Metadata-Version: 2.1
Name: nangs
Version: 0.0.5
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/).

Alternatively, you can use one of our Docker [images](https://hub.docker.com/repository/docker/sensioai/nangs). You will need [Docker](https://docs.docker.com/install/) 19.03 and, if you have an NVIDIA GPU, the NVIDIA Drivers (you do not need CUDA) and [nvidia-docker](https://github.com/NVIDIA/nvidia-docker). We have GPU and CPU images.

## 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
cuda = False
device = "cuda" if torch.cuda.is_available() and cuda else "cpu"

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

# 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(0,1,30)
t = np.linspace(0,1,20)
u = np.array([1.0])
pde.setValues({'x': x, 't': t, 'u': u})

# define input values for testing
x = np.linspace(0,1,25)
t = np.linspace(0,1,15)
pde.setValues({'x': x, 't': t}, 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 = {'layers': 3, 'neurons': 256, 'activations': 'relu'}
pde.buildSolution(mlp)

# set optimization parameters
pde.compile(lr=1e-3, epochs=30, batch_size=32)

# find the solution
hist = pde.solve(device, 'best_solution.pth')
```

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


Epoch 1/30 Losses 0.38618 
 PDE  [ 0.01395 ] 
 boco 0.03983
 initial_condition 0.33240
 Val [ 0.02610 ]<p>Epoch 2/30 Losses 0.27415 
 PDE  [ 0.03017 ] 
 boco 0.05479
 initial_condition 0.18920
 Val [ 0.02214 ]<p>Epoch 3/30 Losses 0.16190 
 PDE  [ 0.03392 ] 
 boco 0.04238
 initial_condition 0.08560
 Val [ 0.03257 ]<p>Epoch 4/30 Losses 0.09087 
 PDE  [ 0.03058 ] 
 boco 0.03144
 initial_condition 0.02886
 Val [ 0.02977 ]<p>Epoch 5/30 Losses 0.06263 
 PDE  [ 0.02181 ] 
 boco 0.02406
 initial_condition 0.01676
 Val [ 0.01644 ]<p>Epoch 6/30 Losses 0.06183 
 PDE  [ 0.01522 ] 
 boco 0.02975
 initial_condition 0.01686
 Val [ 0.01179 ]<p>Epoch 7/30 Losses 0.04133 
 PDE  [ 0.01304 ] 
 boco 0.01755
 initial_condition 0.01074
 Val [ 0.00905 ]<p>Epoch 8/30 Losses 0.03224 
 PDE  [ 0.01251 ] 
 boco 0.01203
 initial_condition 0.00769
 Val [ 0.00972 ]<p>Epoch 9/30 Losses 0.02400 
 PDE  [ 0.00979 ] 
 boco 0.00877
 initial_condition 0.00544
 Val [ 0.00977 ]<p>Epoch 10/30 Losses 0.01900 
 PDE  [ 0.00741 ] 
 boco 0.00734
 initial_condition 0.00426
 Val [ 0.00658 ]<p>Epoch 11/30 Losses 0.01900 
 PDE  [ 0.00640 ] 
 boco 0.00861
 initial_condition 0.00399
 Val [ 0.00672 ]<p>Epoch 12/30 Losses 0.01135 
 PDE  [ 0.00696 ] 
 boco 0.00266
 initial_condition 0.00173
 Val [ 0.00671 ]<p>Epoch 13/30 Losses 0.00613 
 PDE  [ 0.00394 ] 
 boco 0.00127
 initial_condition 0.00092
 Val [ 0.00250 ]<p>Epoch 14/30 Losses 0.01266 
 PDE  [ 0.00258 ] 
 boco 0.00829
 initial_condition 0.00179
 Val [ 0.00277 ]<p>Epoch 15/30 Losses 0.00486 
 PDE  [ 0.00252 ] 
 boco 0.00171
 initial_condition 0.00063
 Val [ 0.00278 ]<p>Epoch 16/30 Losses 0.00310 
 PDE  [ 0.00243 ] 
 boco 0.00033
 initial_condition 0.00035
 Val [ 0.00537 ]<p>Epoch 17/30 Losses 0.00416 
 PDE  [ 0.00333 ] 
 boco 0.00036
 initial_condition 0.00046
 Val [ 0.00219 ]<p>Epoch 18/30 Losses 0.00224 
 PDE  [ 0.00165 ] 
 boco 0.00030
 initial_condition 0.00029
 Val [ 0.00166 ]<p>Epoch 19/30 Losses 0.00181 
 PDE  [ 0.00149 ] 
 boco 0.00014
 initial_condition 0.00018
 Val [ 0.00108 ]<p>Epoch 20/30 Losses 0.00149 
 PDE  [ 0.00097 ] 
 boco 0.00033
 initial_condition 0.00019
 Val [ 0.00114 ]<p>Epoch 21/30 Losses 0.00144 
 PDE  [ 0.00094 ] 
 boco 0.00034
 initial_condition 0.00016
 Val [ 0.00097 ]<p>Epoch 22/30 Losses 0.00172 
 PDE  [ 0.00105 ] 
 boco 0.00048
 initial_condition 0.00018
 Val [ 0.00092 ]<p>Epoch 23/30 Losses 0.00343 
 PDE  [ 0.00079 ] 
 boco 0.00220
 initial_condition 0.00045
 Val [ 0.00116 ]<p>Epoch 24/30 Losses 0.00270 
 PDE  [ 0.00089 ] 
 boco 0.00143
 initial_condition 0.00039
 Val [ 0.00116 ]<p>Epoch 25/30 Losses 0.00114 
 PDE  [ 0.00074 ] 
 boco 0.00026
 initial_condition 0.00013
 Val [ 0.00080 ]<p>Epoch 26/30 Losses 0.00098 
 PDE  [ 0.00078 ] 
 boco 0.00012
 initial_condition 0.00008
 Val [ 0.00190 ]<p>Epoch 27/30 Losses 0.00186 
 PDE  [ 0.00103 ] 
 boco 0.00058
 initial_condition 0.00025
 Val [ 0.00102 ]<p>Epoch 28/30 Losses 0.00268 
 PDE  [ 0.00111 ] 
 boco 0.00115
 initial_condition 0.00041
 Val [ 0.00090 ]<p>Epoch 29/30 Losses 0.00186 
 PDE  [ 0.00092 ] 
 boco 0.00066
 initial_condition 0.00028
 Val [ 0.00130 ]<p>Epoch 30/30 Losses 0.00135 
 PDE  [ 0.00066 ] 
 boco 0.00048
 initial_condition 0.00021
 Val [ 0.00074 ]


    /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_11_0.png)


</div>

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

```python
# evaluate the solution
pde.load_state_dict('best_solution.pth')
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])}, device)
    _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_12_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.


