Feature/pc2

docs/code/surrogates/pce/plot_pce_euler_UQ.py

@@ -0,0 +1,293 @@
+"""
+
+Physics-Informed PCE: Uncertainty Quantification of Euler Beam
+======================================================================
+
+In this example, we use Physics-informed Polynomial Chaos Expansion for approximation and UQ of Euler beam equation.
+"""
+
+# %% md
+#
+# Import necessary libraries.
+
+# %%
+
+from UQpy.distributions import Uniform, JointIndependent
+from UQpy.surrogates import *
+
+# load PC^2
+from UQpy.surrogates.polynomial_chaos.polynomials.TotalDegreeBasis import TotalDegreeBasis
+from UQpy.surrogates.polynomial_chaos.physics_informed.ConstrainedPCE import ConstrainedPCE
+from UQpy.surrogates.polynomial_chaos.physics_informed.PdeData import PdeData
+from UQpy.surrogates.polynomial_chaos.physics_informed.PdePCE import PdePCE
+from UQpy.surrogates.polynomial_chaos.physics_informed.Utilities import *
+from UQpy.surrogates.polynomial_chaos.physics_informed.ReducedPCE import ReducedPCE
+
+
+# %% md
+#
+# We then define functions used for construction of Physically Constrained PCE (PC :math:`^2`)
+
+# %%
+
+# Definition of PDE/ODE
+def pde_func(standardized_sample, pce):
+    der_order = 4
+    deriv_pce = derivative_basis(standardized_sample, pce, derivative_order=der_order, leading_variable=0) * ((2 / 1) ** der_order)
+
+    pde_basis = deriv_pce
+
+    return pde_basis
+
+
+# Definition of the source term
+def pde_res(standardized_sample):
+    load_s = load(standardized_sample)
+
+    return -load_s[:, 0]
+
+
+def load(standardized_sample):
+    return const_load(standardized_sample)
+
+
+def const_load(standardized_sample):
+    l = (1 + (1 + standardized_sample[:, 1]) / 2).reshape(-1, 1)
+    return l
+
+
+# Definition of the function for sampling of boundary conditions
+def bc_sampling(nsim=1000):
+    # BC sampling
+
+    nsim_half = round(nsim / 2)
+    sample = np.zeros((nsim, 2))
+    real_ogrid_1d = ortho_grid(nsim_half, 1, 0.0, 1.0)[:, 0]
+
+    sample[:nsim_half, 0] = np.zeros(nsim_half)
+    sample[:nsim_half, 1] = real_ogrid_1d
+
+    sample[nsim_half:, 0] = np.ones(nsim_half)
+    sample[nsim_half:, 1] = real_ogrid_1d
+
+    return sample
+
+
+# define sampling and evaluation of BC for estimation of error
+def bc_res(nsim, pce):
+    physical_sample= np.zeros((2, 2))
+    physical_sample[1, 0] = 1
+    standardized_sample = polynomial_chaos.Polynomials.standardize_sample(physical_sample, pce.polynomial_basis.distributions)
+
+    der_order = 2
+    deriv_pce = np.sum(
+        derivative_basis(standardized_sample, pce, derivative_order=der_order, leading_variable=0) *
+        ((2 / 1) ** der_order) * np.array(pce.coefficients).T, axis=1)
+
+    return deriv_pce
+
+
+# Definition of the reference solution for an error estimation
+def ref_sol(physical_coordinate, q):
+    return (q + 1) * (-(physical_coordinate ** 4) / 24 + physical_coordinate ** 3 / 12 - physical_coordinate / 24)
+
+
+# %% md
+#
+# Beam equation is parametrized by one deterministic input variable (coordinate :math:`x`) and
+# random load intensity :math:`q`, both modeled as Uniform random variables.
+
+# number of input variables
+
+nrand = 1
+nvar = 1 + nrand
+
+#   definition of a joint probability distribution
+dist1 = Uniform(loc=0, scale=1)
+dist2 = Uniform(loc=0, scale=1)
+marg = [dist1, dist2]
+joint = JointIndependent(marginals=marg)
+
+# %% md
+#
+# The physical domain is defined by :math:`x \in [0,1]`
+
+
+# define geometry of physical space
+geometry_xmin = np.array([0])
+geometry_xmax = np.array([1])
+
+# %% md
+#
+# Boundary conditions are prescribed as
+#
+# .. math:: \frac{d^2 u(0,q)}{d x^2}=0, \qquad \frac{d^2 u(1,q)}{d x^2}=0, \qquad u(0,q)=0, \qquad  u(1,q)=0
+
+# number of BC samples
+nbc = 2 * 10
+
+# derivation orders of prescribed BCs
+der_orders = [0, 2]
+# normals associated to precribed BCs
+bc_normals = np.array([0, 0])
+# sampling of BC points
+
+bc_xtotal = bc_sampling(20)
+bc_ytotal = np.zeros(len(bc_xtotal))
+
+bc_x = [bc_xtotal, bc_xtotal]
+bc_y = [bc_ytotal, bc_ytotal]
+
+# %% md
+#
+# Here we construct an object containing general physical information (geometry, boundary conditions)
+pde_data = PdeData(geometry_xmax, geometry_xmin, der_orders, bc_normals, bc_x, bc_y)
+
+# %% md
+#
+# Further we construct an object containing PDE physical data and PC :math:`^2` definitions of PDE
+
+pde_pce = PdePCE(pde_data, pde_func, pde_res=pde_res, bc_res=bc_res)
+
+# %% md
+#
+# Get dirichlet boundary conditions from PDE data object that are further used for construction of initial PCE object.
+
+dirichlet_bc = pde_data.dirichlet
+x_train = dirichlet_bc[:, :-1]
+y_train = dirichlet_bc[:, -1]
+
+# %% md
+#
+# Finally, PC :math:`^2` is constructed using Karush-Kuhn-Tucker normal equations solved by ordinary least squares and
+# Least Angle Regression.
+
+least_squares = LeastSquareRegression()
+p = 9
+
+PCEorder = p
+polynomial_basis = TotalDegreeBasis(joint, PCEorder, hyperbolic=1)
+
+#  create initial PCE object containing basis, regression method and dirichlet BC
+initpce = PolynomialChaosExpansion(polynomial_basis=polynomial_basis, regression_method=least_squares)
+initpce.set_data(x_train, y_train)
+
+# construct a PC^2 object combining pde_data, pde_pce and initial PCE objects
+pcpc = ConstrainedPCE(pde_data, pde_pce, initpce)
+# get coefficients of PC^2 by least angle regression
+pcpc.lar()
+
+# get coefficients of PC^2 by ordinary least squares
+pcpc.ols()
+
+# evaluate errors of approximations
+real_ogrid = ortho_grid(100, nvar, 0.0, 1.0)
+yy_val_pce = pcpc.lar_pce.predict(real_ogrid).flatten()
+yy_val_pce_ols = pcpc.initial_pce.predict(real_ogrid).flatten()
+yy_val_true = ref_sol(real_ogrid[:, 0], real_ogrid[:, 1]).flatten()
+
+err = np.abs(yy_val_pce - yy_val_true)
+tot_err = np.sum(err)
+print('\nTotal approximation error by PC^2-LAR: ', tot_err)
+
+err_ols = np.abs(yy_val_pce_ols - yy_val_true)
+tot_err_ols = np.sum(err_ols)
+print('Total approximation error by PC^2-OLS: ', tot_err_ols)
+
+# %% md
+#
+# Once the PC :math:`^2` is constructed, we use ReducedPce class to filter out influence of deterministic variable in UQ
+
+reduced_pce = ReducedPCE(pcpc.lar_pce, n_deterministic=1)
+
+# %% md
+#
+# ReducedPce is used for estimation of local statistical moments and quantiles :math:`\pm2\sigma`
+coeff_res = []
+var_res = []
+mean_res = []
+vartot_res = []
+lower_quantiles_modes = []
+upper_quantiles_modes = []
+
+n_derivations = 4
+sigma_mult = 2
+beam_x = np.arange(0, 101) / 100
+
+for x in beam_x:
+    mean = np.zeros(n_derivations + 1)
+    var = np.zeros(n_derivations + 1)
+    variances = np.zeros((n_derivations + 1, nrand))
+    lq = np.zeros((1 + n_derivations, nrand))
+    uq = np.zeros((1 + n_derivations, nrand))
+    for d in range(1 + n_derivations):
+        if d == 0:
+            coeff = (reduced_pce.evaluate_coordinate(np.array(x), return_coefficients=True))
+        else:
+            coeff = reduced_pce.derive_coordinate(np.array(x), derivative_order=d, leading_variable=0,
+                                                  return_coefficients=True,
+                                                  derivative_multiplier=transformation_multiplier(pde_data, 0, d))
+        mean[d] = coeff[0]
+        var[d] = np.sum(coeff[1:] ** 2)
+        variances[d, :] = reduced_pce.variance_contributions(coeff)
+
+        for e in range(nrand):
+            lq[d, e] = mean[d] + sigma_mult * np.sqrt(np.sum(variances[d, :e + 1]))
+            uq[d, e] = mean[d] - sigma_mult * np.sqrt(np.sum(variances[d, :e + 1]))
+
+    lower_quantiles_modes.append(lq)
+    upper_quantiles_modes.append(uq)
+    var_res.append(variances)
+    mean_res.append(mean)
+    vartot_res.append(var)
+
+mean_res = np.array(mean_res)
+vartot_res = np.array(vartot_res)
+var_res = np.array(var_res)
+lower_quantiles_modes = np.array(lower_quantiles_modes)
+upper_quantiles_modes = np.array(upper_quantiles_modes)
+
+# %% md
+#
+# PC :math:`^2` and corresponding Reduced PCE can be further used for local UQ. Obtained results are depicted in figure.
+
+# plot results
+import matplotlib.pyplot as plt
+
+fig, ax = plt.subplots(1, 4, figsize=(14.5, 3))
+colors = ['black', 'blue', 'green', 'red']
+
+for d in range(2, 1 + n_derivations):
+    ax[d - 1].plot(beam_x, mean_res[:, d], color=colors[d - 1])
+    ax[d - 1].plot(beam_x, mean_res[:, d] + sigma_mult * np.sqrt(vartot_res[:, d]), color=colors[d - 1])
+    ax[d - 1].plot(beam_x, mean_res[:, d] - sigma_mult * np.sqrt(vartot_res[:, d]), color=colors[d - 1])
+
+    ax[d - 1].plot(beam_x, lower_quantiles_modes[:, d, 0], '--', alpha=0.7, color=colors[d - 1])
+    ax[d - 1].plot(beam_x, upper_quantiles_modes[:, d, 0], '--', alpha=0.7, color=colors[d - 1])
+    ax[d - 1].fill_between(beam_x, lower_quantiles_modes[:, d, 0], upper_quantiles_modes[:, d, 0],
+                           facecolor=colors[d - 1], alpha=0.05)
+
+    ax[d - 1].plot(beam_x, np.zeros(len(beam_x)), color='black')
+
+ax[0].plot(beam_x, mean_res[:, 0], color=colors[0])
+ax[0].plot(beam_x, mean_res[:, 0] + sigma_mult * np.sqrt(vartot_res[:, 0]), color=colors[0])
+ax[0].plot(beam_x, mean_res[:, 0] - sigma_mult * np.sqrt(vartot_res[:, 0]), color=colors[0])
+
+ax[0].plot(beam_x, lower_quantiles_modes[:, 0, 0], '--', alpha=0.7, color=colors[0])
+ax[0].plot(beam_x, upper_quantiles_modes[:, 0, 0], '--', alpha=0.7, color=colors[0])
+ax[0].fill_between(beam_x, lower_quantiles_modes[:, 0, 0], upper_quantiles_modes[:, 0, 0],
+                   facecolor=colors[0], alpha=0.05)
+
+ax[0].plot(beam_x, np.zeros(len(beam_x)), color='black')
+
+ax[3].invert_yaxis()
+ax[1].invert_yaxis()
+
+ax[3].set_title(r'Load $\frac{\partial^4 w}{\partial x^4}$', y=1.04)
+ax[2].set_title(r'Shear Force $\frac{\partial^3 w}{\partial x^3}$', y=1.04)
+ax[1].set_title(r'Bending Moment $\frac{\partial^2 w}{\partial x^2}$', y=1.04)
+ax[0].set_title(r'Deflection $w$', y=1.04)
+
+for axi in ax.flatten():
+    axi.set_xlabel(r'$x$')
+plt.show()

docs/code/surrogates/pce/plot_pce_sparsity_lars.py

@@ -237,7 +237,7 @@ def ishigami(xx):
 
 # %% md
 #
-# In case of high-dimensional input and/or high :math:P` it is also beneficial to reduce the TD basis set by hyperbolic
+# In case of high-dimensional input and/or high :math:`P` it is also beneficial to reduce the TD basis set by hyperbolic
 # truncation. The hyperbolic truncation reduces higher-order interaction terms in dependence to parameter :math:`q` in
 # interval :math:`(0,1)`. The set of multi indices :math:`\alpha` is reduced as follows:
 #