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Add layer to perform RBF interpolation in reduced order modeling (#315)

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* add RBF implementation in pytorch (in layers)
* add POD-RBF example (baseline for nonintrusive closure)
* Add POD only and POD+RBF implementation
* add POD-RBF in tutorial 8
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Anna Ivagnes
2024-08-12 14:46:22 +02:00
committed by GitHub
parent 16261c9baf
commit d4ced3a7d7
14 changed files with 1111 additions and 224 deletions
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tutorials/tutorial8/tutorial.ipynb vendored
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tutorials/tutorial8/tutorial.py vendored
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@@ -1,20 +1,20 @@
#!/usr/bin/env python
# coding: utf-8
# # Tutorial: Reduced order model (PODNN) for parametric problems
# # Tutorial: Reduced order model (POD-RBF and POD-NN) for parametric problems
# The tutorial aims to show how to employ the **PINA** library in order to apply a reduced order modeling technique [1]. Such methodologies have several similarities with machine learning approaches, since the main goal consists of predicting the solution of differential equations (typically parametric PDEs) in a real-time fashion.
#
# In particular we are going to use the Proper Orthogonal Decomposition with Neural Network (PODNN) [2], which basically perform a dimensional reduction using the POD approach, approximating the parametric solution manifold (at the reduced space) using a NN. In this example, we use a simple multilayer perceptron, but the plenty of different archiutectures can be plugged as well.
#
# The tutorial aims to show how to employ the **PINA** library in order to apply a reduced order modeling technique [1]. Such methodologies have several similarities with machine learning approaches, since the main goal consists in predicting the solution of differential equations (typically parametric PDEs) in a real-time fashion.
#
# In particular we are going to use the Proper Orthogonal Decomposition with either Radial Basis Function Interpolation(POD-RBF) or Neural Network (POD-NN) [2]. Here we basically perform a dimensional reduction using the POD approach, and approximating the parametric solution manifold (at the reduced space) using an interpolation (RBF) or a regression technique (NN). In this example, we use a simple multilayer perceptron, but the plenty of different architectures can be plugged as well.
#
# #### References
# 1. Rozza G., Stabile G., Ballarin F. (2022). Advanced Reduced Order Methods and Applications in Computational Fluid Dynamics, Society for Industrial and Applied Mathematics.
# 1. Rozza G., Stabile G., Ballarin F. (2022). Advanced Reduced Order Methods and Applications in Computational Fluid Dynamics, Society for Industrial and Applied Mathematics.
# 2. Hesthaven, J. S., & Ubbiali, S. (2018). Non-intrusive reduced order modeling of nonlinear problems using neural networks. Journal of Computational Physics, 363, 55-78.
# Let's start with the necessary imports.
# It's important to note the minimum PINA version to run this tutorial is the `0.1`.
# In[29]:
# In[1]:
get_ipython().run_line_magic('matplotlib', 'inline')
@@ -26,7 +26,7 @@ import pina
from pina.geometry import CartesianDomain
from pina.problem import ParametricProblem
from pina.model.layers import PODBlock
from pina.model.layers import PODBlock, RBFBlock
from pina import Condition, LabelTensor, Trainer
from pina.model import FeedForward
from pina.solvers import SupervisedSolver
@@ -36,10 +36,10 @@ print(f'We are using PINA version {pina.__version__}')
# We exploit the [Smithers](www.github.com/mathLab/Smithers) library to collect the parametric snapshots. In particular, we use the `NavierStokesDataset` class that contains a set of parametric solutions of the Navier-Stokes equations in a 2D L-shape domain. The parameter is the inflow velocity.
# The dataset is composed by 500 snapshots of the velocity (along $x$, $y$, and the magnitude) and pressure fields, and the corresponding parameter values.
#
# To visually check the snapshots, let's plot also the data points and the reference solution: this is the expected output of the neural network.
#
# To visually check the snapshots, let's plot also the data points and the reference solution: this is the expected output of our model.
# In[30]:
# In[2]:
from smithers.dataset import NavierStokesDataset
@@ -51,10 +51,10 @@ for ax, p, u in zip(axs, dataset.params[:4], dataset.snapshots['mag(v)'][:4]):
ax.set_title(f'$\mu$ = {p[0]:.2f}')
# The *snapshots* - aka the numerical solutions computed for several parameters - and the corresponding parameters are the only data we need to train the model, in order to predict for any new test parameter the solution.
# The *snapshots* - aka the numerical solutions computed for several parameters - and the corresponding parameters are the only data we need to train the model, in order to predict the solution for any new test parameter.
# To properly validate the accuracy, we initially split the 500 snapshots into the training dataset (90% of the original data) and the testing one (the reamining 10%). It must be said that, to plug the snapshots into **PINA**, we have to cast them to `LabelTensor` objects.
# In[31]:
# In[3]:
u = torch.tensor(dataset.snapshots['mag(v)']).float()
@@ -73,7 +73,7 @@ p_train, p_test = p[:n_train], p[n_train:]
# It is now time to define the problem! We inherit from `ParametricProblem` (since the space invariant typically of this methodology), just defining a simple *input-output* condition.
# In[32]:
# In[4]:
class SnapshotProblem(ParametricProblem):
@@ -81,13 +81,85 @@ class SnapshotProblem(ParametricProblem):
parameter_domain = CartesianDomain({'mu': [0, 100]})
conditions = {
'io': Condition(input_points=p, output_points=u)
'io': Condition(input_points=p_train, output_points=u_train)
}
poisson_problem = SnapshotProblem()
# Then, we define the model we want to use: basically we have a MLP architecture that takes in input the parameter and return the *modal coefficients*, so the reduced dimension representation (the coordinates in the POD space). Such latent variable is the projected to the original space using the POD modes, which are computed and stored in the `PODBlock` object.
# In[33]:
# We can then build a `PODRBF` model (using a Radial Basis Function interpolation as approximation) and a `PODNN` approach (using an MLP architecture as approximation).
# ## POD-RBF reduced order model
# Then, we define the model we want to use, with the POD (`PODBlock`) and the RBF (`RBFBlock`) objects.
# In[5]:
class PODRBF(torch.nn.Module):
"""
Proper orthogonal decomposition with Radial Basis Function interpolation model.
"""
def __init__(self, pod_rank, rbf_kernel):
"""
"""
super().__init__()
self.pod = PODBlock(pod_rank)
self.rbf = RBFBlock(kernel=rbf_kernel)
def forward(self, x):
"""
Defines the computation performed at every call.
:param x: The tensor to apply the forward pass.
:type x: torch.Tensor
:return: the output computed by the model.
:rtype: torch.Tensor
"""
coefficents = self.rbf(x)
return self.pod.expand(coefficents)
def fit(self, p, x):
"""
Call the :meth:`pina.model.layers.PODBlock.fit` method of the
:attr:`pina.model.layers.PODBlock` attribute to perform the POD,
and the :meth:`pina.model.layers.RBFBlock.fit` method of the
:attr:`pina.model.layers.RBFBlock` attribute to fit the interpolation.
"""
self.pod.fit(x)
self.rbf.fit(p, self.pod.reduce(x))
# We can then fit the model and ask it to predict the required field for unseen values of the parameters. Note that this model does not need a `Trainer` since it does not include any neural network or learnable parameters.
# In[6]:
pod_rbf = PODRBF(pod_rank=20, rbf_kernel='thin_plate_spline')
pod_rbf.fit(p_train, u_train)
# In[7]:
u_test_rbf = pod_rbf(p_test)
u_train_rbf = pod_rbf(p_train)
relative_error_train = torch.norm(u_train_rbf - u_train)/torch.norm(u_train)
relative_error_test = torch.norm(u_test_rbf - u_test)/torch.norm(u_test)
print('Error summary for POD-RBF model:')
print(f' Train: {relative_error_train.item():e}')
print(f' Test: {relative_error_test.item():e}')
# ## POD-NN reduced order model
# In[8]:
class PODNN(torch.nn.Module):
@@ -97,10 +169,10 @@ class PODNN(torch.nn.Module):
def __init__(self, pod_rank, layers, func):
"""
"""
super().__init__()
self.pod = PODBlock(pod_rank)
self.nn = FeedForward(
input_dimensions=1,
@@ -108,7 +180,7 @@ class PODNN(torch.nn.Module):
layers=layers,
func=func
)
def forward(self, x):
"""
@@ -130,30 +202,28 @@ class PODNN(torch.nn.Module):
self.pod.fit(x)
# We highlight that the POD modes are directly computed by means of the singular value decomposition (computed over the input data), and not trained using the back-propagation approach. Only the weights of the MLP are actually trained during the optimization loop.
# We highlight that the POD modes are directly computed by means of the singular value decomposition (computed over the input data), and not trained using the backpropagation approach. Only the weights of the MLP are actually trained during the optimization loop.
# In[34]:
# In[9]:
poisson_problem = SnapshotProblem()
pod_nn = PODNN(pod_rank=20, layers=[10, 10, 10], func=torch.nn.Tanh)
pod_nn.fit_pod(u)
pod_nn.fit_pod(u_train)
pinn_stokes = SupervisedSolver(
problem=poisson_problem,
model=pod_nn,
pod_nn_stokes = SupervisedSolver(
problem=poisson_problem,
model=pod_nn,
optimizer=torch.optim.Adam,
optimizer_kwargs={'lr': 0.0001})
# Now that we set the `Problem` and the `Model`, we have just to train the model and use it for predict the test snapshots.
# Now that we have set the `Problem` and the `Model`, we have just to train the model and use it for predicting the test snapshots.
# In[35]:
# In[10]:
trainer = Trainer(
solver=pinn_stokes,
solver=pod_nn_stokes,
max_epochs=1000,
batch_size=100,
log_every_n_steps=5,
@@ -161,47 +231,69 @@ trainer = Trainer(
trainer.train()
# Done! Now the computational expensive part is over, we can load in future the model to infer new parameters (simply loading the checkpoint file automatically created by `Lightning`) or test its performances. We measure the relative error for the training and test datasets, printing the mean one.
# Done! Now that the computational expensive part is over, we can load in future the model to infer new parameters (simply loading the checkpoint file automatically created by `Lightning`) or test its performances. We measure the relative error for the training and test datasets, printing the mean one.
# In[36]:
# In[11]:
u_test_pred = pinn_stokes(p_test)
u_train_pred = pinn_stokes(p_train)
u_test_nn = pod_nn_stokes(p_test)
u_train_nn = pod_nn_stokes(p_train)
relative_error_train = torch.norm(u_train_pred - u_train)/torch.norm(u_train)
relative_error_test = torch.norm(u_test_pred - u_test)/torch.norm(u_test)
relative_error_train = torch.norm(u_train_nn - u_train)/torch.norm(u_train)
relative_error_test = torch.norm(u_test_nn - u_test)/torch.norm(u_test)
print('Error summary:')
print('Error summary for POD-NN model:')
print(f' Train: {relative_error_train.item():e}')
print(f' Test: {relative_error_test.item():e}')
# We can of course also plot the solutions predicted by the `PODNN` model, comparing them to the original ones. We can note here some differences, especially for low velocities, but improvements can be accomplished thanks to longer training.
# ## POD-RBF vs POD-NN
# In[37]:
# We can of course also plot the solutions predicted by the `PODRBF` and by the `PODNN` model, comparing them to the original ones. We can note here, in the `PODNN` model and for low velocities, some differences, but improvements can be accomplished thanks to longer training.
# In[12]:
idx = torch.randint(0, len(u_test_pred), (4,))
u_idx = pinn_stokes(p_test[idx])
idx = torch.randint(0, len(u_test), (4,))
u_idx_rbf = pod_rbf(p_test[idx])
u_idx_nn = pod_nn_stokes(p_test[idx])
import numpy as np
import matplotlib
fig, axs = plt.subplots(3, 4, figsize=(14, 9))
import matplotlib.pyplot as plt
relative_error = np.abs(u_test[idx] - u_idx.detach())
relative_error = np.where(u_test[idx] < 1e-7, 1e-7, relative_error/u_test[idx])
for i, (idx_, u_, err_) in enumerate(zip(idx, u_idx, relative_error)):
cm = axs[0, i].tricontourf(dataset.triang, u_.detach())
fig, axs = plt.subplots(5, 4, figsize=(14, 9))
relative_error_rbf = np.abs(u_test[idx] - u_idx_rbf.detach())
relative_error_rbf = np.where(u_test[idx] < 1e-7, 1e-7, relative_error_rbf/u_test[idx])
relative_error_nn = np.abs(u_test[idx] - u_idx_nn.detach())
relative_error_nn = np.where(u_test[idx] < 1e-7, 1e-7, relative_error_nn/u_test[idx])
for i, (idx_, rbf_, nn_, rbf_err_, nn_err_) in enumerate(
zip(idx, u_idx_rbf, u_idx_nn, relative_error_rbf, relative_error_nn)):
axs[0, i].set_title(f'$\mu$ = {p_test[idx_].item():.2f}')
plt.colorbar(cm)
cm = axs[1, i].tricontourf(dataset.triang, u_test[idx_].flatten())
plt.colorbar(cm)
cm = axs[0, i].tricontourf(dataset.triang, rbf_.detach()) # POD-RBF prediction
plt.colorbar(cm, ax=axs[0, i])
cm = axs[2, i].tripcolor(dataset.triang, err_, norm=matplotlib.colors.LogNorm())
plt.colorbar(cm)
cm = axs[1, i].tricontourf(dataset.triang, nn_.detach()) # POD-NN prediction
plt.colorbar(cm, ax=axs[1, i])
cm = axs[2, i].tricontourf(dataset.triang, u_test[idx_].flatten()) # Truth
plt.colorbar(cm, ax=axs[2, i])
cm = axs[3, i].tripcolor(dataset.triang, rbf_err_, norm=matplotlib.colors.LogNorm()) # Error for POD-RBF
plt.colorbar(cm, ax=axs[3, i])
cm = axs[4, i].tripcolor(dataset.triang, nn_err_, norm=matplotlib.colors.LogNorm()) # Error for POD-NN
plt.colorbar(cm, ax=axs[4, i])
plt.show()
# In[ ]: