Metadata-Version: 2.1
Name: nangs
Version: 0.0.8
Summary: Solving Partial Differential Equations with Neural Networks
Home-page: https://github.com/juansensio/nangs
Author: Juan B. Pedro
Author-email: sensioai@gmail.com
License: Apache Software License 2.0
Keywords: Partial Differential Equations Neural Networks
Platform: UNKNOWN
Classifier: Development Status :: 3 - Alpha
Classifier: Intended Audience :: Developers
Classifier: License :: OSI Approved :: Apache Software License
Classifier: Natural Language :: English
Classifier: Programming Language :: Python :: 3.6
Classifier: Programming Language :: Python :: 3.7
Classifier: Programming Language :: Python :: 3.8
Requires-Python: >=3.6
Description-Content-Type: text/markdown

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### THIS FILE WAS AUTOGENERATED! DO NOT EDIT! ###
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# file to edit: index.ipynb
# command to build the docs after a change: nbdev_build_docs

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# Welcome to nangs

> Solving Partial Differential Equations with Neural Networks.


Nangs is a Python library built on top of Pytorch to solve Partial Differential Equations.

Our objective is to develop a new tool for simulating nature, using Neural Networks as solution approximation to Partial Differential Equations, increasing accuracy and optimziation speed while reducing computational cost.

Read our [paper](https://arxiv.org/abs/1912.04737) to know more.

Read the [docs](https://juansensio.github.io/nangs/).

## Installing

nangs is on PyPI so you can just run:

`pip install nangs`

You will also need to insall [Pytorch](https://pytorch.org/).

## Getting Started

Let's assume we want to solve the following PDE:

![adv1d](pics/adv1d.png)

Different numerical techniques that solve this problem exist, and all of them are based on finding an approximate function that satisfies the PDE. Traditional numerical methods discretize the domain into small elements where a form of the solutions is assumed (for example, a constant) and then the final solution is composed as a piece-wise, discontinuous function.

Nangs uses the property of neural networks (NNs) as universal function approximators to find a continuous and derivable solution to the PDE, that requires significant less computing resources compared with traditional techniques and with the advantage of including the free-parameters as part of the solution.

The independen variables (i.e, *x* and *t*) are used as input values for the NN, and the solution (i.e. *p*) is the output. In order to find the solution, at each step the NN outputs are derived w.r.t the inputs. Then, a loss function that matches the PDE is built and the weights are updated accordingly. If the loss function goes to zero, we can assume that our NN is indeed the solution to our PDE.
<div class="codecell" markdown="1">
<div class="input_area" markdown="1">

```python
import math
import numpy as np 
import matplotlib.pyplot as plt 
import torch

# import nangs
from nangs.pde import PDE
from nangs.bocos import PeriodicBoco, DirichletBoco
from nangs.solutions import MLP

# define custom PDE
class MyPDE(PDE):
    def __init__(self, inputs, outputs, params=None):
        super().__init__(inputs, outputs, params)
    def computePDELoss(self, grads, inputs, outputs, params): 
        # here is where the magic happens
        dpdt, dpdx = grads['p']['t'], grads['p']['x']
        u = params['u']
        return [dpdt + u*dpdx]

# instanciate pde
pde = MyPDE(inputs=['x', 't'], outputs=['p'], params=['u'])

# define input values for training
x = np.linspace(0,1,40)
t = np.linspace(0,1,30)
u = np.array([1.0])
pde.setValues({'x': x, 't': t, 'u': u})

# define input values for testing
x_v = np.linspace(0,1,25)
t_v = np.linspace(0,1,15)
pde.setValues({'x': x_v, 't': t_v}, train=False)

# periodic b.c for the space dimension
x1, x2 = np.array([0]), np.array([1])
boco = PeriodicBoco('boco', {'x': x1, 't': t}, {'x': x2, 't': t})
pde.addBoco(boco)

# initial condition (dirichlet for temporal dimension)
p0 = np.sin(2.*math.pi*x)
boco = DirichletBoco('initial_condition', {'x': x, 't': np.array([0])}, {'p': p0})
pde.addBoco(boco)

# define solution topology
mlp = MLP(pde.n_inputs, pde.n_outputs, 3, 256)
optimizer = torch.optim.Adam(mlp.parameters(), lr=3e-4)
pde.compile(mlp, optimizer)

# find the solution
hist = pde.solve()
```

</div>
<div class="output_area" markdown="1">


Epoch 1/30 Losses 0.40837 
 PDE  [ 0.00777 ] 
 boco 0.02863
 initial_condition 0.37197
 Val [ 0.01869 ]<p>Epoch 2/30 Losses 0.27972 
 PDE  [ 0.02495 ] 
 boco 0.05397
 initial_condition 0.20080
 Val [ 0.02539 ]<p>Epoch 3/30 Losses 0.15535 
 PDE  [ 0.02909 ] 
 boco 0.03968
 initial_condition 0.08659
 Val [ 0.02680 ]<p>Epoch 4/30 Losses 0.08142 
 PDE  [ 0.02092 ] 
 boco 0.03128
 initial_condition 0.02922
 Val [ 0.01671 ]<p>Epoch 5/30 Losses 0.04809 
 PDE  [ 0.01526 ] 
 boco 0.02045
 initial_condition 0.01238
 Val [ 0.01072 ]<p>Epoch 6/30 Losses 0.03523 
 PDE  [ 0.01239 ] 
 boco 0.01436
 initial_condition 0.00848
 Val [ 0.00939 ]<p>Epoch 7/30 Losses 0.02426 
 PDE  [ 0.00980 ] 
 boco 0.00917
 initial_condition 0.00529
 Val [ 0.00886 ]<p>Epoch 8/30 Losses 0.01611 
 PDE  [ 0.00713 ] 
 boco 0.00579
 initial_condition 0.00319
 Val [ 0.00503 ]<p>Epoch 9/30 Losses 0.00978 
 PDE  [ 0.00515 ] 
 boco 0.00300
 initial_condition 0.00164
 Val [ 0.00502 ]<p>Epoch 10/30 Losses 0.00626 
 PDE  [ 0.00340 ] 
 boco 0.00183
 initial_condition 0.00103
 Val [ 0.00265 ]<p>Epoch 11/30 Losses 0.00404 
 PDE  [ 0.00265 ] 
 boco 0.00091
 initial_condition 0.00049
 Val [ 0.00240 ]<p>Epoch 12/30 Losses 0.00291 
 PDE  [ 0.00217 ] 
 boco 0.00046
 initial_condition 0.00028
 Val [ 0.00274 ]<p>Epoch 13/30 Losses 0.00270 
 PDE  [ 0.00216 ] 
 boco 0.00033
 initial_condition 0.00020
 Val [ 0.00160 ]<p>Epoch 14/30 Losses 0.00212 
 PDE  [ 0.00165 ] 
 boco 0.00028
 initial_condition 0.00019
 Val [ 0.00181 ]<p>Epoch 15/30 Losses 0.00193 
 PDE  [ 0.00151 ] 
 boco 0.00026
 initial_condition 0.00015
 Val [ 0.00157 ]<p>Epoch 16/30 Losses 0.00188 
 PDE  [ 0.00150 ] 
 boco 0.00022
 initial_condition 0.00016
 Val [ 0.00158 ]<p>Epoch 17/30 Losses 0.00171 
 PDE  [ 0.00122 ] 
 boco 0.00033
 initial_condition 0.00016
 Val [ 0.00114 ]<p>Epoch 18/30 Losses 0.00121 
 PDE  [ 0.00094 ] 
 boco 0.00019
 initial_condition 0.00008
 Val [ 0.00102 ]<p>Epoch 19/30 Losses 0.00110 
 PDE  [ 0.00085 ] 
 boco 0.00017
 initial_condition 0.00009
 Val [ 0.00096 ]<p>Epoch 20/30 Losses 0.00116 
 PDE  [ 0.00080 ] 
 boco 0.00026
 initial_condition 0.00010
 Val [ 0.00078 ]<p>Epoch 21/30 Losses 0.00092 
 PDE  [ 0.00073 ] 
 boco 0.00012
 initial_condition 0.00007
 Val [ 0.00073 ]<p>Epoch 22/30 Losses 0.00085 
 PDE  [ 0.00068 ] 
 boco 0.00010
 initial_condition 0.00006
 Val [ 0.00074 ]<p>Epoch 23/30 Losses 0.00101 
 PDE  [ 0.00065 ] 
 boco 0.00027
 initial_condition 0.00009
 Val [ 0.00071 ]<p>Epoch 24/30 Losses 0.00090 
 PDE  [ 0.00063 ] 
 boco 0.00019
 initial_condition 0.00008
 Val [ 0.00060 ]<p>Epoch 25/30 Losses 0.00113 
 PDE  [ 0.00080 ] 
 boco 0.00021
 initial_condition 0.00012
 Val [ 0.00059 ]<p>Epoch 26/30 Losses 0.00077 
 PDE  [ 0.00053 ] 
 boco 0.00017
 initial_condition 0.00007
 Val [ 0.00067 ]<p>Epoch 27/30 Losses 0.00107 
 PDE  [ 0.00052 ] 
 boco 0.00043
 initial_condition 0.00013
 Val [ 0.00047 ]<p>Epoch 28/30 Losses 0.00057 
 PDE  [ 0.00044 ] 
 boco 0.00008
 initial_condition 0.00005
 Val [ 0.00052 ]<p>Epoch 29/30 Losses 0.00050 
 PDE  [ 0.00042 ] 
 boco 0.00005
 initial_condition 0.00003
 Val [ 0.00052 ]<p>Epoch 30/30 Losses 0.00066 
 PDE  [ 0.00042 ] 
 boco 0.00018
 initial_condition 0.00006
 Val [ 0.00038 ]


    /opt/conda/lib/python3.7/site-packages/numpy/core/fromnumeric.py:3257: RuntimeWarning: Mean of empty slice.
      out=out, **kwargs)
    /opt/conda/lib/python3.7/site-packages/numpy/core/_methods.py:161: RuntimeWarning: invalid value encountered in double_scalars
      ret = ret.dtype.type(ret / rcount)


</div>

</div>
<div class="codecell" markdown="1">
<div class="input_area" markdown="1">

```python
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15,5))
ax1.plot(hist['train_loss'], label="train_loss")
ax1.plot(hist['val_loss'], label="val_loss")
ax1.grid(True)
ax1.set_yscale("log")
ax1.legend()
for boco in pde.bocos:
    ax2.plot(hist['bocos'][boco.name], label=boco.name)
ax2.legend()
ax2.grid(True)
ax2.set_yscale("log")
plt.show()
```

</div>
<div class="output_area" markdown="1">


![png](docs/images/output_10_0.png)


</div>

</div>
<div class="codecell" markdown="1">
<div class="input_area" markdown="1">

```python
# evaluate the solution
x = np.linspace(0,1,50)
t = np.linspace(0,1,100)
p, p0, l2 = [], [], []
for _t in t:
    _p0 = np.sin(2.*math.pi*(x-u*_t)) 
    pde.evaluate({'x': x, 't': np.array([_t])})
    _p = pde.outputs['p']
    _l2 = np.mean((_p - _p0)**2)
    p.append(_p)
    p0.append(_p0)
    l2.append(_l2)

from matplotlib import animation, rc
rc('animation', html='html5')

def plot(x, p, p0, t, l2):
    ax.clear()
    tit = ax.set_title(f"t = {t:.2f}, l2 = {l2:.5f}", fontsize=14)
    ax.plot(x, p0, "-k", label="Exact")
    ax.plot(x, p, "g^", label="NN")
    ax.set_xlabel("x", fontsize=14)
    ax.set_ylabel("p", fontsize=14, rotation=np.pi/2)
    ax.legend(loc="upper left")
    ax.grid(True)
    ax.set_xlim([0, 1])
    ax.set_ylim([-1.2, 1.2])
    return [tit]

def get_anim(fig, ax, x, p, p0, t, l2):
    def anim(i):
        return plot(x, p[i], p0[i], t[i], l2[i])
    return anim

fig = plt.figure(figsize=(10,5))
ax = fig.add_subplot(111, autoscale_on=False)
animate = get_anim(fig, ax, x, p, p0, t, l2)
anim = animation.FuncAnimation(fig, animate, frames=len(t), interval=100, blit=True)
```

</div>
<div class="output_area" markdown="1">


![png](docs/images/output_11_0.png)


</div>

</div>
<div class="codecell" markdown="1">
<div class="input_area" markdown="1">

```python
anim
```

</div>
<div class="output_area" markdown="1">




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">
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</video>



</div>

</div>

## Examples

Check the [examples](https://github.com/juansensio/nangs/tree/master/examples) to learn more about using nangs to solve PDEs with NNs.

## Copyright

Copyright 2020 onwards, SensioAI. Licensed under the Apache License, Version 2.0 (the "License"); you may not use this project's files except in compliance with the License. A copy of the License is provided in the LICENSE file in this repository.


