Metadata-Version: 2.1
Name: nangs
Version: 0.0.7
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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# 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(epochs=50)
```

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


Epoch 1/50 Losses 0.40488 
 PDE  [ 0.00865 ] 
 boco 0.02703
 initial_condition 0.36920
 Val [ 0.02271 ]<p>Epoch 2/50 Losses 0.27958 
 PDE  [ 0.02658 ] 
 boco 0.05469
 initial_condition 0.19832
 Val [ 0.02176 ]<p>Epoch 3/50 Losses 0.16315 
 PDE  [ 0.03167 ] 
 boco 0.03852
 initial_condition 0.09297
 Val [ 0.02226 ]<p>Epoch 4/50 Losses 0.08097 
 PDE  [ 0.02417 ] 
 boco 0.02951
 initial_condition 0.02729
 Val [ 0.02281 ]<p>Epoch 5/50 Losses 0.04978 
 PDE  [ 0.01821 ] 
 boco 0.01974
 initial_condition 0.01183
 Val [ 0.01231 ]<p>Epoch 6/50 Losses 0.03304 
 PDE  [ 0.01291 ] 
 boco 0.01263
 initial_condition 0.00749
 Val [ 0.00973 ]<p>Epoch 7/50 Losses 0.02165 
 PDE  [ 0.00996 ] 
 boco 0.00743
 initial_condition 0.00425
 Val [ 0.01086 ]<p>Epoch 8/50 Losses 0.01205 
 PDE  [ 0.00668 ] 
 boco 0.00337
 initial_condition 0.00201
 Val [ 0.00553 ]<p>Epoch 9/50 Losses 0.00777 
 PDE  [ 0.00500 ] 
 boco 0.00162
 initial_condition 0.00115
 Val [ 0.00421 ]<p>Epoch 10/50 Losses 0.00519 
 PDE  [ 0.00386 ] 
 boco 0.00079
 initial_condition 0.00054
 Val [ 0.00359 ]<p>Epoch 11/50 Losses 0.00389 
 PDE  [ 0.00290 ] 
 boco 0.00062
 initial_condition 0.00037
 Val [ 0.00259 ]<p>Epoch 12/50 Losses 0.00303 
 PDE  [ 0.00237 ] 
 boco 0.00042
 initial_condition 0.00025
 Val [ 0.00190 ]<p>Epoch 13/50 Losses 0.00257 
 PDE  [ 0.00200 ] 
 boco 0.00035
 initial_condition 0.00021
 Val [ 0.00201 ]<p>Epoch 14/50 Losses 0.00209 
 PDE  [ 0.00176 ] 
 boco 0.00020
 initial_condition 0.00013
 Val [ 0.00157 ]<p>Epoch 15/50 Losses 0.00195 
 PDE  [ 0.00156 ] 
 boco 0.00024
 initial_condition 0.00014
 Val [ 0.00164 ]<p>Epoch 16/50 Losses 0.00165 
 PDE  [ 0.00134 ] 
 boco 0.00019
 initial_condition 0.00012
 Val [ 0.00129 ]<p>Epoch 17/50 Losses 0.00186 
 PDE  [ 0.00132 ] 
 boco 0.00039
 initial_condition 0.00016
 Val [ 0.00129 ]<p>Epoch 18/50 Losses 0.00143 
 PDE  [ 0.00116 ] 
 boco 0.00018
 initial_condition 0.00009
 Val [ 0.00221 ]<p>Epoch 19/50 Losses 0.00201 
 PDE  [ 0.00140 ] 
 boco 0.00042
 initial_condition 0.00019
 Val [ 0.00088 ]<p>Epoch 20/50 Losses 0.00135 
 PDE  [ 0.00096 ] 
 boco 0.00026
 initial_condition 0.00013
 Val [ 0.00088 ]<p>Epoch 21/50 Losses 0.00133 
 PDE  [ 0.00094 ] 
 boco 0.00027
 initial_condition 0.00011
 Val [ 0.00071 ]<p>Epoch 22/50 Losses 0.00095 
 PDE  [ 0.00073 ] 
 boco 0.00015
 initial_condition 0.00007
 Val [ 0.00064 ]<p>Epoch 23/50 Losses 0.00079 
 PDE  [ 0.00063 ] 
 boco 0.00011
 initial_condition 0.00006
 Val [ 0.00081 ]<p>Epoch 24/50 Losses 0.00090 
 PDE  [ 0.00064 ] 
 boco 0.00018
 initial_condition 0.00008
 Val [ 0.00081 ]<p>Epoch 25/50 Losses 0.00126 
 PDE  [ 0.00066 ] 
 boco 0.00046
 initial_condition 0.00015
 Val [ 0.00056 ]<p>Epoch 26/50 Losses 0.00109 
 PDE  [ 0.00058 ] 
 boco 0.00038
 initial_condition 0.00013
 Val [ 0.00055 ]<p>Epoch 27/50 Losses 0.00065 
 PDE  [ 0.00057 ] 
 boco 0.00005
 initial_condition 0.00004
 Val [ 0.00062 ]<p>Epoch 28/50 Losses 0.00080 
 PDE  [ 0.00058 ] 
 boco 0.00015
 initial_condition 0.00007
 Val [ 0.00058 ]<p>Epoch 29/50 Losses 0.00098 
 PDE  [ 0.00050 ] 
 boco 0.00036
 initial_condition 0.00012
 Val [ 0.00043 ]<p>Epoch 30/50 Losses 0.00083 
 PDE  [ 0.00054 ] 
 boco 0.00021
 initial_condition 0.00009
 Val [ 0.00048 ]<p>Epoch 31/50 Losses 0.00060 
 PDE  [ 0.00042 ] 
 boco 0.00012
 initial_condition 0.00006
 Val [ 0.00046 ]<p>Epoch 32/50 Losses 0.00099 
 PDE  [ 0.00047 ] 
 boco 0.00039
 initial_condition 0.00014
 Val [ 0.00061 ]<p>Epoch 33/50 Losses 0.00086 
 PDE  [ 0.00045 ] 
 boco 0.00031
 initial_condition 0.00011
 Val [ 0.00037 ]<p>Epoch 34/50 Losses 0.00044 
 PDE  [ 0.00038 ] 
 boco 0.00003
 initial_condition 0.00003
 Val [ 0.00034 ]<p>Epoch 35/50 Losses 0.00045 
 PDE  [ 0.00035 ] 
 boco 0.00006
 initial_condition 0.00004
 Val [ 0.00045 ]<p>Epoch 36/50 Losses 0.00092 
 PDE  [ 0.00040 ] 
 boco 0.00040
 initial_condition 0.00012
 Val [ 0.00047 ]<p>Epoch 37/50 Losses 0.00066 
 PDE  [ 0.00036 ] 
 boco 0.00022
 initial_condition 0.00008
 Val [ 0.00029 ]<p>Epoch 38/50 Losses 0.00069 
 PDE  [ 0.00031 ] 
 boco 0.00028
 initial_condition 0.00009
 Val [ 0.00033 ]<p>Epoch 39/50 Losses 0.00074 
 PDE  [ 0.00032 ] 
 boco 0.00033
 initial_condition 0.00010
 Val [ 0.00048 ]<p>Epoch 40/50 Losses 0.00054 
 PDE  [ 0.00031 ] 
 boco 0.00017
 initial_condition 0.00007
 Val [ 0.00032 ]<p>Epoch 41/50 Losses 0.00055 
 PDE  [ 0.00032 ] 
 boco 0.00017
 initial_condition 0.00006
 Val [ 0.00041 ]<p>Epoch 42/50 Losses 0.00059 
 PDE  [ 0.00040 ] 
 boco 0.00013
 initial_condition 0.00006
 Val [ 0.00041 ]<p>Epoch 43/50 Losses 0.00084 
 PDE  [ 0.00033 ] 
 boco 0.00040
 initial_condition 0.00011
 Val [ 0.00039 ]<p>Epoch 44/50 Losses 0.00061 
 PDE  [ 0.00037 ] 
 boco 0.00017
 initial_condition 0.00007
 Val [ 0.00042 ]<p>Epoch 45/50 Losses 0.00050 
 PDE  [ 0.00030 ] 
 boco 0.00015
 initial_condition 0.00006
 Val [ 0.00037 ]<p>Epoch 46/50 Losses 0.00040 
 PDE  [ 0.00026 ] 
 boco 0.00010
 initial_condition 0.00004
 Val [ 0.00040 ]<p>Epoch 47/50 Losses 0.00077 
 PDE  [ 0.00028 ] 
 boco 0.00036
 initial_condition 0.00013
 Val [ 0.00029 ]<p>Epoch 48/50 Losses 0.00045 
 PDE  [ 0.00028 ] 
 boco 0.00012
 initial_condition 0.00005
 Val [ 0.00028 ]<p>Epoch 49/50 Losses 0.00067 
 PDE  [ 0.00027 ] 
 boco 0.00031
 initial_condition 0.00010
 Val [ 0.00030 ]<p>Epoch 50/50 Losses 0.00062 
 PDE  [ 0.00030 ] 
 boco 0.00025
 initial_condition 0.00008
 Val [ 0.00030 ]


    /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.


