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tutorial validation (#185)

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Co-authored-by: Ben Volokh <89551265+benv123@users.noreply.github.com>
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Nicola Demo
2023-10-17 10:54:31 +02:00
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docs/source/_rst/tutorial3/tutorial.rst
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@@ -1,8 +1,8 @@
Tutorial 3: resolution of wave equation with hard constraint PINNs.
===================================================================
The problem solution
~~~~~~~~~~~~~~~~~~~~
The problem definition
----------------------
In this tutorial we present how to solve the wave equation using hard
constraint PINNs. For doing so we will build a costum torch model and
@@ -10,13 +10,15 @@ pass it to the ``PINN`` solver.
The problem is written in the following form:
:raw-latex:`\begin{equation}
\begin{cases}
\Delta u(x,y,t) = \frac{\partial^2}{\partial t^2} u(x,y,t) \quad \text{in } D, \\\\
u(x, y, t=0) = \sin(\pi x)\sin(\pi y), \\\\
u(x, y, t) = 0 \quad \text{on } \Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4,
\end{cases}
\end{equation}`
.. raw:: latex
\begin{equation}
\begin{cases}
\Delta u(x,y,t) = \frac{\partial^2}{\partial t^2} u(x,y,t) \quad \text{in } D, \\\\
u(x, y, t=0) = \sin(\pi x)\sin(\pi y), \\\\
u(x, y, t) = 0 \quad \text{on } \Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4,
\end{cases}
\end{equation}
where :math:`D` is a square domain :math:`[0,1]^2`, and
:math:`\Gamma_i`, with :math:`i=1,...,4`, are the boundaries of the
@@ -80,21 +82,24 @@ predicted one.
problem = Wave()
Hard Constraint Model
---------------------
After the problem, a **torch** model is needed to solve the PINN.
Usually many models are already implemented in ``PINA``, but the user
has the possibility to build his/her own model in ``pyTorch``. The hard
constraint we impose are on the boundary of the spatial domain.
Specificly our solution is written as:
Usually, many models are already implemented in ``PINA``, but the user
has the possibility to build his/her own model in ``PyTorch``. The hard
constraint we impose is on the boundary of the spatial domain.
Specifically, our solution is written as:
.. math:: u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t),
where :math:`NN` is the neural net output. This neural network takes as
input the coordinates (in this case :math:`x`, :math:`y` and :math:`t`)
and provides the unkwown field of the Wave problem. By construction it
is zero on the boundaries. The residual of the equations are evaluated
at several sampling points (which the user can manipulate using the
method ``discretise_domain``) and the loss minimized by the neural
network is the sum of the residuals.
and provides the unknown field :math:`u`. By construction, it is zero on
the boundaries. The residuals of the equations are evaluated at several
sampling points (which the user can manipulate using the method
``discretise_domain``) and the loss minimized by the neural network is
the sum of the residuals.
.. code:: ipython3
@@ -114,6 +119,9 @@ network is the sum of the residuals.
hard = x.extract(['x'])*(1-x.extract(['x']))*x.extract(['y'])*(1-x.extract(['y']))
return hard*self.layers(x)
Train and Inference
-------------------
In this tutorial, the neural network is trained for 3000 epochs with a
learning rate of 0.001 (default in ``PINN``). Training takes
approximately 1 minute.
@@ -128,10 +136,20 @@ approximately 1 minute.
.. parsed-literal::
GPU available: False, used: False
/u/n/ndemo/.local/lib/python3.9/site-packages/torch/cuda/__init__.py:546: UserWarning: Can't initialize NVML
warnings.warn("Can't initialize NVML")
GPU available: True (cuda), used: True
TPU available: False, using: 0 TPU cores
IPU available: False, using: 0 IPUs
HPU available: False, using: 0 HPUs
Missing logger folder: /u/n/ndemo/PINA/tutorials/tutorial3/lightning_logs
2023-10-17 10:24:02.163746: I tensorflow/core/util/port.cc:110] oneDNN custom operations are on. You may see slightly different numerical results due to floating-point round-off errors from different computation orders. To turn them off, set the environment variable `TF_ENABLE_ONEDNN_OPTS=0`.
2023-10-17 10:24:02.218849: I tensorflow/core/platform/cpu_feature_guard.cc:182] This TensorFlow binary is optimized to use available CPU instructions in performance-critical operations.
To enable the following instructions: AVX2 AVX512F AVX512_VNNI FMA, in other operations, rebuild TensorFlow with the appropriate compiler flags.
2023-10-17 10:24:07.063047: W tensorflow/compiler/tf2tensorrt/utils/py_utils.cc:38] TF-TRT Warning: Could not find TensorRT
/opt/sissa/apps/intelpython/2022.0.2/intelpython/latest/lib/python3.9/site-packages/scipy/__init__.py:138: UserWarning: A NumPy version >=1.16.5 and <1.23.0 is required for this version of SciPy (detected version 1.26.0)
warnings.warn(f"A NumPy version >={np_minversion} and <{np_maxversion} is required for this version of "
LOCAL_RANK: 0 - CUDA_VISIBLE_DEVICES: [0]
| Name | Type | Params
----------------------------------------
@@ -144,20 +162,17 @@ approximately 1 minute.
0.002 Total estimated model params size (MB)
.. parsed-literal::
Epoch 2999: : 1it [00:00, 79.33it/s, v_num=5, mean_loss=0.00119, D_loss=0.00542, t0_loss=0.0017, gamma1_loss=0.000, gamma2_loss=0.000, gamma3_loss=0.000, gamma4_loss=0.000]
Training: 0it [00:00, ?it/s]
.. parsed-literal::
`Trainer.fit` stopped: `max_epochs=3000` reached.
.. parsed-literal::
Epoch 2999: : 1it [00:00, 68.62it/s, v_num=5, mean_loss=0.00119, D_loss=0.00542, t0_loss=0.0017, gamma1_loss=0.000, gamma2_loss=0.000, gamma3_loss=0.000, gamma4_loss=0.000]
Notice that the loss on the boundaries of the spatial domain is exactly
zero, as expected! After the training is completed one can now plot some
results using the ``Plotter`` class of **PINA**.
@@ -177,14 +192,13 @@ results using the ``Plotter`` class of **PINA**.
.. image:: tutorial_files/tutorial_12_0.png
.. image:: output_14_0.png
.. image:: tutorial_files/tutorial_12_1.png
.. image:: output_14_1.png
.. image:: tutorial_files/tutorial_12_2.png
.. image:: output_14_2.png