Tutorial: Reduced order model (POD-RBF or POD-NN) for parametric problems ========================================================================= |Open In Colab| .. |Open In Colab| image:: https://colab.research.google.com/assets/colab-badge.svg :target: https://colab.research.google.com/github/mathLab/PINA/blob/master/tutorials/tutorial8/tutorial.ipynb 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. 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``. .. code:: ipython3 ## routine needed to run the notebook on Google Colab try: import google.colab IN_COLAB = True except: IN_COLAB = False if IN_COLAB: !pip install "pina-mathlab" %matplotlib inline import matplotlib.pyplot as plt import torch import pina from pina.geometry import CartesianDomain from pina.problem import ParametricProblem from pina.model.layers import PODBlock, RBFBlock from pina import Condition, LabelTensor, Trainer from pina.model import FeedForward from pina.solvers import SupervisedSolver print(f'We are using PINA version {pina.__version__}') .. parsed-literal:: We are using PINA version 0.1.1 We exploit the `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 :math:`x`, :math:`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 our model. .. code:: ipython3 from smithers.dataset import NavierStokesDataset dataset = NavierStokesDataset() fig, axs = plt.subplots(1, 4, figsize=(14, 3)) for ax, p, u in zip(axs, dataset.params[:4], dataset.snapshots['mag(v)'][:4]): ax.tricontourf(dataset.triang, u, levels=16) ax.set_title(f'$\mu$ = {p[0]:.2f}') .. image:: tutorial_files/tutorial_5_0.png 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. .. code:: ipython3 u = torch.tensor(dataset.snapshots['mag(v)']).float() p = torch.tensor(dataset.params).float() p = LabelTensor(p, labels=['mu']) u = LabelTensor(u, labels=[f's{i}' for i in range(u.shape[1])]) ratio_train_test = 0.9 n = u.shape n_train = int(u.shape[0] * ratio_train_test) n_test = u - n_train u_train, u_test = u[:n_train], u[n_train:] 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. .. code:: ipython3 class SnapshotProblem(ParametricProblem): output_variables = [f's{i}' for i in range(u.shape[1])] parameter_domain = CartesianDomain({'mu': [0, 100]}) conditions = { 'io': Condition(input_points=p_train, output_points=u_train) } poisson_problem = SnapshotProblem() 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. .. code:: ipython3 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. .. code:: ipython3 pod_rbf = PODRBF(pod_rank=20, rbf_kernel='thin_plate_spline') pod_rbf.fit(p_train, u_train) .. code:: ipython3 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}') .. parsed-literal:: Error summary for POD-RBF model: Train: 1.287801e-03 Test: 1.217041e-03 POD-NN reduced order model -------------------------- .. code:: ipython3 class PODNN(torch.nn.Module): """ Proper orthogonal decomposition with neural network model. """ def __init__(self, pod_rank, layers, func): """ """ super().__init__() self.pod = PODBlock(pod_rank) self.nn = FeedForward( input_dimensions=1, output_dimensions=pod_rank, layers=layers, func=func ) 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.nn(x) return self.pod.expand(coefficents) def fit_pod(self, x): """ Just call the :meth:`pina.model.layers.PODBlock.fit` method of the :attr:`pina.model.layers.PODBlock` attribute. """ 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 backpropagation approach. Only the weights of the MLP are actually trained during the optimization loop. .. code:: ipython3 pod_nn = PODNN(pod_rank=20, layers=[10, 10, 10], func=torch.nn.Tanh) pod_nn.fit_pod(u_train) pod_nn_stokes = SupervisedSolver( problem=poisson_problem, model=pod_nn, optimizer=torch.optim.Adam, optimizer_kwargs={'lr': 0.0001}) 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. .. code:: ipython3 trainer = Trainer( solver=pod_nn_stokes, max_epochs=1000, batch_size=100, log_every_n_steps=5, accelerator='cpu') trainer.train() .. parsed-literal:: GPU available: True (cuda), used: False TPU available: False, using: 0 TPU cores IPU available: False, using: 0 IPUs HPU available: False, using: 0 HPUs /u/a/aivagnes/anaconda3/lib/python3.8/site-packages/pytorch_lightning/trainer/setup.py:187: GPU available but not used. You can set it by doing `Trainer(accelerator='gpu')`. | Name | Type | Params ---------------------------------------- 0 | _loss | MSELoss | 0 1 | _neural_net | Network | 460 ---------------------------------------- 460 Trainable params 0 Non-trainable params 460 Total params 0.002 Total estimated model params size (MB) /u/a/aivagnes/anaconda3/lib/python3.8/site-packages/torch/cuda/__init__.py:152: UserWarning: Found GPU0 Quadro K600 which is of cuda capability 3.0. PyTorch no longer supports this GPU because it is too old. The minimum cuda capability supported by this library is 3.7. warnings.warn(old_gpu_warn % (d, name, major, minor, min_arch // 10, min_arch % 10)) .. parsed-literal:: Training: | | 0/? [00:00