def f(t, y):
dx = 1
y_x = diff(y, dx, endpoints=True)
y_xx = diff2(y, dx, endpoints=True)
y2_x = diff(y**2, dx, endpoints=True)
return np.array([y, y**2, y**3, y_x, y_xx, y_x * y, y_x * y_x, y_x * y_xx
]).reshape(-1, len(y)).T @ xi
def lib_at_c(I, c):
dx = 1
dy = 1
I_x = diff(I, dx, axis=0, endpoints=True)
I_y = diff(I, dy, axis=1, endpoints=True)
I_xx = diff2(I, dx, axis=0, endpoints=True)
I_xy = pdiff(I, [dx, dy], axis=[0, 1], endpoints=True)
I_yy = diff2(I, dy, axis=1, endpoints=True)
return (np.array([I, I_x, I_y, I_xx, I_xy, I_yy]).reshape(-1, M * N).T @ c)
def f(t, y):
dx = 1
dy = 1
I = np.reshape(y, (M, N))
I_x = diff(I, dx, axis=0, endpoints=True)
I_y = diff(I, dx, axis=1, endpoints=True)
I_xx = diff2(I, dx, axis=0, endpoints=True)
I_xy = pdiff(I, [dx, dx], axis=[0, 1], endpoints=True)
I_yy = diff2(I, dx, axis=1, endpoints=True)
return np.array([I, I_x, I_y, I_xx, I_xy, I_yy]).reshape(-1, M * N).T @ c
def diff_model(t_fine, P_, max_deg, num_refinements):
# compute time derivative of data
dt = t_fine[1] - t_fine[0]
dP = diff(P_, dt, endpoints=True, axis=1)
# make dictionary of polynomials in both population1
A = bin_exp(P_, max_deg)
weights = np.sum(A, axis=0)
# compute coefficeints
c = np.linalg.lstsq(A, dP.T)[0]
# c[:,0] = lasso(A,dP[0].T,alpha=100)
# c[:,1] = lasso(A,dP[1].T,alpha=100)
params = np.ones(np.shape(c))
# iteritively remove some functions from library
for i in range(2):
for j in range(num_refinements[i]):
# find nonzero entries
nz = np.nonzero(params[:, i])[0]
# find index of lowest weight of these entires
x = np.argmin(weights[nz] * c[nz, i])
# zero this index
params[nz[x], i] = 0
# compute new coefficeints
c[:, i] = np.linalg.lstsq(A * params[:, i], dP[i].T)[0]
# solve solution from given coefficeints
def rhs(t, xy):
return bin_exp(xy, max_deg) @ c
ivp = integrate.solve_ivp(rhs, [0, 60], P_[:, 0], dense_output=True)
return {
"lib": lambda xy: bin_exp(xy, max_deg) @ c,
"model": ivp.sol,
"num_params": np.sum(params, axis=0),
"num_data": len(t_fine)
}
from data_driven_modeling import diff, diff2, lasso
from scipy import io
burgers_mat = io.loadmat('burgers.mat')
#%%
x = burgers_mat['x'][0]
t = burgers_mat['t'][:, 0]
usol = burgers_mat['usol']
u = np.real(usol)
[m, n] = np.shape(u)
dt = t[1] - t[0]
u_t = diff(u, dt, axis=1, endpoints=True)
# %%
'''
D = np.diag(np.ones(m-1),1)-np.diag(np.ones(m-1),-1)
D[0,m-1] = -1
D[m-1,0] = 1
D /= 2*dx
D2 = -2*np.diag(np.ones(m))+np.diag(np.ones(m-1),1)+np.diag(np.ones(m-1),-1)
D2 /= (dx)**2
D2[m-1,0]=1;
D2[0,m-1]=1;
ux = D@X[:,1:-1]
plt.axis('image')
plt.savefig('img/BZ/slice_' + str(t) + '.pdf')
#%% TRY 1D SLICES
u = I[np.arange(130), np.arange(130)]
#u = BZ_tensor[np.arange(320,220,-1),np.arange(340,440),400:]
[M, T] = np.shape(u)
plt.figure()
plt.scatter(np.arange(M), u[:, 200])
plt.show()
dt = 1
u_t = diff(u, dt, axis=1, endpoints=True)
u_tt = diff2(u, dt, axis=1, endpoints=True)
dx = 1
u_x = diff(u, dx, endpoints=True)
u_xx = diff2(u, dx, endpoints=True)
u2_x = diff(u**2, dx, endpoints=True)
A = np.array([u, u**2, u**3, u_x, u_xx, u_x * u, u_x * u_x,
u_x * u_xx]).reshape(-1, M * T).T
#%% regress
xi = np.linalg.lstsq(A, u_t.reshape(-1))[0]
#%%