def basis_vectors(N, equation):
xx = legslbndm(N)
lepolys = gen_lepolys(N, xx)
lepoly_x = dx(N, xx, lepolys)
lepoly_xx = dxx(N, xx, lepolys)
phi = basis(N, lepolys, equation)
phi_x = basis_x(N, phi, lepoly_x, equation)
phi_xx = basis_xx(N, phi_x, lepoly_xx, equation)
return xx, lepolys, lepoly_x, lepoly_xx, phi, phi_x, phi_xx
def diff(N, T, D):
x = legslbndm(N + 1)
# D = torch.from_numpy(legslbdiff(N+1, x)).to(device).float()
T_ = T.clone()
for i in range(T.shape[0]):
element = torch.mm(D, T[i, :].reshape(T.shape[1],
1)).reshape(T.shape[1], )
T_[i, :] = element
T = T_.clone()
del T_
return T
def weak_form(eps,
N,
f,
u,
alphas,
lepolys,
phi,
phi_x,
equation,
nbfuncs,
index=0):
B, i, j = u.shape
N -= 1
LHS = torch.zeros((B, 1, 1)).to(device).float()
RHS = torch.zeros((B, 1, 1)).to(device).float()
phi = torch.transpose(phi, 0, 1)
denom = torch.square(torch.from_numpy(lepolys[N]).to(device).float())
denom = torch.transpose(denom, 0, 1)
diffusion = 6 * eps * alphas[:, :, index]
if equation == 'Standard':
u_x = reconstruct(alphas, phi_x)
ux_phi = u_x * phi[:, index]
convection = torch.sum(ux_phi * 2 / (N * (N + 1)) / denom, axis=2)
LHS[:, 0] = diffusion - convection
RHS[:, 0] = torch.sum(2 * f * phi[:, index] / (N * (N + 1)) / denom,
axis=2)
elif equation == 'Burgers':
phi_x = torch.transpose(phi_x, 0, 1)
convection = torch.sum(u**2 * phi_x[:, index] / (N * (N + 1)) / denom,
axis=2)
LHS[:, 0] = diffusion - convection
RHS[:, 0] = torch.sum(2 * f * phi[:, index] / (N * (N + 1)) / denom,
axis=2)
elif equation == 'Helmholtz':
ku = 3.5
x = legslbndm(N + 1)
D_ = torch.from_numpy(legslbdiff(N + 1, x)).to(device).float()
D = torch.zeros((B, N + 1, N + 1)).to(device).float()
D[:, :, :] = D_
phi_x = torch.transpose(phi_x, 0, 1)
u_ = torch.transpose(u, 1, 2)
temp = torch.bmm(D, u_)
temp = torch.transpose(temp, 1, 2)
diffusion = torch.sum(2 * temp * phi_x[:, index] / (N * (N + 1)) /
denom,
axis=2)
reaction = ku * torch.sum(
2 * u * phi[:, index] / (N * (N + 1)) / denom, axis=2)
LHS[:, 0] = -diffusion + reaction
RHS[:, 0] = torch.sum(2 * f * phi[:, index] / (N * (N + 1)) / denom,
axis=2)
return LHS, RHS
LHS, RHS = 0, 0 for _ in range(10): phi_0 = lepolys[_] - lepolys[_+2] phi_x = D@phi_0 diffusion = -epsilon*(4*_+6)*(-1)*alphas[_] denom = lepolys[N]**2 convection = np.sum(u**2*phi_x/(N*(N+1))/denom) LHS += diffusion - convection RHS += np.sum(2*f*phi_0/(N*(N+1))/denom) else: u, f, alphas, params = generate(x, D, a, b, lepolys, epsilon, equation, sd, forcing) data.append([u, f, alphas, params, epsilon]) return data x = sem.legslbndm(N+1) D = sem.legslbdiff(N+1, x) lepolys = gen_lepolys(N, x) # pprint(lepolys) if equation == 'Standard2D': lepolysx = gen_lepolysx(N, x, lepolys) # pprint(lepolysx) data = create_fast(N, epsilon, size, eps_flag, equation, sd, forcing) data = np.array(data, dtype=object) def save_obj(data, name, equation, kind): cwd = os.getcwd() path = os.path.join(cwd,'data', equation, kind)
def create_fast(N:int, epsilon:float, size:int, eps_flag=False):
def func(t: float) -> float:
# Random force
m = 2*np.random.rand(4) - 1
f = m[0]*np.sin(m[1]*np.pi*t) + m[2]*np.cos(m[3]*np.pi*t)
return f, m
def gen_lepolys(N, x):
lepolys = {}
for i in range(N+3):
lepolys[i] = sem.lepoly(i, x)
return lepolys
def generate(x, D, a, b, lepolys, epsilon):
f, params = func(x)
s_diag = np.zeros((N-1,1))
M = np.zeros((N-1,N-1))
for ii in range(1, N):
k = ii - 1
s_diag[ii-1] = -(4*k+6)*b
phi_k_M = D@(lepolys[k] + a*lepolys[k+1] + b*lepolys[k+2])
for jj in range(1,N):
if abs(ii-jj) <=2:
l = jj-1
psi_l_M = lepolys[l] + a*lepolys[l+1] + b*lepolys[l+2]
M[jj-1,ii-1] = np.sum((psi_l_M*phi_k_M)*2/(N*(N+1))/(lepolys[N]**2))
S = s_diag*np.eye(N-1)
g = np.zeros((N+1,))
for i in range(1,N+1):
k = i-1
g[i-1] = (2*k+1)/(N*(N+1))*np.sum(f*(lepolys[k])/(lepolys[N]**2))
g[N-1] = 1/(N+1)*np.sum(f/lepolys[N])
bar_f = np.zeros((N-1,))
for i in range(1,N):
k = i-1
bar_f[i-1] = g[i-1]/(k+1/2) + a*g[i]/(k+3/2) + b*g[i+1]/(k+5/2)
Mass = epsilon*S-M
u = np.linalg.solve(Mass, bar_f)
alphas = np.copy(u)
g[0], g[1] = u[0], u[1] + a*u[0]
for i in range(3, N):
k = i - 1
g[i-1] = u[i-1] + a*u[i-2] + b*u[i-3]
g[N-1] = a*u[N-2] + b*u[N-3]
g[N] = b*u[N-2]
u = np.zeros((N+1,))
for i in range(1,N+2):
_ = 0
for j in range(1, N+2):
k = j-1
L = lepolys[k]
_ += g[j-1]*L[i-1]
_ = _[0]
u[i-1] = _
return u, f, alphas, params
def loop(N, epsilon, size, lepolys, eps_flag):
if eps_flag == True:
epsilons = np.random.uniform(1E0, 1E-6, SIZE)
data = []
U, F, ALPHAS, PARAMS = [], [], [], []
for n in tqdm(range(size)):
if eps_flag == True:
epsilon = epsilons[n]
u, f, alphas, params = generate(x, D, a, b, lepolys, epsilon)
data.append([u, f, alphas, params, epsilon])
return data
x = sem.legslbndm(N+1)
D = sem.legslbdiff(N+1, x)
a, b = 0, -1
lepolys = gen_lepolys(N, x)
return loop(N, epsilon, size, lepolys, eps_flag)
def create_fast(N: int,
epsilon: float,
size: int,
eps_flag=False,
equation='Standard',
sd=1,
forcing='uniform'):
def func(x: np.ndarray, equation: str, sd: float,
forcing: str) -> np.ndarray:
if forcing == 'uniform':
m = 3 + 2 * np.random.rand(2)
n = np.pi * (1 + 2 * np.random.rand(2))
f = m[0] * np.sin(n[0] * x) + m[1] * np.cos(n[1] * x)
m = np.array([m[0], m[1], n[0], n[1]])
elif forcing == 'normal':
m = np.random.normal(0, sd, 4)
f = m[0] * np.sin(m[1] * np.pi * x) + m[2] * np.cos(
m[3] * np.pi * x)
return f, m
def gen_lepolys(N, x):
lepolys = {}
for i in range(N + 3):
lepolys[i] = sem.lepoly(i, x)
return lepolys
def generate(x, D, a, b, lepolys, epsilon, equation, sd, forcing):
f, params = func(x, equation, sd, forcing)
s_diag = np.zeros((N - 1, 1))
if equation == 'Standard':
M = np.zeros((N - 1, N - 1))
for ii in range(1, N):
k = ii - 1
s_diag[ii - 1] = -(4 * k + 6) * b
phi_k_M = D @ (lepolys[k] + a * lepolys[k + 1] +
b * lepolys[k + 2])
for jj in range(1, N):
if np.abs(ii - jj) <= 2:
l = jj - 1
psi_l_M = lepolys[l] + a * lepolys[
l + 1] + b * lepolys[l + 2]
M[jj - 1,
ii - 1] = np.sum((psi_l_M * phi_k_M) * 2 /
(N * (N + 1)) / (lepolys[N]**2))
S = s_diag * np.eye(N - 1)
g = np.zeros((N + 1, ))
for i in range(1, N + 1):
k = i - 1
g[i - 1] = (2 * k + 1) / (N *
(N + 1)) * np.sum(f * (lepolys[k]) /
(lepolys[N]**2))
g[N - 1] = 1 / (N + 1) * np.sum(f / lepolys[N])
bar_f = np.zeros((N - 1, ))
for i in range(1, N):
k = i - 1
bar_f[i - 1] = g[i - 1] / (k + 1 / 2) + a * g[i] / (
k + 3 / 2) + b * g[i + 1] / (k + 5 / 2)
Mass = epsilon * S - M
u = np.linalg.solve(Mass, bar_f)
alphas = np.copy(u)
g[0], g[1] = u[0], u[1] + a * u[0]
for i in range(3, N):
k = i - 1
g[i - 1] = u[i - 1] + a * u[i - 2] + b * u[i - 3]
g[N - 1] = a * u[N - 2] + b * u[N - 3]
g[N] = b * u[N - 2]
u = np.zeros((N + 1, ))
for i in range(1, N + 2):
_ = 0
for j in range(1, N + 2):
k = j - 1
L = lepolys[k]
_ += g[j - 1] * L[i - 1]
u[i - 1] = _[0]
elif equation == 'Burgers':
for ii in range(1, N):
k = ii - 1
s_diag[k] = -(4 * k + 6) * b
S = s_diag * np.eye(N - 1)
Mass = epsilon * S
error, tolerance, u_old, force = 1, 1E-9, 0 * f.copy(), f.copy()
iterations = 0
while error > tolerance:
f_ = force - u_old * (D @ u_old)
g = np.zeros((N + 1, ))
for i in range(1, N + 1):
k = i - 1
g[k] = (2 * k + 1) / (N *
(N + 1)) * np.sum(f_ * (lepolys[k]) /
(lepolys[N]**2))
g[N - 1] = 1 / (N + 1) * np.sum(f_ / lepolys[N])
bar_f = np.zeros((N - 1, ))
for i in range(1, N):
k = i - 1
bar_f[k] = g[k] / (k + 1 / 2) + a * g[k + 1] / (
k + 3 / 2) + b * g[k + 2] / (k + 5 / 2)
alphas = np.linalg.solve(Mass, bar_f)
u_sol = np.zeros((N + 1, 1))
for ij in range(1, N):
i_ind = ij - 1
u_sol += alphas[i_ind] * (lepolys[i_ind] +
a * lepolys[i_ind + 1] +
b * lepolys[i_ind + 2])
error = np.max(u_sol - u_old)
u_old = u_sol
iterations += 1
u = u_sol
elif equation == 'BurgersT':
M = np.zeros((N - 1, N - 1))
tol, T, dt = 1E-9, 5E-4, 1E-4
t_f = int(T / dt)
u_pre, u_ans, f_ans, alphas_ans = np.sin(np.pi * x), [], [], []
for ii in range(1, N):
k = ii - 1
s_diag[k] = -(4 * k + 6) * b
phi_k_M = lepolys[k] + a * lepolys[k + 1] + b * lepolys[k + 2]
for jj in range(1, N):
if np.abs(ii - jj) <= 2:
l = jj - 1
psi_l_M = lepolys[l] + a * lepolys[
l + 1] + b * lepolys[l + 2]
entry = psi_l_M * phi_k_M * 2 / (N *
(N + 1)) / (lepolys[N]
**2)
M[l, k] = np.sum(entry)
S = s_diag * np.eye(N - 1)
Mass = epsilon * S + (1 / dt) * M
for t_idx in np.linspace(1, t_f, t_f, endpoint=True):
error, tolerance, u_old, force = 1, tol, u_pre, np.cos(
t_idx * dt) * f
iterations = 0
while error > tolerance:
f_ = force - u_old * (D @ u_old) + (1 / dt) * u_pre
g = np.zeros((N + 1, ))
for i in range(1, N + 1):
k = i - 1
g[k] = (2 * k + 1) / (N * (N + 1)) * np.sum(
f_ * (lepolys[k]) / (lepolys[N]**2))
g[N - 1] = 1 / (N + 1) * np.sum(f_ / lepolys[N])
bar_f = np.zeros((N - 1, ))
for i in range(1, N):
k = i - 1
bar_f[k] = g[k] / (k + 1 / 2) + a * g[k + 1] / (
k + 3 / 2) + b * g[k + 2] / (k + 5 / 2)
alphas = np.linalg.solve(Mass, bar_f)
u_sol = np.zeros((N + 1, 1))
for ij in range(1, N):
i_ind = ij - 1
u_sol += alphas[i_ind] * (lepolys[i_ind] +
a * lepolys[i_ind + 1] +
b * lepolys[i_ind + 2])
error = np.max(u_sol - u_old)
u_old = u_sol.copy()
iterations += 1
u_ans.append(u_sol)
f_ans.append(force)
alphas_ans.append(alphas)
u_pre = u_sol
u, f, alphas = u_ans, f_ans, alphas_ans
elif equation == 'Helmholtz':
ku = 3.5
M = np.zeros((N - 1, N - 1))
for ii in range(1, N):
k = ii - 1
s_diag[k] = -(4 * k + 6) * b[k]
phi_k_M = lepolys[k] + a[k] * lepolys[k + 1] + b[k] * lepolys[
k + 2]
for jj in range(1, N):
if np.abs(ii - jj) <= 2:
l = jj - 1
psi_l_M = lepolys[l] + a[l] * lepolys[
l + 1] + b[l] * lepolys[l + 2]
entry = psi_l_M * phi_k_M * 2 / (N *
(N + 1)) / (lepolys[N]
**2)
M[l, k] = np.sum(entry)
S = s_diag * np.eye(N - 1)
g = np.zeros((N + 1, ))
for i in range(1, N + 1):
k = i - 1
g[k] = (2 * k + 1) / (N * (N + 1)) * np.sum(f * (lepolys[k]) /
(lepolys[N]**2))
g[N] = 1 / (N + 1) * np.sum(f / lepolys[N])
bar_f = np.zeros((N - 1, ))
for i in range(1, N):
k = i - 1
bar_f[k] = g[k] / (k + 1 / 2) + a[k] * g[k + 1] / (
k + 3 / 2) + b[k] * g[k + 2] / (k + 5 / 2)
Mass = -S + ku * M
alphas = np.linalg.solve(Mass, bar_f)
u = np.zeros((N + 1, 1))
for ij in range(1, N):
i_ind = ij - 1
u += alphas[i_ind] * (lepolys[i_ind] +
a[i_ind] * lepolys[i_ind + 1] +
b[i_ind] * lepolys[i_ind + 2])
return u, f, alphas, params
def loop(N, epsilon, size, lepolys, eps_flag, equation, a, b, forcing):
if eps_flag == True:
epsilons = np.random.uniform(1E0, 1E-6, SIZE)
data = []
U, F, ALPHAS, PARAMS = [], [], [], []
for n in tqdm(range(size)):
if eps_flag == True:
epsilon = epsilons[n]
if equation == 'BurgersT':
u, f, alphas, params = generate(x, D, a, b, lepolys, epsilon,
equation, sd, forcing)
for i, u_ in enumerate(u):
if i < len(u):
data.append([u[i], f[i], alphas[i], params, epsilon])
elif equation == 'Burgers':
u, f, alphas, params = generate(x, D, a, b, lepolys, epsilon,
equation, sd, forcing)
LHS, RHS = 0, 0
for _ in range(10):
phi_0 = lepolys[_] - lepolys[_ + 2]
phi_x = D @ phi_0
diffusion = -epsilon * (4 * _ + 6) * (-1) * alphas[_]
denom = lepolys[N]**2
convection = np.sum(u**2 * phi_x / (N * (N + 1)) / denom)
LHS += diffusion - convection
RHS += np.sum(2 * f * phi_0 / (N * (N + 1)) / denom)
while np.abs(LHS - RHS) > 1E-5 and np.linalg.norm(
u, ord=2) < 1E-2:
u, f, alphas, params = generate(x, D, a, b, lepolys,
epsilon, equation, sd,
forcing)
LHS, RHS = 0, 0
for _ in range(10):
phi_0 = lepolys[_] - lepolys[_ + 2]
phi_x = D @ phi_0
diffusion = -epsilon * (4 * _ + 6) * (-1) * alphas[_]
denom = lepolys[N]**2
convection = np.sum(u**2 * phi_x / (N * (N + 1)) /
denom)
LHS += diffusion - convection
RHS += np.sum(2 * f * phi_0 / (N * (N + 1)) / denom)
else:
u, f, alphas, params = generate(x, D, a, b, lepolys, epsilon,
equation, sd, forcing)
data.append([u, f, alphas, params, epsilon])
return data
x = sem.legslbndm(N + 1)
D = sem.legslbdiff(N + 1, x)
lepolys = gen_lepolys(N, x)
if equation == 'Helmholtz':
a, b = np.zeros((N + 1, )), np.zeros((N + 1, ))
for i in range(1, N + 2):
k = i - 1
b[k] = -k * (k + 1) / ((k + 2) * (k + 3))
else:
a, b = 0, -1
return loop(N, epsilon, size, lepolys, eps_flag, equation, a, b, forcing)
z,
cmap=cm.coolwarm,
linewidth=0,
antialiased=False)
plt.xlim(-1, 1)
plt.ylim(-1, 1)
plt.grid(alpha=0.618)
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.show()
# exit()
if __name__ == '__main__':
start = time()
x = legslbndm(N + 1)
y = x.copy()
D = legslbdiff(N + 1, x)
D2 = D @ D
D2 = D2[1:-1, 1:-1]
lepolys = gen_lepolys(N, x)
lepolysx = gen_lepolysx(N, x, lepolys)
I = np.eye(N - 1)
L = np.kron(I, D2) + np.kron(D2, I)
x1, y1 = np.meshgrid(x, x)
f = f2D(x1, y1)
f_ = f[1:-1, 1:-1]
f_ = f_.ravel()
u = np.linalg.solve(-L, f_)
xx, yy = np.meshgrid(x[1:-1], x[1:-1])
uu = np.zeros_like(x1)
def plotter(xx, sample, epoch, a=None, u=None, DE=None, title='alpha', ks=7):
def relative_l2(measured, theoretical):
return np.linalg.norm(measured - theoretical, ord=2) / np.linalg.norm(
theoretical, ord=2)
def relative_linf(measured, theoretical):
return np.linalg.norm(measured - theoretical,
ord=np.inf) / np.linalg.norm(theoretical,
ord=np.inf)
def mae(measured, theoretical):
return np.linalg.norm(measured - theoretical, ord=1) / len(theoretical)
aa = sample['a'][0, 0, :].to('cpu').detach().numpy()
uu = sample['u'][0, 0, :].to('cpu').detach().numpy()
ff = sample['f'][0, 0, :].to('cpu').detach().numpy()
x_ = legslbndm(len(xx) - 2)
xxx = np.linspace(-1, 1, len(ff), endpoint=True)
if a is not None:
ahat = a[0, :].to('cpu').detach().numpy()
mae_error_a = mae(ahat, aa)
l2_error_a = relative_l2(ahat, aa)
linf_error_a = relative_linf(ahat, aa)
plt.figure(1, figsize=(10, 6))
plt.title(f'Alphas Example Epoch {epoch}\n'\
f'Alphas MAE Error: {np.round(mae_error_a, 6)}\n'\
f'Alphas Rel. $L_2$ Error: {np.round(float(l2_error_a), 6)}\n'\
f'Alphas Rel. $L_\\infty$ Error: {np.round(float(linf_error_a), 6)}')
plt.plot(x_, aa, 'r-', mfc='none', label='$\\alpha$')
plt.plot(x_, ahat, 'bo', mfc='none', label='$\\hat{\\alpha}$')
# plt.plot(xxx, ff, 'g-', label='$f$')
plt.xlim(-1, 1)
plt.grid(alpha=0.618)
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.legend(shadow=True)
plt.savefig(
f'./pics/{title}_ks{ks}_epoch{str(epoch).zfill(5)}_alphas.png',
bbox_inches='tight')
plt.close(1)
if u is not None:
uhat = u[0, :].to('cpu').detach().numpy()
mae_error_u = mae(uhat, uu)
l2_error_u = relative_l2(uhat, uu)
linf_error_u = relative_linf(uhat, uu)
plt.figure(2, figsize=(10, 6))
plt.title(f'Reconstruction Example Epoch {epoch}\n'\
f'Reconstruction MAE Error: {np.round(mae_error_u, 6)}\n'\
f'Reconstruction Rel. $L_2$ Error: {np.round(float(l2_error_u), 6)}\n'\
f'Reconstruction Rel. $L_\\infty$ Error: {np.round(float(linf_error_u), 6)}')
plt.plot(xx, uu, 'r-', mfc='none', label='$u$')
plt.plot(xx, uhat.T, 'bo', mfc='none', label='$\\hat{u}$')
plt.xlim(-1, 1)
plt.grid(alpha=0.618)
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.legend(shadow=True)
plt.savefig(
f'./pics/{title}_ks{ks}_epoch{str(epoch).zfill(5)}_reconstruction.png',
bbox_inches='tight')
# plt.show()
plt.close(2)
if DE is not None:
de = DE[0, :].to('cpu').detach().numpy()
plt.figure(3, figsize=(10, 6))
mae_error_de = mae(de, ff)
l2_error_de = relative_l2(de, ff)
linf_error_de = relative_linf(de, ff)
plt.title(f'DE Example Epoch {epoch}\n'\
f'DE MAE Error: {np.round(mae_error_de, 6)}\n'\
f'DE Rel. $L_2$ Error: {np.round(float(l2_error_de), 6)}\n'\
f'DE Rel. $L_\\infty$ Error: {np.round(float(linf_error_de), 6)}')
plt.plot(xxx, ff, 'g-', label='$f$')
plt.plot(xxx, de, 'co', mfc='none', label='ODE')
plt.xlim(-1, 1)
plt.grid(alpha=0.618)
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.legend(shadow=True)
plt.savefig(f'./pics/{title}_ks{ks}_epoch{str(epoch).zfill(5)}_DE.png',
bbox_inches='tight')
# plt.show()
plt.close(3)