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
Description: <!--
        
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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.
        
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
