff(y[i_d]).diff(y[i_D])
expr = expr.subs({d(D(f)): dD_f})
for d in _derivatives:
i_d = d.grad_index
d_f = fnew(*args).diff(y[i_d])
expr = expr.subs({d(f): d_f})
elif isinstance(y, Symbol):
# 1D case, we only use dx
expr = expr.subs({dx(dx(f)): fnew(*args).diff(y).diff(y)})
expr = expr.subs({dx(f): fnew(*args).diff(y)})
else:
raise TypeError('expecting Tuple or Symbol')
# ...
# partial derivatives must be sorted from high to low
ops = sort_partial_derivatives(expr)
for i in ops:
if not (len(i.args) == 1):
raise ValueError(
'expecting only one argument for partial derivatives')
# if i = dx(u) then type(i) is dx
d = type(i)
a = i.args[0]
# terms like dx(Derivative(..))
if isinstance(a, Derivative):
if isinstance(y, Tuple):
i_d = d.grad_index
expr = expr.subs({i: a.diff(y[i_d])})
elif isinstance(y, Symbol):
expr = expr.subs({i: a.diff(y)})
# terms like dx(u(..))
# TODO this is not good, since we don't know if the function is an
# unkown => store unknown names in a list and pass it recursively?
elif isinstance(a, Function) and not (isinstance(a, _derivatives)):
if isinstance(y, Tuple):
i_d = d.grad_index
expr = expr.subs({i: a.diff(y[i_d])})
elif isinstance(y, Symbol):
expr = expr.subs({i: a.diff(y)})
# terms like dx(dx(Derivative(..)))
elif isinstance(a, _derivatives):
if not (len(a.args) == 1):
raise ValueError(
'expecting only one argument for partial derivatives')
D = type(a)
b = a.args[0]
if isinstance(y, Tuple):
i_d = d.grad_index
i_D = D.grad_index
expr = expr.subs({i: b.diff(y[i_d]).diff(y[i_D])})
elif isinstance(y, Symbol):
expr = expr.subs({i: b.diff(y).diff(y)})
else:
raise TypeError('expecting a Derivative or partial derivative,'
' given {} :: {}'.format(a, type(a)))
# finally, we replace u by u(xi)
fn = [i for i in expr.free_symbols if isinstance(i, Unknown)]
for u in fn:
fnew = Function(u.name)
expr = expr.subs({u: fnew(*args)})
return expr
from mlhiphy.calculus import Unknown
from mlhiphy.kernels import compute_kernel, generic_kernel
from mlhiphy.templates import template_scalar, template_header_scalar
from sympy import expand
from sympy import Lambda
from sympy import symbols
from sympy import exp
x, xi, xj = symbols('x xi xj')
alpha = Constant('alpha')
beta = Constant('beta')
u = Unknown('u')
expr = alpha * u + dx(u)
print('> generic_kernel := ', expand(generic_kernel(expr, u, (xi, xj))))
xi, xj, theta = symbols('xi xj theta')
kuu = theta * exp(-0.5 * ((xi - xj)**2))
import os
def mkdir_p(dir):
if os.path.isdir(dir):
return
os.makedirs(dir)
