From c298c7d4d1657ffd108f586ba55b579cdd5e80e7 Mon Sep 17 00:00:00 2001 From: Dario Coscia Date: Tue, 21 Nov 2023 09:53:21 +0100 Subject: [PATCH 1/3] tut8 begin --- tutorials/tutorial8/tutorial.ipynb | 147 +++++++++++++++++++++++++++++ 1 file changed, 147 insertions(+) create mode 100644 tutorials/tutorial8/tutorial.ipynb diff --git a/tutorials/tutorial8/tutorial.ipynb b/tutorials/tutorial8/tutorial.ipynb new file mode 100644 index 00000000..d9107bbe --- /dev/null +++ b/tutorials/tutorial8/tutorial.ipynb @@ -0,0 +1,147 @@ +{ + "cells": [ + { + "attachments": {}, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Tutorial: Advection Equation with DeepONet neural operator\n", + "\n", + "In this tutorial we will show how to solve a neural operator problem using `DeepONet`. Specifically, ....\n", + "\n", + "We import the relevant modules first." + ] + }, + { + "cell_type": "code", + "execution_count": 62, + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "from pina.geometry import CartesianDomain\n", + "from pina import LabelTensor, Trainer, Condition\n", + "from pina.equation import Equation\n", + "from pina.problem import TimeDependentProblem, SpatialProblem\n", + "import torch\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Generating the data\n", + "\n", + "We first generate the data we will use for training." + ] + }, + { + "cell_type": "code", + "execution_count": 85, + "metadata": {}, + "outputs": [], + "source": [ + "# fixing mu ~ U[0.05, 1], and sigma=0.0002\n", + "r1 = 0.05\n", + "r2 = 1.\n", + "mu = (r1 - r2) * torch.rand(1) + r2\n", + "sigma = torch.tensor([0.0002])\n", + "\n", + "# initial condition\n", + "def initial_condition(pts):\n", + " return torch.exp(-(pts - mu)**2 / sigma) * (sigma * torch.pi) ** (-0.5)\n", + "\n", + "def solution(pts, t):\n", + " return torch.stack([pts, t.repeat(pts.shape, 1), initial_condition(pts - t)]).T\n", + "\n", + "def compute_solution(pts, times):\n", + " # fixing mu ~ U[0.05, 1], and sigma=0.0002\n", + " mu = (0.05 - 1) * torch.rand(1) + 1\n", + " sigma = torch.tensor([0.0002])\n", + " # compute solution for each timestep\n", + " solutions = torch.stack([solution(pts, t) for t in times])\n", + " \n", + " return solutions\n" + ] + }, + { + "cell_type": "code", + "execution_count": 94, + "metadata": {}, + "outputs": [], + "source": [ + "t = torch.linspace(0, 1, 100)\n", + "x = torch.linspace(0, 2, 256)\n", + "solutions = LabelTensor(torch.stack([compute_solution(x, t) for _ in range(1000)]), ['x', 't', 'u'])" + ] + }, + { + "cell_type": "code", + "execution_count": 112, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "torch.Size([1000, 100, 256, 1])" + ] + }, + "execution_count": 112, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "solutions.extract(['u']).shape" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "class AdvectionProblem(SpatialProblem, TimeDependentProblem):\n", + "\n", + " output_variables = ['u']\n", + " input_variables = ['x', 't']\n", + "\n", + " # define initial condition\n", + " def initial_condition(input_, output_):\n", + " u_expected = -torch.sin(torch.pi*input_.extract(['x']))\n", + " return output_.extract(['u']) - u_expected\n", + "\n", + " # assign output/ spatial and temporal variables\n", + " output_variables = ['u']\n", + " spatial_domain = CartesianDomain({'x': [-1, 1]})\n", + " temporal_domain = CartesianDomain({'t': [0, 1]})\n", + "\n", + " # problem condition statement\n", + " conditions = {\n", + " 'data': Condition(input_pts = ),\n", + " }" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.16" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} From ac232e562d5af518c7ad1dbbcb9effe2cae900fd Mon Sep 17 00:00:00 2001 From: Dario Coscia Date: Tue, 28 Nov 2023 13:45:55 +0100 Subject: [PATCH 2/3] tut8 update/ deeponet.py update/ test_deeponet.py update --- docs/source/_rst/_tutorial.rst | 1 + .../_rst/tutorials/tutorial8/tutorial.rst | 343 +++++++++++++++ .../tutorial_files/tutorial_17_0.png | Bin 0 -> 11037 bytes .../tutorial_files/tutorial_17_1.png | Bin 0 -> 11465 bytes .../tutorial_files/tutorial_17_2.png | Bin 0 -> 11239 bytes .../tutorial_files/tutorial_19_0.png | Bin 0 -> 68900 bytes .../tutorial8/tutorial_files/tutorial_5_0.png | Bin 0 -> 43969 bytes pina/model/deeponet.py | 45 +- tests/test_model/test_deeponet.py | 23 +- tutorials/README.md | 1 + .../tutorial8/advection_input_testing.pt | Bin 0 -> 400987 bytes .../tutorial8/advection_input_training.pt | Bin 0 -> 40990 bytes .../tutorial8/advection_output_testing.pt | Bin 0 -> 400990 bytes .../tutorial8/advection_output_training.pt | Bin 0 -> 40993 bytes tutorials/tutorial8/deeponet.png | Bin 0 -> 359826 bytes tutorials/tutorial8/tutorial.ipynb | 406 +++++++++++++++--- tutorials/tutorial8/tutorial.py | 237 ++++++++++ 17 files changed, 976 insertions(+), 80 deletions(-) create mode 100644 docs/source/_rst/tutorials/tutorial8/tutorial.rst create mode 100644 docs/source/_rst/tutorials/tutorial8/tutorial_files/tutorial_17_0.png create mode 100644 docs/source/_rst/tutorials/tutorial8/tutorial_files/tutorial_17_1.png create mode 100644 docs/source/_rst/tutorials/tutorial8/tutorial_files/tutorial_17_2.png create mode 100644 docs/source/_rst/tutorials/tutorial8/tutorial_files/tutorial_19_0.png create mode 100644 docs/source/_rst/tutorials/tutorial8/tutorial_files/tutorial_5_0.png create mode 100644 tutorials/tutorial8/advection_input_testing.pt create mode 100644 tutorials/tutorial8/advection_input_training.pt create mode 100644 tutorials/tutorial8/advection_output_testing.pt create mode 100644 tutorials/tutorial8/advection_output_training.pt create mode 100644 tutorials/tutorial8/deeponet.png create mode 100644 tutorials/tutorial8/tutorial.py diff --git a/docs/source/_rst/_tutorial.rst b/docs/source/_rst/_tutorial.rst index 87e6d9f6..82a33ada 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 + One dimensional advection problem with data driven DeepONet neural operator Supervised Learning ------------------- diff --git a/docs/source/_rst/tutorials/tutorial8/tutorial.rst b/docs/source/_rst/tutorials/tutorial8/tutorial.rst new file mode 100644 index 00000000..91ae11f0 --- /dev/null +++ b/docs/source/_rst/tutorials/tutorial8/tutorial.rst @@ -0,0 +1,343 @@ +Tutorial: Advection Equation with data driven DeepONet neural operator +====================================================================== + +In this tutorial we will show how to solve the advection neural operator +problem using ``DeepONet``. Specifically, we will follow the original +formulation of Lu Lu, et al. in `DeepONet: Learning nonlinear operators +for identifying differential equations based on the universal +approximation theorem of operator `__. + +We import the relevant modules first. + +.. code:: ipython3 + + import matplotlib.pyplot as plt + from pina import Trainer, Condition + from pina.model import FeedForward, DeepONet + from pina.solvers import SupervisedSolver + from pina.problem import AbstractProblem + from pina.loss import LpLoss + import torch + +The advection mathematical problem and data loading +--------------------------------------------------- + +Consider the one dimensional advection problem: + +.. math:: + + + \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} = 0, \quad x\in[0,2], \;t\in[0,1] + +with periodic boundary condition. We choose the initial condition as a +Gaussian wave centered in :math:`\mu` and of width +:math:`\sigma^2=0.02`. We choose :math:`\mu` to be distributed according +to a Uniform distribution in :math:`[0.05, 1]`. Hence the initial +condition is: + +.. math:: + + + u_0(x) = \frac{1}{\sqrt{\pi\sigma^2}}e^{-\frac{(x - \mu)^2}{2\sigma^2}}, \quad \mu\sim U(0.05, 1), \; x\in[0,2]. + +The final objective is to learn the operator: + +.. math:: \mathcal{G} : u_0(x) → u(x, t = \delta) = u_0(x - \delta) + +where we set :math:`\delta=0.5` just for sake of the tutorial. Notice +that a general problem is to lear the operator mapping the initial +condition to a specific time instance. By fixing :math:`\delta` dataset +is composed of trajectories containing initial conditions as input, and +the same trajectories after :math:`\delta` time as output. + +The input/output shape will be ``[T, Nx, D]``, where ``T`` is the number +of trajectories, ``Nx`` the discretization along the spatial dimension, +and ``D`` the dimension of the input. In this case ``D=1`` since the +input is the one dimensional field value ``u``. The data for training +are stored in ``advection_input_training.pt``, +``advection_output_training.pt``; while the one for testing in +``advection_input_testing.pt``, ``advection_output_testing.pt``. We +train on ``T=100`` trajectories, and test on ``T=1000``. Notice that the +discretization in the spatial dimension is fixed to ``Nx=100``. + +We start by loading the data and perform some plotting. + +.. code:: ipython3 + + # loading training data + data_0_training = torch.load('advection_input_training.pt') + data_dt_training = torch.load('advection_output_training.pt') + + # loading testing data + data_0_testing = torch.load('advection_input_testing.pt') + data_dt_testing = torch.load('advection_output_testing.pt') + +The data are loaded, let’s visualize a few of the initial conditions! + +.. code:: ipython3 + + # storing the discretization in space: + Nx = data_0_training.shape[1] + + for idx, i in enumerate(torch.randint(0, data_0_training.shape[0], (3,))): + u0 = data_0_training[i].extract('ui') + u = data_dt_training[i].extract('u') + x = torch.linspace(0, 2, Nx) # the discretization in the spatial dimension is fixed + plt.subplot(3, 1, idx+1) + plt.plot(x, u0.flatten(), label=fr'$u_0(x)$') + plt.plot(x, u.flatten(), label=fr'$u(x, t=\delta)$') + plt.xlabel(fr'$x$') + plt.tight_layout() + plt.legend() + + + +.. image:: tutorial_files/tutorial_5_0.png + + +Great! We have created a travellig wave and we have visualized few +examples. Now we are going to use the data to train the network. + +DeepONet +-------- + +The classical formulation of DeepONet has two networks, namely +**trunck** and **branch** net (see below figure). + +.. raw:: html + +
+ +.. raw:: html + +
+ +.. raw:: html + +
+ +Image from: Moya, C.; Lin, G. Fed-DeepONet: Stochastic Gradient-Based +Federated Training of Deep Operator Networks. Algorithms 2022, 15, 325. + +.. raw:: html + +
+ +In our example the branch net will take as input a vector ``[B, Nx]`` +representing for each trajectory the field solution at time zero +discretize by ``Nx``. The trunk net will take as input ``[B, 1]`` +representing the temporal coordinate to evaluate the network. Here ``B`` +is the number of training samples in a batch of the total trajectories. + +Let us now write the advection problem. + +.. code:: ipython3 + + class AdvectionProblem(AbstractProblem): + + output_variables = ['u'] + input_variables = ['ui'] + + # problem condition statement + conditions = { + 'data': Condition(input_points = data_0_training, output_points = data_dt_training) + } + +Notice that the problem inherits from ``AbstractProblem``, since we only +train our model in a datadriven mode. + +We now proceede to create the trunk and branch networks. + +.. code:: ipython3 + + # create Trunck model + class TrunkNet(torch.nn.Module): + def __init__(self, **kwargs): + super().__init__() + self.trunk = FeedForward(**kwargs) + def forward(self, x): + t = torch.zeros(size=(x.shape[0], 1), requires_grad=False) + 0.5 # create an input of only 0.5 + return self.trunk(t) + + # create Branch model + class BranchNet(torch.nn.Module): + def __init__(self, **kwargs): + super().__init__() + self.branch = FeedForward(**kwargs) + def forward(self, x): + return self.branch(x.flatten(-2)) + +The ``TrunckNet`` is a simple ``FeedForward`` neural network with a +modified ``forward`` pass. In the forward pass we simply create a tensor +of :math:`0.5` repeated for each trajectory. + +The ``BranchNet`` is a simple ``FeedForward`` neural network with a +modified ``forward`` pass as well. The input is flatten across the last +dimension to obtain a vector of the same dimension as the +discretization, which represents the initial condition at the sensor +points. + +We now proceed to create the DeepONet model using the ``DeepONet`` class +from ``pina.model`` + +.. code:: ipython3 + + # initialize truck and branch net + trunk = TrunkNet( + layers=[256] * 4, + output_dimensions=Nx, # output the spatial discretization, must be equal to the one of branch net + input_dimensions=1, # time variable + func=torch.nn.ReLU + ) + branch = BranchNet( + layers=[256] * 4, + output_dimensions=Nx, # output the spatial discretization, must be equal to the one of trunck net + input_dimensions=Nx, # spatial discretization (supposing) data is alligned + func=torch.nn.ReLU + ) + + # initialize the DeepONet model + model = DeepONet(branch_net=branch, + trunk_net=trunk, + input_indeces_branch_net=['ui'], + input_indeces_trunk_net=['ui'], + reduction = lambda x : x, + aggregator = lambda x : torch.einsum('bm,bm->bm', x[0], x[1]) + ) + +The aggregation + reduction functions combine the output of the two +networks. In this case the output of the networks ar multiplied element +wise and the corresponding vector is not reduced (i.e. we obtain as +output the same as the trunck and branch output). + +The solver, as in the `FNO +tutorial `__, +is simply a ``SupervisedSolver`` trained with ``MSE`` loss. In the next +lines we define the solver first, and then we define the trainer which +is used for training the solver. + +The neural network training as been shown to better converge if some +tricks are applied during training. As an example, we will use batch +training with batch accumulation; and gradient clipping of the norm +(acting as regularizer to prevent overfitting). Thanks to the `PyTorch +Lightning `__ API, this +complex techniques are achieved by only changing some flags in the +Trainer. Notice that we do not hypersearch the best values, and we do +this optimizations just for demonstration. + +.. code:: ipython3 + + # define solver + solver = SupervisedSolver(problem=AdvectionProblem(), model=model) + + # define the trainer and train + # we train on cpu -> accelerator='cpu' + # we do not display model summary -> enable_model_summary=False + # we train for 100 epochs -> max_epochs=100 + # we use batch size 5 -> batch_size=5 + # we accumulate the gradient accross 20 batches -> accumulate_grad_batches=20 + # we clip the gradient norm to a max of 1 -> gradient_clip_algorithm='norm', gradient_clip_val=1 + trainer = Trainer(solver=solver, max_epochs=500, enable_model_summary=False, + accelerator='cpu', accumulate_grad_batches=20, batch_size=5, + gradient_clip_algorithm='norm', gradient_clip_val=1) + trainer.train() + + +.. parsed-literal:: + + GPU available: False, used: False + TPU available: False, using: 0 TPU cores + IPU available: False, using: 0 IPUs + HPU available: False, using: 0 HPUs + /Users/dariocoscia/anaconda3/envs/pina/lib/python3.9/site-packages/torch/_tensor.py:1295: UserWarning: The use of `x.T` on tensors of dimension other than 2 to reverse their shape is deprecated and it will throw an error in a future release. Consider `x.mT` to transpose batches of matrices or `x.permute(*torch.arange(x.ndim - 1, -1, -1))` to reverse the dimensions of a tensor. (Triggered internally at /Users/runner/work/_temp/anaconda/conda-bld/pytorch_1682343673238/work/aten/src/ATen/native/TensorShape.cpp:3575.) + ret = func(*args, **kwargs) + + +.. parsed-literal:: + + Epoch 499: : 20it [00:00, 364.36it/s, v_num=31, mean_loss=3.09e-5] + +.. parsed-literal:: + + `Trainer.fit` stopped: `max_epochs=500` reached. + + +.. parsed-literal:: + + Epoch 499: : 20it [00:00, 318.64it/s, v_num=31, mean_loss=3.09e-5] + + +Let us see the final train and test errors: + +.. code:: ipython3 + + # the l2 error + l2 = LpLoss() + + with torch.no_grad(): + train_err = l2(trainer.solver.neural_net(data_0_training), data_dt_training) + test_err = l2(trainer.solver.neural_net(data_0_testing), data_dt_testing) + + print(f'Training error: {float(train_err.mean()):.2%}') + print(f'Tresting error: {float(test_err.mean()):.2%}') + + +.. parsed-literal:: + + Training error: 0.39% + Tresting error: 1.03% + + +We can see that the testing error is slightly higher than the training +one, maybe due to overfitting. We now plot some results trajectories. + +.. code:: ipython3 + + for i in [1, 2, 3]: + plt.subplot(3, 1, i) + plt.plot(torch.linspace(0, 2, Nx), solver.neural_net(data_0_training)[10*i].detach().flatten(), label=r'$u_{NN}$') + plt.plot(torch.linspace(0, 2, Nx), data_dt_training[10*i].extract('u').flatten(), label=r'$u$') + plt.xlabel(r'$x$') + plt.legend(loc='upper right') + plt.show() + + + +.. image:: tutorial_files/tutorial_17_0.png + + + +.. image:: tutorial_files/tutorial_17_1.png + + + +.. image:: tutorial_files/tutorial_17_2.png + + +As we can see, they are barely indistinguishable. To better understand +the difference, we now plot the residuals, i.e. the difference of the +exact solution and the predicted one. + +.. code:: ipython3 + + for i in [1, 2, 3]: + plt.subplot(3, 1, i) + plt.plot(torch.linspace(0, 2, Nx), data_dt_training[10*i].extract('u').flatten() - solver.neural_net(data_0_training)[10*i].detach().flatten(), label=r'$u - u_{NN}$') + plt.xlabel(r'$x$') + plt.tight_layout() + plt.legend(loc='upper right') + + + +.. image:: tutorial_files/tutorial_19_0.png + + +What’s next? +------------ + +We have made a very simple example on how to use the ``DeepONet`` for +learning the advection neural operator. We suggest to extend the +tutorial using more complex problems and train for longer, to see the +full potential of neural operators. 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z2!9@fag^J8P7xujkWjRHssV8O_KFz;rzm8HauQS^;TVB(Zxzze;5^_IXaoNlCI3sz zwP}wv7$0|_GK!JZy@6?+NCZJXjggs|jI3<+*U#tNgc{%NGLggx^uXw+pN^x(zPN~E z@OH2bK5;w{zfJpJ<^E=%Q~$TCj^BLhw>> branch_net1 = FeedForward(input_dimensons=1, output_dimensions=10) >>> branch_net2 = FeedForward(input_dimensons=2, output_dimensions=10) @@ -112,17 +118,18 @@ def __init__(self, check_consistency(scale, bool) check_consistency(translation, bool) - # check trunk branch nets consistency - shapes = [] - for key, value in networks.items(): - check_consistency(value, (str, int)) - check_consistency(key, torch.nn.Module) - input_ = torch.rand(10, len(value)) - shapes.append(key(input_).shape[-1]) + # TODO: Fix it does not work + # # check trunk branch nets consistency + # shapes = [] + # for key, value in networks.items(): + # check_consistency(value, (str, int)) + # check_consistency(key, torch.nn.Module) + # input_ = torch.rand(10, len(value)) + # shapes.append(key(input_).shape[-1]) - if not all(map(lambda x: x == shapes[0], shapes)): - raise ValueError('The passed networks have not the same ' - 'output dimension.') + # if not all(map(lambda x: x == shapes[0], shapes)): + # raise ValueError('The passed networks have not the same ' + # 'output dimension.') # assign trunk and branch net with their input indeces self.models = torch.nn.ModuleList(networks.keys()) @@ -153,9 +160,11 @@ def _symbol_functions(**kwargs): } def _init_aggregator(self, aggregator): - aggregator_funcs = DeepONet._symbol_functions(dim=2) + aggregator_funcs = DeepONet._symbol_functions(dim=-1) if aggregator in aggregator_funcs: - aggregator_func = aggregator_funcs[aggregator] + def aggregator_func(tensors): + out_ = torch.stack(tensors, dim=-1) + return aggregator_funcs[aggregator](out_) elif isinstance(aggregator, nn.Module) or is_function(aggregator): aggregator_func = aggregator else: @@ -207,10 +216,13 @@ def forward(self, x): ] # aggregation - aggregated = self._aggregator(torch.dstack(output_)) + aggregated = self._aggregator(output_) + + # before apply the reduction we clone the tensor for backprop + aggregated = aggregated.as_subclass(torch.Tensor).clone() # reduce - output_ = self._reduction(aggregated).reshape(-1, 1) + output_ = self._reduction(aggregated).unsqueeze(-1) # scale and translate output_ *= self._scale @@ -330,6 +342,11 @@ def __init__(self, only a list of integers can be passed for ``input_indeces_branch_net`` and ``input_indeces_trunk_net``. + .. warning:: + In the forward pass we do not check if the network outputs of the single networks + are the same, and if this does not happen unwanted behaviour could result in the + forward pass not correctly extecuted. + :Example: >>> branch_net = FeedForward(input_dimensons=1, output_dimensions=10) >>> trunk_net = FeedForward(input_dimensons=1, output_dimensions=10) diff --git a/tests/test_model/test_deeponet.py b/tests/test_model/test_deeponet.py index cfba6149..3892251c 100644 --- a/tests/test_model/test_deeponet.py +++ b/tests/test_model/test_deeponet.py @@ -21,16 +21,19 @@ def test_constructor(): aggregator='*') -def test_constructor_fails_when_invalid_inner_layer_size(): - branch_net = FeedForward(input_dimensions=1, output_dimensions=10) - trunk_net = FeedForward(input_dimensions=2, output_dimensions=8) - with pytest.raises(ValueError): - DeepONet(branch_net=branch_net, - trunk_net=trunk_net, - input_indeces_branch_net=['a'], - input_indeces_trunk_net=['b', 'c'], - reduction='+', - aggregator='*') +# This test is wrong! The user could define a custom network and do a +# reshape at the end! A more general way to check input and output is +# needed +# def test_constructor_fails_when_invalid_inner_layer_size(): +# branch_net = FeedForward(input_dimensions=1, output_dimensions=10) +# trunk_net = FeedForward(input_dimensions=2, output_dimensions=8) +# with pytest.raises(ValueError): +# DeepONet(branch_net=branch_net, +# trunk_net=trunk_net, +# input_indeces_branch_net=['a'], +# input_indeces_trunk_net=['b', 'c'], +# reduction='+', +# aggregator='*') def test_forward_extract_str(): diff --git a/tutorials/README.md b/tutorials/README.md index 601b5e16..447a5f0b 100644 --- a/tutorials/README.md +++ b/tutorials/README.md @@ -21,6 +21,7 @@ Resolution of a 2D Poisson inverse problem |[[.ipynb](tutorial7/tutorial.ipynb), | Description | Tutorial | |---------------|-----------| Two dimensional Darcy flow using the Fourier Neural Operator         |[[.ipynb](tutorial5/tutorial.ipynb), [.py](tutorial5/tutorial.py), [.html](http://mathlab.github.io/PINA/_rst/tutorial5/tutorial.html)]| +One dimensional advection problem with data driven DeepONet neural operator         |[[.ipynb](tutorial8/tutorial.ipynb), [.py](tutorial8/tutorial.py), [.html](http://mathlab.github.io/PINA/_rst/tutorial8/tutorial.html)]| ## Supervised Learning | Description | Tutorial | diff --git a/tutorials/tutorial8/advection_input_testing.pt b/tutorials/tutorial8/advection_input_testing.pt new file mode 100644 index 0000000000000000000000000000000000000000..d0e469d940198465d93837a6d7341b518cf9d6a7 GIT binary patch literal 400987 zcmcG$2UJwik~Z4poO5VW6Pj-5oco-!w+IL#B1%$0lAs8PL`AZq5)>5>kem@z%%WmK z$rwQq6)}qnim0Fn$j_bo=DoGv``xGDS{nEN z5hI)p&VOTEP+&|*#8%&st=nQ_d}D&5V?wrWqHgq$@uzGH4YRkwaO?g{k}4V)6A=}- znX+hmYji}EtZ!6MKx|0ZM&ID5h;ZMS_-#SHaSYi=9P!`vVq-$WqW>SYe}3)XbVLHT zg>3bwg!u;qh5fs^75{kQ_D|E1qC}#X)Z!0~UOc7$rl3f%f7+C_4~y`RS@X{~Wh2GC zL>6^ukrG~Fi^sriv5}I?EBA`vVj`tnop8<{#hf-eiR_D%{>R}TPh>I!10%x1{y9+5 zvMx~@gQ9{q+J*$i$VSRqu8x$8iIh)@RQR_u`~KU-MMo<7L@IfSE}D-GiB$HFj)_!R 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