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Exploring connections between automatic differentiation and smooth infinitesimal analysis, or smooth algebras

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# Smooth Infinitesimal Analysis, or Automatic Differentiation with higher infinitesimals ## Background

A technique of automatic differentiation (AD) is in wide use in the realm of scientific computation and machine learning. One prominent implementation of AD in Haskell is Edward Kmett's ad package.

One method to implement AD is the forward mode, using the notion of dual numbers, $\mathbb{R}[\varepsilon] \cong \mathbb{R}[X]/X^2$. Theoretically, the ring of dual numbers could be regarded as an example of $C^\infty$-rings, or more specifically Weil algebras, which are used in the area called Smooth Infinitesimal Analysis (SIA) and Synthetic Differential Geometry (SDG). Weil algebras could be regarded as a real line with additional (higher) infinitesimal structures, in the sense we can regard $\mathcal{R}[\epsilon]$ as real line augmented with nilpotent infinitesimal of order two. Weil algebras allow much more abundance of such infinitesimals; for example, we can consider nilpotent infinitesimals of order three, or mutually nilpotent infinitesimals, and so on.

In this package, we will explore the possibility to extend ADs with higher infinitesimals exploiting Weil algebra structure.

## How to Play with?

First, you need to setup Haskell environment.

There are two options:

### Using Local Environment

Although you can use cabal-install, we recommend to use stack as it is the tool we use officially to develop this project.

First, compile all the dependencies once:

$cd smooth$ stack build

Then, you have two options to play with: REPL and HLS.

#### Playing with REPL

If you want to run the examples in app/Main.hs, you can specify the target explicitly by:

$stack ghci smooth:exe:smooth-exe If you want to play directly with library, you can invoke: $ stack ghci smooth:exe:smooth-exe --no-load

Then import and feed whatever you want:

>>> import Numeric.Algebra.Smooth
>>> import Numeric.Algebra.Smooth.Weil

-- Setting needed extensions:
>>> :set -Wno-type-defaults -XDataKinds -XPolyKinds -XGADTs -XTypeOperators
>>> :set -XRankNTypes -XFlexibleContexts

>>> diffUpTo 5 (\x -> sin (x/2) * exp (x^2)) (pi/4)
fromList [(0,0.7091438342369428),(1,1.9699328611326816),(2,5.679986037666626),(3,19.85501973096302),(4,73.3133870997595),(5,299.9934189752827)]

>>> sin (pi/3 + di 0) * exp (pi/6 + di 1) :: Weil (DOrder 2 |*| DOrder 3) Double
0.42202294874111723 d(0) d(1)^2 + 0.8440458974822345 d(0) d(1)
+ 0.7309651891796508 d(1)^2 + 0.8440458974822345 d(0)
+ 1.4619303783593016 d(1)   + 1.4619303783593016

-- Computing with general Weil algebra
>>> :m +Algebra.Ring.Polynomial Algebra.Ring.Ideal
>>> import qualified Algebra.Prelude.Core as AP
>>> let [x,y] = vars :: [Polynomial AP.Rational 2]
>>> let theIdeal = toIdeal [x^3 - 2 * y^2, y^3, x^2*y]
>>> withWeil (toIdeal [x  ^ 3 - 2 * y^2, y^3, x^2*y]) (sin (pi/3 + di 0) * exp (pi/6 + di 1))
Just (0.5438504802710591*X_0*X_1^2 - 0.7309651891796508*X_0^2 + 0.8440458974822345*X_0*X_1 + 0.44961655668557265*X_1^2 + 0.8440458974822345*X_0 + 1.4619303783593016*X_1 + 1.4619303783593016)

#### Playing with Haskell Language Server

If you use LSP-supported editors, you can use haskell-language-server's Eval plugin. It allows you to evaluate repl lines (comment string starting with >>>) embedded in Haskell code. Due to the GHC's bug, it might fail with GHC >= 8.10.3, 8.10.4; so I recommend you to use GHC 8.10.5. (If you don't get the meaning of this instruction, we recommend to use REPL.)

### Playing with Jupyter Notebook

There are two options:

#### Running Jupyter Locally

You first have to install jupyter:

$pip install jupyter Then within smooth directory, you must install all dependencies and packages explicitly: $ stack build ihaskell ihaskell-blaze smooth symbolic

If this is the first time to use IHaskell backend with Jupyter Notebook, you have to install Haskell kernel into jupyter with Stack support:

$stack exec -- ihaskell --stack install Then in the top-level directory of smooth, you can invoke jupyter: $ jupyter notebook # or: jupyter lab

Browse to notebooks directory and open demote.ipynb in Jupyter. You can now edit/evaluate something as you like. Enjoy!

#### Jupyter With Docker

We are providing the docker image containing the library and Jupyter Notebooks to play with: konn/smooth-jupyter. It contains all the dependencies and Jupyter Notebook environment to play with.

You can run the following to download the appropriate docker image and run Jupyter Lab server in the isolated container:

\$ docker run --rm -p 8888:8888 konn/smooth-jupyter:latest

Open https://127.0.0.1:8888 in the browse and open notebooks/demo.ipynb to play with. You can now evaluate cells in a top-down manner, re-edit cells, re-evaluate it, etc.

Exploring connections between automatic differentiation and smooth infinitesimal analysis, or smooth algebras

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