Metadata-Version: 2.1
Name: nangs
Version: 0.0.6
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/).
        
        ## 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 0.38070 
         PDE  [ 0.01580 ] 
         boco 0.03874
         initial_condition 0.32616
         Val [ 0.02477 ]<p>Epoch 2/30 Losses 0.24440 
         PDE  [ 0.02929 ] 
         boco 0.05514
         initial_condition 0.15997
         Val [ 0.03093 ]<p>Epoch 3/30 Losses 0.14186 
         PDE  [ 0.03262 ] 
         boco 0.04536
         initial_condition 0.06388
         Val [ 0.02594 ]<p>Epoch 4/30 Losses 0.08540 
         PDE  [ 0.02532 ] 
         boco 0.03465
         initial_condition 0.02543
         Val [ 0.02103 ]<p>Epoch 5/30 Losses 0.07090 
         PDE  [ 0.01604 ] 
         boco 0.03478
         initial_condition 0.02009
         Val [ 0.01865 ]<p>Epoch 6/30 Losses 0.04612 
         PDE  [ 0.01343 ] 
         boco 0.02101
         initial_condition 0.01168
         Val [ 0.00770 ]<p>Epoch 7/30 Losses 0.02798 
         PDE  [ 0.00895 ] 
         boco 0.01199
         initial_condition 0.00704
         Val [ 0.00675 ]<p>Epoch 8/30 Losses 0.02767 
         PDE  [ 0.00834 ] 
         boco 0.01428
         initial_condition 0.00504
         Val [ 0.00693 ]<p>Epoch 9/30 Losses 0.01288 
         PDE  [ 0.00616 ] 
         boco 0.00448
         initial_condition 0.00224
         Val [ 0.00389 ]<p>Epoch 10/30 Losses 0.00616 
         PDE  [ 0.00346 ] 
         boco 0.00139
         initial_condition 0.00131
         Val [ 0.00252 ]<p>Epoch 11/30 Losses 0.00526 
         PDE  [ 0.00291 ] 
         boco 0.00140
         initial_condition 0.00095
         Val [ 0.00318 ]<p>Epoch 12/30 Losses 0.00509 
         PDE  [ 0.00246 ] 
         boco 0.00196
         initial_condition 0.00067
         Val [ 0.00168 ]<p>Epoch 13/30 Losses 0.00370 
         PDE  [ 0.00202 ] 
         boco 0.00119
         initial_condition 0.00049
         Val [ 0.00206 ]<p>Epoch 14/30 Losses 0.00258 
         PDE  [ 0.00148 ] 
         boco 0.00084
         initial_condition 0.00026
         Val [ 0.00162 ]<p>Epoch 15/30 Losses 0.00269 
         PDE  [ 0.00159 ] 
         boco 0.00080
         initial_condition 0.00029
         Val [ 0.00129 ]<p>Epoch 16/30 Losses 0.00511 
         PDE  [ 0.00107 ] 
         boco 0.00345
         initial_condition 0.00060
         Val [ 0.00132 ]<p>Epoch 17/30 Losses 0.00233 
         PDE  [ 0.00109 ] 
         boco 0.00092
         initial_condition 0.00032
         Val [ 0.00088 ]<p>Epoch 18/30 Losses 0.00188 
         PDE  [ 0.00088 ] 
         boco 0.00077
         initial_condition 0.00022
         Val [ 0.00103 ]<p>Epoch 19/30 Losses 0.00168 
         PDE  [ 0.00074 ] 
         boco 0.00073
         initial_condition 0.00022
         Val [ 0.00084 ]<p>Epoch 20/30 Losses 0.00303 
         PDE  [ 0.00091 ] 
         boco 0.00179
         initial_condition 0.00032
         Val [ 0.00111 ]<p>Epoch 21/30 Losses 0.00211 
         PDE  [ 0.00091 ] 
         boco 0.00085
         initial_condition 0.00035
         Val [ 0.00107 ]<p>Epoch 22/30 Losses 0.00492 
         PDE  [ 0.00073 ] 
         boco 0.00363
         initial_condition 0.00056
         Val [ 0.00091 ]<p>Epoch 23/30 Losses 0.00270 
         PDE  [ 0.00108 ] 
         boco 0.00126
         initial_condition 0.00036
         Val [ 0.00093 ]<p>Epoch 24/30 Losses 0.00130 
         PDE  [ 0.00081 ] 
         boco 0.00032
         initial_condition 0.00017
         Val [ 0.00078 ]<p>Epoch 25/30 Losses 0.00097 
         PDE  [ 0.00070 ] 
         boco 0.00014
         initial_condition 0.00013
         Val [ 0.00115 ]<p>Epoch 26/30 Losses 0.00415 
         PDE  [ 0.00079 ] 
         boco 0.00229
         initial_condition 0.00107
         Val [ 0.00079 ]<p>Epoch 27/30 Losses 0.00288 
         PDE  [ 0.00074 ] 
         boco 0.00138
         initial_condition 0.00076
         Val [ 0.00071 ]<p>Epoch 28/30 Losses 0.00175 
         PDE  [ 0.00071 ] 
         boco 0.00079
         initial_condition 0.00025
         Val [ 0.00096 ]<p>Epoch 29/30 Losses 0.00165 
         PDE  [ 0.00064 ] 
         boco 0.00083
         initial_condition 0.00018
         Val [ 0.00064 ]<p>Epoch 30/30 Losses 0.00193 
         PDE  [ 0.00060 ] 
         boco 0.00112
         initial_condition 0.00021
         Val [ 0.00049 ]
        
        
            /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
        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_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.
        
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
