From 98f7694d6fd47b77cce47994f177146f4ab61146 Mon Sep 17 00:00:00 2001 From: Dario Coscia Date: Thu, 14 Mar 2024 19:16:48 +0100 Subject: [PATCH] tut10 --- docs/source/_rst/_tutorial.rst | 1 + .../_rst/tutorials/tutorial10/tutorial.rst | 347 +++++++++++++++ .../tutorial_files/tutorial_11_0.png | Bin 0 -> 276829 bytes .../tutorial_files/tutorial_5_0.png | Bin 0 -> 43679 bytes pina/model/avno.py | 76 ++-- tests/test_model/test_avno.py | 160 +++++-- tutorials/README.md | 1 + tutorials/tutorial10/dat/Data_KS.mat | Bin 0 -> 22528456 bytes tutorials/tutorial10/dat/Data_KS2.mat | Bin 0 -> 22528456 bytes tutorials/tutorial10/tutorial.ipynb | 412 ++++++++++++++++++ tutorials/tutorial10/tutorial.py | 252 +++++++++++ 11 files changed, 1171 insertions(+), 78 deletions(-) create mode 100644 docs/source/_rst/tutorials/tutorial10/tutorial.rst create mode 100644 docs/source/_rst/tutorials/tutorial10/tutorial_files/tutorial_11_0.png create mode 100644 docs/source/_rst/tutorials/tutorial10/tutorial_files/tutorial_5_0.png create mode 100644 tutorials/tutorial10/dat/Data_KS.mat create mode 100644 tutorials/tutorial10/dat/Data_KS2.mat create mode 100644 tutorials/tutorial10/tutorial.ipynb create mode 100644 tutorials/tutorial10/tutorial.py diff --git a/docs/source/_rst/_tutorial.rst b/docs/source/_rst/_tutorial.rst index 5bed177..ad9a986 100644 --- a/docs/source/_rst/_tutorial.rst +++ b/docs/source/_rst/_tutorial.rst @@ -30,6 +30,7 @@ Neural Operator Learning :titlesonly: Two dimensional Darcy flow using the Fourier Neural Operator + Time dependent Kuramoto Sivashinsky equation using the Averaging Neural Operator Supervised Learning ------------------- diff --git a/docs/source/_rst/tutorials/tutorial10/tutorial.rst b/docs/source/_rst/tutorials/tutorial10/tutorial.rst new file mode 100644 index 0000000..de0d6d5 --- /dev/null +++ b/docs/source/_rst/tutorials/tutorial10/tutorial.rst @@ -0,0 +1,347 @@ +Tutorial: Averaging Neural Operator for solving Kuramoto Sivashinsky equation +============================================================================= + +In this tutorial we will build a Neural Operator using the +``AveragingNeuralOperator`` model and the ``SupervisedSolver``. At the +end of the tutorial you will be able to train a Neural Operator for +learning the operator of time dependent PDEs. + +First of all, some useful imports. Note we use ``scipy`` for i/o +operations. + +.. code:: ipython3 + + import torch + import matplotlib.pyplot as plt + from scipy import io + from pina import Condition, LabelTensor + from pina.problem import AbstractProblem + from pina.model import AveragingNeuralOperator + from pina.solvers import SupervisedSolver + from pina.trainer import Trainer + +Data Generation +--------------- + +We will focus on solving a specific PDE, the **Kuramoto Sivashinsky** +(KS) equation. The KS PDE is a fourth-order nonlinear PDE with the +following form: + +.. math:: + + + \frac{\partial u}{\partial t}(x,t) = -u(x,t)\frac{\partial u}{\partial x}(x,t)- \frac{\partial^{4}u}{\partial x^{4}}(x,t) - \frac{\partial^{2}u}{\partial x^{2}}(x,t). + +In the above :math:`x\in \Omega=[0, 64]` represents a spatial location, +:math:`t\in\mathbb{T}=[0,50]` the time and :math:`u(x, t)` is the value +of the function :math:`u:\Omega \times\mathbb{T}\in\mathbb{R}`. We +indicate with :math:`\mathbb{U}` a suitable space for :math:`u`, i.e. we +have that the solution :math:`u\in\mathbb{U}`. + +We impose Dirichlet boundary conditions on the derivative of :math:`u` +on the border of the domain :math:`\partial \Omega` + +.. math:: + + + \frac{\partial u}{\partial x}(x,t)=0 \quad \forall (x,t)\in \partial \Omega\times\mathbb{T}. + + +Initial conditions are sampled from a distribution over truncated +Fourier series with random coefficients +:math:`\{A_k, \ell_k, \phi_k\}_k` as + +.. math:: + + + u(x,0) = \sum_{k=1}^N A_k \sin(2 \pi \ell_k x / L + \phi_k) \ , + +where :math:`A_k \in [-0.4, -0.3]`, :math:`\ell_k = 2`, +:math:`\phi_k = 2\pi \quad \forall k=1,\dots,N`. + +We have already generated some data for differenti initial conditions, +and our objective will be to build a Neural Operator that, given +:math:`u(x, t)` will output :math:`u(x, t+\delta)`, where :math:`\delta` +is a fixed time step. We will come back on the Neural Operator +architecture, for now we first need to import the data. + +**Note:** *The numerical integration is obtained by using pseudospectral +method for spatial derivative discratization and implicit Runge Kutta 5 +for temporal dynamics.* + +.. code:: ipython3 + + # load data + data=io.loadmat("dat/Data_KS.mat") + + # converting to label tensor + initial_cond_train = LabelTensor(torch.tensor(data['initial_cond_train'], dtype=torch.float), ['t','x','u0']) + initial_cond_test = LabelTensor(torch.tensor(data['initial_cond_test'], dtype=torch.float), ['t','x','u0']) + sol_train = LabelTensor(torch.tensor(data['sol_train'], dtype=torch.float), ['u']) + sol_test = LabelTensor(torch.tensor(data['sol_test'], dtype=torch.float), ['u']) + + print('Data Loaded') + print(f' shape initial condition: {initial_cond_train.shape}') + print(f' shape solution: {sol_train.shape}') + + +.. parsed-literal:: + + Data Loaded + shape initial condition: torch.Size([100, 12800, 3]) + shape solution: torch.Size([100, 12800, 1]) + + +The data are saved in the form ``B \times N \times D``, where ``B`` is +the batch_size (basically how many initial conditions we sample), ``N`` +the number of points in the mesh (which is the product of the +discretization in ``x`` timese the one in ``t``), and ``D`` the +dimension of the problem (in this case we have three variables +``[u, t, x]``). + +We are now going to plot some trajectories! + +.. code:: ipython3 + + # helper function + def plot_trajectory(coords, real, no_sol=None): + # find the x-t shapes + dim_x = len(torch.unique(coords.extract('x'))) + dim_t = len(torch.unique(coords.extract('t'))) + # if we don't have the Neural Operator solution we simply plot the real one + if no_sol is None: + fig, axs = plt.subplots(1, 1, figsize=(15, 5), sharex=True, sharey=True) + c = axs.imshow(real.reshape(dim_t, dim_x).T.detach(),extent=[0, 50, 0, 64], cmap='PuOr_r', aspect='auto') + axs.set_title('Real solution') + fig.colorbar(c, ax=axs) + axs.set_xlabel('t') + axs.set_ylabel('x') + # otherwise we plot the real one, the Neural Operator one, and their difference + else: + fig, axs = plt.subplots(1, 3, figsize=(15, 5), sharex=True, sharey=True) + axs[0].imshow(real.reshape(dim_t, dim_x).T.detach(),extent=[0, 50, 0, 64], cmap='PuOr_r', aspect='auto') + axs[0].set_title('Real solution') + axs[1].imshow(no_sol.reshape(dim_t, dim_x).T.detach(),extent=[0, 50, 0, 64], cmap='PuOr_r', aspect='auto') + axs[1].set_title('NO solution') + c = axs[2].imshow((real - no_sol).abs().reshape(dim_t, dim_x).T.detach(),extent=[0, 50, 0, 64], cmap='PuOr_r', aspect='auto') + axs[2].set_title('Absolute difference') + fig.colorbar(c, ax=axs.ravel().tolist()) + for ax in axs: + ax.set_xlabel('t') + ax.set_ylabel('x') + plt.show() + + # a sample trajectory (we use the sample 5, feel free to change) + sample_number = 20 + plot_trajectory(coords=initial_cond_train[sample_number].extract(['x', 't']), + real=sol_train[sample_number].extract('u')) + + + + +.. image:: tutorial_files/tutorial_5_0.png + + +As we can see, as the time progresses the solution becomes chaotic, +which makes it really hard to learn! We will now focus on building a +Neural Operator using the ``SupervisedSolver`` class to tackle the +problem. + +Averaging Neural Operator +------------------------- + +We will build a neural operator :math:`\texttt{NO}` which takes the +solution at time :math:`t=0` for any :math:`x\in\Omega`, the time +:math:`(t)` at which we want to compute the solution, and gives back the +solution to the KS equation :math:`u(x, t)`, mathematically: + +.. math:: + + + \texttt{NO}_\theta : \mathbb{U} \rightarrow \mathbb{U}, + +such that + +.. math:: + + + \texttt{NO}_\theta[u(t=0)](x, t) \rightarrow u(x, t). + +There are many ways on approximating the following operator, e.g. by 2D +`FNO `__ (for +regular meshes), a +`DeepOnet `__, +`Continuous Convolutional Neural +Operator `__, +`MIONet `__. In +this tutorial we will use the *Averaging Neural Operator* presented in +`The Nonlocal Neural Operator: Universal +Approximation `__ which is a `Kernel +Neural +Operator `__ +with integral kernel: + +.. math:: + + + K(v) = \sigma\left(Wv(x) + b + \frac{1}{|\Omega|}\int_\Omega v(y)dy\right) + +where: + +- :math:`v(x)\in\mathbb{R}^{\rm{emb}}` is the update for a function + :math:`v` with :math:`\mathbb{R}^{\rm{emb}}` the embedding (hidden) + size +- :math:`\sigma` is a non-linear activation +- :math:`W\in\mathbb{R}^{\rm{emb}\times\rm{emb}}` is a tunable matrix. +- :math:`b\in\mathbb{R}^{\rm{emb}}` is a tunable bias. + +If PINA many Kernel Neural Operators are already implemented, and the +modular componets of the `Kernel Neural +Operator `__ +class permits to create new ones by composing base kernel layers. + +**Note:**\ \* We will use the already built class\* +``AveragingNeuralOperator``, *as constructive excercise try to use the* +`KernelNeuralOperator `__ +*class for building a kernel neural operator from scratch. You might +employ the different layers that we have in pina, e.g.* +`FeedForward `__, +*and* +`AveragingNeuralOperator `__ +*layers*. + +.. code:: ipython3 + + class SIREN(torch.nn.Module): + def forward(self, x): + return torch.sin(x) + + embedding_dimesion = 40 # hyperparameter embedding dimension + input_dimension = 3 # ['u', 'x', 't'] + number_of_coordinates = 2 # ['x', 't'] + lifting_net = torch.nn.Linear(input_dimension, embedding_dimesion) # simple linear layers for lifting and projecting nets + projecting_net = torch.nn.Linear(embedding_dimesion + number_of_coordinates, 1) + model = AveragingNeuralOperator(lifting_net=lifting_net, + projecting_net=projecting_net, + coordinates_indices=['x', 't'], + field_indices=['u0'], + n_layers=4, + func=SIREN + ) + +Super easy! Notice that we use the ``SIREN`` activation function, more +on `Implicit Neural Representations with Periodic Activation +Functions `__. + +Solving the KS problem +---------------------- + +We will now focus on solving the KS equation using the +``SupervisedSolver`` class and the ``AveragingNeuralOperator`` model. As +done in the `FNO +tutorial `__ +we now create the ``NeuralOperatorProblem`` class with +``AbstractProblem``. + +.. code:: ipython3 + + # expected running time ~ 1 minute + + class NeuralOperatorProblem(AbstractProblem): + input_variables = initial_cond_train.labels + output_variables = sol_train.labels + conditions = {'data' : Condition(input_points=initial_cond_train, + output_points=sol_train)} + + + # initialize problem + problem = NeuralOperatorProblem() + # initialize solver + solver = SupervisedSolver(problem=problem, model=model,optimizer_kwargs={"lr":0.001}) + # train, only CPU and avoid model summary at beginning of training (optional) + trainer = Trainer(solver=solver, max_epochs=40, accelerator='cpu', enable_model_summary=False, log_every_n_steps=-1, batch_size=5) # we train on CPU and avoid model summary at beginning of training (optional) + trainer.train() + + + +.. parsed-literal:: + + GPU available: True (mps), used: False + TPU available: False, using: 0 TPU cores + IPU available: False, using: 0 IPUs + HPU available: False, using: 0 HPUs + + +.. parsed-literal:: + + Epoch 39: 100%|██████████| 20/20 [00:01<00:00, 13.59it/s, v_num=3, mean_loss=0.118] + +.. parsed-literal:: + + `Trainer.fit` stopped: `max_epochs=40` reached. + + +.. parsed-literal:: + + Epoch 39: 100%|██████████| 20/20 [00:01<00:00, 13.56it/s, v_num=3, mean_loss=0.118] + + +We can now see some plots for the solutions + +.. code:: ipython3 + + sample_number = 2 + no_sol = solver(initial_cond_test) + plot_trajectory(coords=initial_cond_test[sample_number].extract(['x', 't']), + real=sol_test[sample_number].extract('u'), + no_sol=no_sol[5]) + + + +.. image:: tutorial_files/tutorial_11_0.png + + +As we can see we can obtain nice result considering the small trainint +time and the difficulty of the problem! Let’s see how the training and +testing error: + +.. code:: ipython3 + + from pina.loss import PowerLoss + + error_metric = PowerLoss(p=2) # we use the MSE loss + + with torch.no_grad(): + no_sol_train = solver(initial_cond_train) + err_train = error_metric(sol_train.extract('u'), no_sol_train).mean() # we average the error over trajectories + no_sol_test = solver(initial_cond_test) + err_test = error_metric(sol_test.extract('u'),no_sol_test).mean() # we average the error over trajectories + print(f'Training error: {float(err_train):.3f}') + print(f'Testing error: {float(err_test):.3f}') + + +.. parsed-literal:: + + Training error: 0.128 + Testing error: 0.119 + + +as we can see the error is pretty small, which agrees with what we can +see from the previous plots. + +What’s next? +------------ + +Now you know how to solve a time dependent neural operator problem in +**PINA**! There are multiple directions you can go now: + +1. Train the network for longer or with different layer sizes and assert + the finaly accuracy + +2. We left a more challenging dataset + `Data_KS2.mat `__ where + :math:`A_k \in [-0.5, 0.5]`, :math:`\ell_k \in [1, 2, 3]`, + :math:`\phi_k \in [0, 2\pi]` for loger training + +3. 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