In this notebook, we show how to train a GP regression model in GPyTorch of a 2-dimensional function given function values and derivative observations. We consider modeling the Franke function where the values and derivatives are contaminated with independent $\mathcal{N}(0, 0.5)$ distributed noise.
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import torch
import gpytorch
import math
from matplotlib import cm
from matplotlib import pyplot as plt
import numpy as np
%matplotlib inline
%load_ext autoreload
%autoreload 2
The following is a vectorized implementation of the 2-dimensional Franke function (https://www.sfu.ca/~ssurjano/franke2d.html)
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def franke(X, Y):
term1 = .75*torch.exp(-((9*X - 2).pow(2) + (9*Y - 2).pow(2))/4)
term2 = .75*torch.exp(-((9*X + 1).pow(2))/49 - (9*Y + 1)/10)
term3 = .5*torch.exp(-((9*X - 7).pow(2) + (9*Y - 3).pow(2))/4)
term4 = .2*torch.exp(-(9*X - 4).pow(2) - (9*Y - 7).pow(2))
f = term1 + term2 + term3 - term4
dfx = -2*(9*X - 2)*9/4 * term1 - 2*(9*X + 1)*9/49 * term2 + \
-2*(9*X - 7)*9/4 * term3 + 2*(9*X - 4)*9 * term4
dfy = -2*(9*Y - 2)*9/4 * term1 - 9/10 * term2 + \
-2*(9*Y - 3)*9/4 * term3 + 2*(9*Y - 7)*9 * term4
return f, dfx, dfy
In [3]:
xv, yv = torch.meshgrid([torch.linspace(0, 1, 10), torch.linspace(0, 1, 10)])
train_x = torch.cat((
xv.contiguous().view(xv.numel(), 1),
yv.contiguous().view(yv.numel(), 1)),
dim=1
)
f, dfx, dfy = franke(train_x[:, 0], train_x[:, 1])
train_y = torch.stack([f, dfx, dfy], -1).squeeze(1)
train_y += 0.05 * torch.randn(train_y.size())
A GP prior on the function values implies a multi-output GP prior on the function values and the partial derivatives, see 9.4 in http://www.gaussianprocess.org/gpml/chapters/RW9.pdf for more details. This allows using a MultitaskMultivariateNormal and MultitaskGaussianLikelihood to train a GP model from both function values and gradients. The resulting RBF kernel that models the covariance between the values and partial derivatives has been implemented in RBFKernelGrad and the extension of a constant mean is implemented in ConstantMeanGrad.
The RBFKernelGrad is generally worse conditioned than the RBFKernel, so we place a lower bound on the noise parameter to keep the smallest eigenvalues of the kernel matrix away from zero.
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class GPModelWithDerivatives(gpytorch.models.ExactGP):
def __init__(self, train_x, train_y, likelihood):
super(GPModelWithDerivatives, self).__init__(train_x, train_y, likelihood)
self.mean_module = gpytorch.means.ConstantMeanGrad()
self.base_kernel = gpytorch.kernels.RBFKernelGrad()
self.covar_module = gpytorch.kernels.ScaleKernel(self.base_kernel)
def forward(self, x):
mean_x = self.mean_module(x)
covar_x = self.covar_module(x)
return gpytorch.distributions.MultitaskMultivariateNormal(mean_x, covar_x)
likelihood = gpytorch.likelihoods.MultitaskGaussianLikelihood(num_tasks=3) # Value + x-derivative + y-derivative
likelihood.initialize(noise=0.01*train_y.std()) # Require 1% noise (approximately)
likelihood.raw_noise.requires_grad = False
model = GPModelWithDerivatives(train_x, train_y, likelihood)
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# Find optimal model hyperparameters
model.train()
likelihood.train()
# Use the adam optimizer
optimizer = torch.optim.Adam([
{'params': model.parameters()}, # Includes GaussianLikelihood parameters
], lr=0.1)
# "Loss" for GPs - the marginal log likelihood
mll = gpytorch.mlls.ExactMarginalLogLikelihood(likelihood, model)
n_iter = 50
for i in range(n_iter):
optimizer.zero_grad()
output = model(train_x)
loss = -mll(output, train_y)
loss.backward()
print('Iter %d/%d - Loss: %.3f lengthscale: %.3f noise: %.3f' % (
i + 1, n_iter, loss.item(),
model.covar_module.base_kernel.lengthscale.item(),
model.likelihood.noise.item()
))
optimizer.step()
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# Set into eval mode
model.eval()
likelihood.eval()
# Initialize plots
fig, ax = plt.subplots(2, 3, figsize=(14, 10))
# Test points
n1, n2 = 50, 50
xv, yv = torch.meshgrid([torch.linspace(0, 1, n1), torch.linspace(0, 1, n2)])
f, dfx, dfy = franke(xv, yv)
# Make predictions
with torch.no_grad(), gpytorch.settings.max_cg_iterations(50):
test_x = torch.stack([xv.reshape(n1*n2, 1), yv.reshape(n1*n2, 1)], -1).squeeze(1)
predictions = likelihood(model(test_x))
mean = predictions.mean
extent = (xv.min(), xv.max(), yv.max(), yv.min())
ax[0, 0].imshow(f, extent=extent, cmap=cm.jet)
ax[0, 0].set_title('True values')
ax[0, 1].imshow(dfx, extent=extent, cmap=cm.jet)
ax[0, 1].set_title('True x-derivatives')
ax[0, 2].imshow(dfy, extent=extent, cmap=cm.jet)
ax[0, 2].set_title('True y-derivatives')
ax[1, 0].imshow(mean[:, 0].detach().numpy().reshape(n1, n2), extent=extent, cmap=cm.jet)
ax[1, 0].set_title('Predicted values')
ax[1, 1].imshow(mean[:, 1].detach().numpy().reshape(n1, n2), extent=extent, cmap=cm.jet)
ax[1, 1].set_title('Predicted x-derivatives')
ax[1, 2].imshow(mean[:, 2].detach().numpy().reshape(n1, n2), extent=extent, cmap=cm.jet)
ax[1, 2].set_title('Predicted y-derivatives')
None
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