Build and query a simple reduced order model

In this tutorial we will show the typical workflow for the construcion of the Reduced Order Model based only on the outputs of the higher-order model. In detail, we consider here a POD-RBF framework (Proper Orthogonal Decomposition for dimensionality reduction and Radial Basis Function for manifold approximation), but the tutorial can be easily extended to other methods thanks to the modularity nature of EZyRB.

We consider a parametric steady heat conduction problem in a two-dimensional domain \(\Omega\). While in this tutorial we are going to focus on the data-driven approach, the same problem can be tackled in an intrusive manner (with the Reduced Basis method) using the RBniCS, as demonstrated in this RBniCS tutorial. This book is therefore exhaustively discussed in the book Certified reduced basis methods for parametrized partial differential equations, J.S. Hesthaven, G. Rozza, B. Stamm, 2016, Springer. An additional description is available also at `https://rbnics.gitlab.io/RBniCS-jupyter/tutorial_thermal_block.html <>`__.

Since the good documentation already available for this problem and since the data-driven methodologies we will take into consideration, we just summarize the model to allow a better understanding.

The domain is depicted below:

where: - the first parameter \(\mu_o\) controls the conductivity in the circular subdomain \(\Omega_0\); - the second parameter \(\mu_1\) controls the flux over \(\Gamma_\text{base}\).

Initial setting

First of all import the required packages: we need the standard Numpy and Matplotlib, and some classes from EZyRB. In the EZyRB framework, we need three main ingredients to construct a reduced order model: - an initial database where the snapshots are stored; - a reduction method to reduce the dimensionality of the system, in this tutorial we will use the proper orthogonal decomposition (POD) method; - an approximation method to extrapolate the parametric solution for new parameters, in this tutorial we will use a radial basis function (RBF) interpolation.

!pip install ezyrb datasets

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[1m[[0m[34;49mnotice[0m[1;39;49m][0m[39;49m A new release of pip is available: [0m[31;49m24.3.1[0m[39;49m -> [0m[32;49m25.3[0m
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import numpy as np
import matplotlib.tri as mtri
import matplotlib.pyplot as plt
from ezyrb import POD, RBF, Database
from ezyrb import ReducedOrderModel as ROM
%matplotlib inline

Offline phase

In the offline phase, we need some samples of the parametric high-fidelity model. In this case, we extract 8 snapshots from the numerical model implemented in FEniCS, and we import them and the related parameters.

from datasets import load_dataset
data_path = "kshitij-pandey/termal_dataset"
snapshots_hf = load_dataset(data_path, "snapshots", split="train")
param_hf = load_dataset(data_path, "params", split="train")
triangles_hf    = load_dataset(data_path, "triangles", split="train")
coords_hf = load_dataset(data_path, "coords",     split="train")


# convert the dict files into numpy

import pandas as pd

def hf_to_numpy(ds):
    return ds.to_pandas().to_numpy()


snapshots = hf_to_numpy(snapshots_hf)
param = hf_to_numpy(param_hf)
triangles = hf_to_numpy(triangles_hf)
coords = hf_to_numpy(coords_hf)
print(snapshots.shape, param.shape)

(8, 304) (8, 2)

Moreover, to visualize the solution (both the higher-order one and the reduced one), we import also the mesh information to be able to create the triangulation. We underline this additional step is related only to plotting purpose, and not mandatory for the reduced space generation.

x, y  = coords
from matplotlib.tri import Triangulation
triang = Triangulation(x, y, triangles)
triang = triang

For the sake of clarity the snapshots are plotted.

fig, ax = plt.subplots(nrows=2, ncols=4, figsize=(16, 6), sharey=True, sharex=True)
ax = ax.flatten()
for i in range(8):
    ax[i].triplot(triang, 'b-', lw=0.1)
    cm = ax[i].tripcolor(triang, snapshots[i])
    fig.colorbar(cm, ax=ax[i])
    ax[i].set_title('($\mu_0={:5.2f}, \mu_1={:5.2f})$'.format(*param[i]))

First of all, we create a Database object from the parameters and the snapshots.

db = Database(param, snapshots)

Then we need a reduction object. In this case we use the proper orthogonal decomposition so we create a POD object. We use here all the default parameters, but for the complete list of available arguments we refer to original documentation of POD class.

pod = POD('svd')

Then we instantiate the RBF class for interpolating the solution manifold. Also in this case, RBF documentation is the perfect starting point to explore such class.

rbf = RBF()

Few lines of code and our reduced model is created! To complete everything, we create the ReducedOrderModel (aliased to ROM in this tutorial) object by passing the already created objects. For clarity, we puntualize that we need to pass the instances and not the classes. Simply changing such line (with different objects) allows to test different frameworks in a very modular way. The fit() function computes the reduced model, meaning that the original snapshots in the database are projected onto the POD space and the RBF interpolator is created.

rom = ROM(db, pod, rbf)
rom.fit();

Online phase

In the online phase we can query our model in order to predict the solution for a new parameter \(\mu_\text{new}\) that is not in the training set. We just need to pass the new parameters as input of the predict() function.

new_mu = [8,   1]
pred_sol = rom.predict(new_mu)

We can so plot the predicted solution for a fixed parameter…

plt.figure(figsize=(7, 5))
plt.triplot(triang, 'b-', lw=0.1)
plt.tripcolor(triang, *pred_sol)
plt.colorbar();

Error Approximation & Improvement

At the moment, we used a database which is composed by 8 files. we would have an idea of the approximation accuracy we are able to reach with these high-fidelity solutions. Using the leave-one-out strategy, an error is computed for each parametric point in our database and these values are returned as array.

for pt, error in zip(rom.database.parameters_matrix, rom.loo_error()):
    print(pt, error)

[ 0.5 -0.2] 0.3830555986412087
[8.6 0.1] 0.5972596749801533
[5.3 0.8] 0.8082744257222089
[9.4 0.1] 0.4105803285232253
[ 7.3 -0.8] 0.5505863544054451
[0.2 0.8] 0.07567485849711765
[ 3.5 -0.5] 0.66949247698686
[0.3 0.6] 0.06478619218562698

Moreover, we can use the information about the errors to locate the parametric points where we have to compute the new high-fidelity solutions and add these to the database in order to optimally improve the accuracy.

rom.optimal_mu()

array([[ 5.2487694 , -0.06339911]])

These function can be used to achieve the wanted (estimated) accuracy.