Metadata-Version: 2.1
Name: nangs
Version: 0.0.4
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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        # 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). 
        
        ## Getting Started
        
        We have GPU and CPU images
        
        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, 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': 100, 'activations': 'relu'}
        pde.buildSolution(mlp)
        
        # set optimization parameters
        pde.compile(lr=0.01, epochs=20, batch_size=32)
        
        # find the solution
        hist = pde.solve(device, 'best_solution.pth')
        ```
        
        </div>
        <div class="output_area" markdown="1">
        
        
        Epoch 1/20 Losses 0.40368 
         PDE  0.01568
         boco 0.04150
         initial_condition 0.34651
         Val 0.03113 <p>Epoch 2/20 Losses 0.26363 
         PDE  0.03761
         boco 0.05959
         initial_condition 0.16643
         Val 0.02898 <p>Epoch 3/20 Losses 0.14324 
         PDE  0.03913
         boco 0.04351
         initial_condition 0.06060
         Val 0.04284 <p>Epoch 4/20 Losses 0.08736 
         PDE  0.02872
         boco 0.03349
         initial_condition 0.02515
         Val 0.02061 <p>Epoch 5/20 Losses 0.05867 
         PDE  0.01603
         boco 0.02591
         initial_condition 0.01674
         Val 0.01160 <p>Epoch 6/20 Losses 0.04793 
         PDE  0.00891
         boco 0.02590
         initial_condition 0.01311
         Val 0.00647 <p>Epoch 7/20 Losses 0.03881 
         PDE  0.00493
         boco 0.01901
         initial_condition 0.01487
         Val 0.00607 <p>Epoch 8/20 Losses 0.01393 
         PDE  0.00450
         boco 0.00616
         initial_condition 0.00327
         Val 0.00378 <p>Epoch 9/20 Losses 0.00570 
         PDE  0.00183
         boco 0.00265
         initial_condition 0.00121
         Val 0.00114 <p>Epoch 10/20 Losses 0.01189 
         PDE  0.00136
         boco 0.00679
         initial_condition 0.00375
         Val 0.00126 <p>Epoch 11/20 Losses 0.00379 
         PDE  0.00088
         boco 0.00197
         initial_condition 0.00094
         Val 0.00067 <p>Epoch 12/20 Losses 0.00154 
         PDE  0.00067
         boco 0.00057
         initial_condition 0.00030
         Val 0.00047 <p>Epoch 13/20 Losses 0.00076 
         PDE  0.00048
         boco 0.00013
         initial_condition 0.00015
         Val 0.00039 <p>Epoch 14/20 Losses 0.00071 
         PDE  0.00033
         boco 0.00022
         initial_condition 0.00016
         Val 0.00028 <p>Epoch 15/20 Losses 0.00072 
         PDE  0.00030
         boco 0.00024
         initial_condition 0.00018
         Val 0.00029 <p>Epoch 16/20 Losses 0.01209 
         PDE  0.00050
         boco 0.00769
         initial_condition 0.00391
         Val 0.00049 <p>Epoch 17/20 Losses 0.00379 
         PDE  0.00043
         boco 0.00219
         initial_condition 0.00118
         Val 0.00026 <p>Epoch 18/20 Losses 0.00163 
         PDE  0.00029
         boco 0.00090
         initial_condition 0.00044
         Val 0.00022 <p>Epoch 19/20 Losses 0.00054 
         PDE  0.00020
         boco 0.00018
         initial_condition 0.00016
         Val 0.00027 <p>Epoch 20/20 Losses 0.00056 
         PDE  0.00018
         boco 0.00027
         initial_condition 0.00010
         Val 0.00013 
        
        
        </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.legend()
        for boco in pde.bocos:
            ax2.plot(hist['bocos'][boco.name], label=boco.name)
        ax2.legend()
        ax2.grid(True)
        plt.show()
        ```
        
        </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
        # 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))
            _p = pde.evaluate({'x': x, 't': np.array([_t])}, device)
            _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, 1])
            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_13_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
