{
"info": {
"author": "Data Analysis Center",
"author_email": "akoriagin@nes.ru",
"bugtrack_url": null,
"classifiers": [
"Development Status :: 4 - Beta",
"Intended Audience :: Developers",
"Intended Audience :: Science/Research",
"License :: OSI Approved :: Apache Software License",
"Operating System :: OS Independent",
"Programming Language :: Python",
"Programming Language :: Python :: 3",
"Programming Language :: Python :: 3.5",
"Programming Language :: Python :: 3.6",
"Programming Language :: Python :: 3.7",
"Topic :: Scientific/Engineering"
],
"description": "[](https://www.apache.org/licenses/LICENSE-2.0)\n[](https://python.org)\n[](https://pytorch.org)\n[](https://app.shippable.com/github/analysiscenter/pydens)\n\n# PyDEns\n\n**PyDEns** is a framework for solving Ordinary and Partial Differential Equations (ODEs & PDEs) using neural networks. With **PyDEns** one can solve\n - PDEs & ODEs from a large family including [heat-equation](https://en.wikipedia.org/wiki/Heat_equation), [poisson equation](https://en.wikipedia.org/wiki/Poisson%27s_equation) and [wave-equation](https://en.wikipedia.org/wiki/Wave_equation)\n - parametric families of PDEs\n - PDEs with trainable coefficients.\n\nThis page outlines main capabilities of **PyDEns**. To get an in-depth understanding we suggest you to also read [the tutorial](https://github.com/analysiscenter/pydens/blob/master/tutorials/0.%20Theory.ipynb).\n\n## Getting started with **PyDEns**: solving common PDEs\nLet's solve poisson equation\n\n
\n\n
\n\n\nusing simple feed-forward neural network. Let's start by importing `Solver`-class along with other needed libraries:\n\n```python\nfrom pydens import Solver, NumpySampler\nimport numpy as np\nimport torch\n\n```\n\nYou can now set up a **PyDEns**-model for solving the task at hand. For this you need to supply the equation into a `Solver`-instance. Note the use of differentiation token `D`:\n\n```python\n# Define the equation as a callable.\ndef pde(f, x, y):\n return D(D(f, x), x) + D(D(f, y), y) - 5 * torch.sin(np.pi * (x + y))\n\n# Supply the equation, initial condition, the number of variables (`ndims`)\n# and the configration of neural network in Solver-instance.\nsolver = Solver(equation=pde, ndims=2, boundary_condition=1,\n layout='fa fa fa f', activation='Tanh', units=[10, 12, 15, 1])\n\n```\n\nNote that we defined the architecture of the neural network by supplying `layout`, `activation` and `units` parameters. Here `layout` configures the sequence of layers: `fa fa fa f` stands for `f`ully connected architecture with four layers and three `a`ctivations. In its turn, `units` and `activation` cotrol the number of units in dense layers and activation-function. When defining neural network this way use [`ConvBlock`](https://analysiscenter.github.io/batchflow/api/batchflow.models.torch.layers.html?highlight=baseconvblock#batchflow.models.torch.layers.BaseConvBlock) from [`BatchFlow`](https://github.com/analysiscenter/batchflow).\n\nIt's time to run the optimization procedure\n\n```python\nsolver.fit(batch_size=100, niters=1500)\n```\nin a fraction of second we've got a mesh-free approximation of the solution on **[0, 1]X[0, 1]**-square:\n\n
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\n\n## Going deeper into **PyDEns**-capabilities\n**PyDEns** allows to do much more than just solve common PDEs: it also deals with (i) parametric families of PDEs and (ii) PDEs with trainable coefficients.\n\n### Solving parametric families of PDEs\nConsider a *family* of ordinary differential equations\n\n
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\n\nClearly, the solution is a **sin** wave with a phase parametrized by \u03f5:\n\n
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\n\nSolving this problem is just as easy as solving common PDEs. You only need to introduce parameter `e` in the equation and supply the number of parameters (`nparams`) into a `Solver`-instance:\n\n```python\ndef odeparam(f, x, e):\n return D(f, x) - e * np.pi * torch.cos(e * np.pi * x)\n\n# One for argument, one for parameter\ns = NumpySampler('uniform') & NumpySampler('uniform', low=1, high=5)\n\nsolver = Solver(equation=odeparam, ndims=1, nparams=1, initial_condition=1)\nsolver.fit(batch_size=1000, sampler=s, niters=5000, lr=0.01)\n# solving the whole family takes no more than a couple of seconds!\n```\n\nCheck out the result:\n\n
\n\n
\n\n### Solving PDEs with trainable coefficients\n\nWith **PyDEns** things can get even more interesting! Assume that the *initial state of the system is unknown and yet to be determined*:\n\n
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\n\nOf course, without additional information, [the problem is undefined](https://en.wikipedia.org/wiki/Initial_value_problem). To make things better, let's fix the state of the system at some other point:\n\n
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\n\nSetting this problem requires a [slightly more complex configuring](https://github.com/analysiscenter/pydens/blob/master/tutorials/PDE_solving.ipynb). Note the use of `V`-token, that stands for trainable variable, in the initial condition of the problem. Also pay attention to the additional constraint supplied into the `Solver` instance. This constraint binds the final solution to zero at `t=0.5`:\n\n```python\ndef odevar(u, t):\n return D(u, t) - 2 * np.pi * torch.cos(2 * np.pi * t)\ndef initial(*args):\n return V('init', data=torch.Tensor([3.0]))\n\nsolver = Solver(odevar, ndims=1, initial_condition=initial,\n constraints=lambda u, t: u(torch.tensor([0.5])))\n```\nWhen tackling this problem, `pydens` will not only solve the equation, but also adjust the variable (initial condition) to satisfy the additional constraint.\nHence, model-fitting comes in two parts now: (i) solving the equation and (ii) adjusting initial condition to satisfy the additional constraint. Inbetween\nthe steps we need to freeze layers of the network to adjust only the adjustable variable:\n\n```python\nsolver.fit(batch_size=150, niters=100, lr=0.05)\nsolver.model.freeze_layers(['fc1', 'fc2', 'fc3'], ['log_scale'])\nsolver.fit(batch_size=150, niters=100, lr=0.05)\n```\n\nCheck out the results:\n\n
![](\"https://raw.githubusercontent.com/analysiscenter/pydens/master/imgs/poisson_eq.png?invert_in_darkmode\")
\n![](\"https://raw.githubusercontent.com/analysiscenter/pydens/master/imgs/poisson_sol.png?invert_in_darkmode\")
\n![](\"https://raw.githubusercontent.com/analysiscenter/pydens/master/imgs/sinus_eq.png?invert_in_darkmode\")
\n![](\"https://raw.githubusercontent.com/analysiscenter/pydens/master/imgs/sinus_sol_expr.png?invert_in_darkmode\")
\n![](\"https://raw.githubusercontent.com/analysiscenter/pydens/master/imgs/sinus_parametric.gif?invert_in_darkmode\")
\n![](\"https://raw.githubusercontent.com/analysiscenter/pydens/master/imgs/sinus_eq_trainable.png?invert_in_darkmode\")
\n![](\"https://raw.githubusercontent.com/analysiscenter/pydens/master/imgs/sinus_eq_middle_fix.png?invert_in_darkmode\")
\n![](\"https://raw.githubusercontent.com/analysiscenter/pydens/master/imgs/converging_sol.gif?invert_in_darkmode\")
\n