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1d_basis.py
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1d_basis.py
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"""
Function: data-driven reveal SDE by the weak form of FKE: 1. linear regression 2. adversarial update
@author pi square
@email: [email protected]
created in Oct 11, 2021
update log:
0. Oct 11, 2021: created
1. Oct 15, 2021: change time index loop to Tensor form including the time index as the first dimension
2. Oct 28, 2021: fix bugs of self.t_number -> self.bash_size
3. Dec 9, 2021: modifies the adversarial.py to weak_gaussian_sampling for sampling in the test function
"""
import torch
import torch.nn as nn
import numpy as np
from collections import OrderedDict
from data.GenerateData_fun import DataSet
import time
import utils
import scipy.io
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
class Gaussian(torch.nn.Module):
def __init__(self, mu, sigma):
super(Gaussian, self).__init__()
self.mu = mu
self.sigma = sigma
def gaussB(self, x):
func = 1/(self.sigma*torch.sqrt(2*torch.tensor(torch.pi))) * torch.exp(-0.5*(x-self.mu)**2/self.sigma**2)
return func
def gaussZero(self, x):
func = 1
for d in range(x.shape[2]):
func = func * self.gaussB(x[:, :, d])
return func
def gaussFirst(self, x, g0):
func = torch.zeros([x.shape[0], x.shape[1], x.shape[2]])
for k in range(x.shape[2]):
func[:, :, k] = -(x[:, :, k] - self.mu)/self.sigma**2 * g0
return func
def gaussSecond(self, x, g0):
func = torch.zeros([x.shape[0], x.shape[1], x.shape[2], x.shape[2]])
for k in range(x.shape[2]):
for j in range(x.shape[2]):
if k == j:
func[:, :, k, j] = (
-1/self.sigma**2 + (-(x[:, :, k]-self.mu)/self.sigma**2)
* (-(x[:, :, j]-self.mu)/self.sigma**2)
) * g0
else:
func[:, :, k, j] = (-(x[:, :, k]-self.mu)/self.sigma**2)*(
-(x[:, :, j]-self.mu)/self.sigma**2
) * g0
return func
def forward(self, x, diff_order=0):
g0 = self.gaussZero(x)
if diff_order == 0:
return g0
elif diff_order == 1:
return self.gaussFirst(x, g0)
elif diff_order == 2:
return self.gaussSecond(x, g0)
else:
raise RuntimeError("higher order derivatives of the gaussian has not bee implemented!")
class Model(object):
"""A ``Model`` solve the true coefficients of the basis on the data by the outloop for linear regression and
and the inner loop of increasing the parameters in the test function TestNet.
Args:
t : `` t'' vector read from the file
data: ``data`` matrix read from the file.
testFunc: ``DNN`` instance.
"""
def __init__(self, t, data, testFunc):
self.t = t
self.itmax = len(t)
self.data = data
self.net = testFunc
self.basis = None # given by build_basis
self.A = None # given by build_A
self.b = None # given by build_b
self.dimension = None
self.basis_number = None
self.basis_order = None
self.bash_size = data.shape[1]
self.zeta = None # coefficients of the unknown function
self.error_tolerance = None
self.max_iter = None
self.loss = None
# self.batch_size = None # tbd
# self.train_state = TrainState() # tbd
# self.losshistory = LossHistory() # tbd
def _get_data_t(self, it):
X = self.data[it,:,:]
return X
@utils.timing # decorator
@torch.no_grad()
def build_basis(self): # \Lambda matrix
"""build the basis list for the different time snapshot
"""
self.t_number = len(self.t)
self.basis_number = (self.polynomial_order+1) * (self.cos_order+1) * (self.exp_order+1)
basis = []
for it in range(self.t_number):
X = self._get_data_t(it)
basis_count = 0
Theta = torch.zeros(X.size(0), self.basis_number)
for i in range(self.polynomial_order+1):
for j in range(self.cos_order+1):
for k in range(self.exp_order+1):
Theta[:, basis_count] = X[:, 0]**i * torch.cos(j*X[:, 0]) * torch.exp(k*X[:, 0])
basis_count = basis_count + 1
# print("basis_count:", basis_count, "basis_number:", self.basis_number)
assert basis_count == self.basis_number
basis.append(Theta)
# print("X", X)
# print("theta", Theta.shape)
self.basis = torch.stack(basis)
print("self.basis.shape", self.basis.shape)
def computeLoss(self):
return (torch.matmul(self.A, torch.tensor(self.zeta).to(torch.float).unsqueeze(-1))-self.b.unsqueeze(-1)).norm(2)
def computeTrueLoss(self):
return (torch.matmul(self.A, self.zeta_true)-self.b.unsqueeze(-1)).norm(2)
def computeAb(self, gauss):
H_number = self.dimension * self.basis_number
F_number = self.dimension * self.dimension #* self.basis_number
A = torch.zeros([self.t_number, H_number+F_number])
rb = torch.zeros(self.t_number)
b = torch.zeros(self.t_number)
# ##########################################################
# Tensor form of computing A and b for parallel computing
# ##########################################################
TX = self.data
TX.requires_grad = True
# Phi = self.net(TX)
gauss0 = gauss(TX, diff_order=0)
gauss1 = gauss(TX, diff_order=1)
gauss2 = gauss(TX, diff_order=2)
# print("self.basis_number", self.basis_number)
# print("self.dimension", self.dimension)
for kd in range(self.dimension):
for jb in range(self.basis_number):
# print("gauss1[:, :, %s]" % kd, gauss1[:, :, kd].size())
H = 1/self.bash_size * torch.sum(
gauss1[:, :, kd]
*
self.basis[:, :, jb], dim=1
)
A[:, kd*self.basis_number+jb] = H
# compute A by F_lkj
for ld in range(self.dimension):
for kd in range(self.dimension):
F = 1/self.bash_size * torch.sum(
gauss2[:, :, ld, kd], dim=1
)
A[:, H_number] = F
rb = 1/self.bash_size * torch.sum(gauss0, dim=1).squeeze()
dt = (torch.max(self.t)-torch.min(self.t)) / (self.t_number - 1)
# print("b", rb)
# b = torch.tensor(torch.enable_grad()(utils.compute_b)(rb, dt, time_diff='Tik'))
# print("b.shape", b.shape)
# plt.clf()
# plt.plot(rb.detach().numpy(),'-*')
# plt.plot(b.detach().numpy(),'-o')
# plt.draw()
# plt.pause(1)
# print("b", b)
# print("A.shape", A.shape)
if self.type == 'PDEFind':
b = torch.tensor(torch.enable_grad()(utils.compute_b)(rb, dt, time_diff='Tik'))
return A, b
if self.type == 'LMM_2':
AA = torch.ones(A.size(0) - 1, A.size(1))
for i in range(AA.size(0)):
AA[i, :] = (A[i, :] + A[i + 1, :]) / 2
bb = torch.from_numpy(np.diff(rb.numpy())) / dt
return AA, bb
if self.type == 'LMM_3':
AA = torch.ones(A.size(0) - 2, A.size(1))
bb = torch.ones(A.size(0) - 2)
for i in range(AA.size(0)):
AA[i, :] = (A[i, :] + 4*A[i + 1, :] + A[i + 2, :]) * dt / 3
bb[i] = rb[i + 2] - rb[i]
return AA, bb
if self.type == 'LMM_6':
AA = torch.ones(A.size(0) - 5, A.size(1))
bb = torch.ones(A.size(0) - 5)
for i in range(AA.size(0)):
AA[i, :] = (
A[i+1, :] +
1/2 * (A[i+2, :] + A[i+1, :]) +
5/12 * A[i+3, :] + 8/12 * A[i+2, :] - 1/12 * A[i+1, :] +
9/24 * A[i+4, :] + 19/24 * A[i+3, :] - 5/24 * A[i+2, :] + 1/24 * A[i+1, :] +
251/720 * A[i + 5, :] + 646/720 * A[i + 4, :] - 264/720 * A[i + 3, :] + 106/720 * A[i + 2, :] - 19/720 * A[i + 1, :]
) * dt
bb[i] = rb[i + 5] - rb[i]
return AA, bb
if self.type == 'bdf2':
AA = torch.ones(A.size(0) - 2, A.size(1))
bb = torch.ones(A.size(0) - 2)
for i in range(AA.size(0)):
AA[i, :] = (A[i, :] + 4*A[i + 1, :] + A[i + 2, :]) * dt / 3
bb[i] = rb[i + 2] - rb[i]
return AA, bb
if self.type == 'LMM_2_nonequal':
AA = torch.ones(A.size(0) - 1, A.size(1))
bb = torch.ones(A.size(0) - 1)
ht = torch.from_numpy(np.diff(self.t.numpy()))
for i in range(AA.size(0)):
AA[i, :] = (A[i, :] + A[i + 1, :]) / 2 * ht[i]
bb[i] = rb[i + 1] - rb[i]
return AA, bb
if self.type == 'non-equal3':
AA = torch.ones(A.size(0) - 2, A.size(1))
bb = torch.ones(A.size(0) - 2)
ht = torch.from_numpy(np.diff(self.t.numpy()))
# print("ht: ", ht)
wt = torch.tensor([ht[i + 1] / ht[i] for i in range(ht.size(0) - 1)])
# print("wt: ", wt)
for i in range(AA.size(0)):
print("ht[i + 1]", ht[i + 1], "wt[i]", wt[i])
AA[i, :] = ht[i + 1] * (1 + wt[i]) / (1 + 2 * wt[i]) * A[i + 2, :]
bb[i] = rb[i + 2] - (1 + wt[i]) ** 2 / (1 + 2 * wt[i]) * rb[i + 1] + wt[i] ** 2 / (1 + 2 * wt[i]) * rb[i]
return AA, bb
if self.type == 'non-equal-adams':
AA = torch.ones(A.size(0) - 2, A.size(1))
bb = torch.ones(A.size(0) - 2)
ht = torch.from_numpy(np.diff(self.t.numpy()))
# print("ht: ", ht)
wt = torch.tensor([ht[i + 1] / ht[i] for i in range(ht.size(0) - 1)])
# print("wt: ", wt)
for i in range(AA.size(0)):
AA[i, :] = ht[i + 1] / (6 * (1 + wt[i])) * (
(3 + 2 * wt[i]) * A[i + 2, :]
+ (3 + wt[i]) * (1 + wt[i]) * A[i + 1, :]
- wt[i] ** 2 * A[i, :])
bb[i] = rb[i + 2] - rb[i + 1]
return AA, bb
def sampleTestFunc(self, samp_number):
# for i in range(self.sampling_number):
if self.gauss_samp_way == 'lhs':
mu_list = self.lhs_ratio * torch.rand(samp_number)*(self.data.max()-self.data.min()) + self.data.min()
if self.gauss_samp_way == 'SDE':
if samp_number <= self.bash_size:
index = np.arange(self.bash_size)
np.random.shuffle(index)
mu_list = data[-1, index[0: samp_number], :]
# print("mu_list", mu_list)
sigma_list = torch.ones(samp_number)*self.variance
return mu_list, sigma_list
def buildLinearSystem(self, samp_number):
mu_list, sigma_list = self.sampleTestFunc(samp_number)
A_list = []
b_list = []
for i in range(mu_list.shape[0]):
mu = mu_list[i]
sigma = sigma_list[i]
gauss = self.net(mu, sigma)
A, b = self.computeAb(gauss)
A_list.append(A)
b_list.append(b)
# print("A_list", A_list)
# print("b_list", b_list)
self.A = torch.cat(A_list, dim=0) # 2-dimension
self.b = torch.cat(b_list, dim=0).unsqueeze(-1) # 1-dimension
@utils.timing
def solveLinearRegress(self):
self.zeta = torch.tensor(np.linalg.lstsq(self.A.detach().numpy(), self.b.detach().numpy())[0])
# TBD sparse regression
@utils.timing
def STRidge(self, X0, y, lam, maxit, tol, normalize = 0, print_results = False):
"""
Sequential Threshold Ridge Regression algorithm for finding (hopefully) sparse
approximation to X^{-1}y. The idea is that this may do better with correlated observables.
This assumes y is only one column
"""
n,d = X0.shape
X = np.zeros((n,d), dtype=np.complex64)
# First normalize data
if normalize != 0:
Mreg = np.zeros((d,1))
for i in range(0,d):
Mreg[i] = 1.0/(np.linalg.norm(X0[:,i],normalize))
X[:,i] = Mreg[i]*X0[:,i]
else: X = X0
# Get the standard ridge esitmate
if lam != 0: w = np.linalg.lstsq(X.T.dot(X) + lam*np.eye(d),X.T.dot(y),rcond=None)[0]
else: w = np.linalg.lstsq(X,y,rcond=None)[0]
num_relevant = d
biginds = np.where(abs(w) > tol)[0]
# Threshold and continue
for j in range(maxit):
# Figure out which items to cut out
smallinds = np.where(abs(w) < tol)[0]
print("STRidge_j: ", j)
print("smallinds", smallinds)
new_biginds = [i for i in range(d) if i not in smallinds]
# If nothing changes then stop
if num_relevant == len(new_biginds):
print("here1")
break
else: num_relevant = len(new_biginds)
# Also make sure we didn't just lose all the coefficients
if len(new_biginds) == 0:
if j == 0:
print("here2")
#if print_results: print "Tolerance too high - all coefficients set below tolerance"
return w
else:
print("here3")
break
biginds = new_biginds
# Otherwise get a new guess
w[smallinds] = 0
if lam != 0: w[biginds] = np.linalg.lstsq(X[:, biginds].T.dot(X[:, biginds]) + lam*np.eye(len(biginds)),X[:, biginds].T.dot(y),rcond=None)[0]
else: w[biginds] = np.linalg.lstsq(X[:, biginds],y,rcond=None)[0]
# Now that we have the sparsity pattern, use standard least squares to get w
if biginds != []: w[biginds] = np.linalg.lstsq(X[:, biginds],y,rcond=None)[0]
if normalize != 0: return np.multiply(Mreg,w)
else: return w
@utils.timing
def compile(self, basis_order, gauss_variance, type, drift_term, diffusion_term, gauss_samp_way, lhs_ratio):
self.dimension = self.data.shape[-1]
self.polynomial_order = basis_order[0]
self.cos_order = basis_order[1]
self.exp_order = basis_order[2]
self.build_basis()
self.variance = gauss_variance
self.type = type
self.drift = drift_term
self.diffusion = diffusion_term
self.gauss_samp_way = gauss_samp_way
self.lhs_ratio = lhs_ratio if self.gauss_samp_way == 'lhs' else 1
@utils.timing
@torch.no_grad()
def train(self, gauss_samp_number, lam, STRidge_threshold):
self.buildLinearSystem(samp_number=gauss_samp_number)
print("A: ", self.A.size(), "b: ", self.b.size())
self.zeta = torch.tensor(self.STRidge(self.A.detach().numpy(), self.b.detach().numpy(), lam, 100, STRidge_threshold)).to(torch.float)
# pinv_norm = np.linalg.norm(np.linalg.pinv(self.A.detach().numpy()), ord=np.inf)
# print("The pinv_norm of A is:", pinv_norm)
print("zeta: ", self.zeta.squeeze()[:-1].view(self.polynomial_order+1, self.cos_order+1, self.exp_order+1))
self.zeta[-1] = torch.sqrt(self.zeta[-1]*2)
print("Diffusion term: ", self.zeta[-1])
if __name__ == '__main__':
np.random.seed(100)
torch.manual_seed(100)
dt = 0.0001
t = torch.tensor([0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1])
# t = torch.arange(0, 10.1, 0.1)
def drift(x):
return x + torch.cos(3*x)
def diffusion(x):
return 1
samples = 10000
dataset = DataSet(t, dt=dt, samples_num=samples, dim=1, drift_fun=drift, diffusion_fun=diffusion,
initialization=torch.normal(mean=0., std=0.1, size=(samples, 1)), explosion_prevention=False)
data = dataset.get_data(plot_hist=False)
print("data: ", data.shape, data.max(), data.min())
testFunc = Gaussian
model = Model(t, data, testFunc)
model.compile(basis_order=[3, 3, 0], gauss_variance=1, type='LMM_2_nonequal', drift_term=drift, diffusion_term=diffusion,
gauss_samp_way='lhs', lhs_ratio=0.91)
model.train(gauss_samp_number=20, lam=0.0, STRidge_threshold=0.2)
# drift=x - cosx - e^x, samples=10000, gauss=50, lam=0.01, STRidge=0.2, order=[3, 3, 3]