https://github.com/rjrosati/SymbolicTensors.jl

Manipulate tensors symbolically in Julia! Currently needs a SymPy dependency, but work is ongoing to change the backend to SymbolicUtils.jl
https://github.com/rjrosati/SymbolicTensors.jl

julia physics symbolic-computation symbolic-manipulation symbolic-math tensor-algebra

Last synced: 3 months ago
JSON representation

Manipulate tensors symbolically in Julia! Currently needs a SymPy dependency, but work is ongoing to change the backend to SymbolicUtils.jl

• Host: GitHub
• URL: https://github.com/rjrosati/SymbolicTensors.jl
• Owner: rjrosati
• License: mit
• Created: 2020-05-08T22:29:09.000Z (over 4 years ago)
• Default Branch: master
• Last Pushed: 2024-04-12T19:27:50.000Z (7 months ago)
• Last Synced: 2024-05-22T07:35:37.227Z (6 months ago)
• Topics: julia, physics, symbolic-computation, symbolic-manipulation, symbolic-math, tensor-algebra
• Language: Julia
• Homepage:
• Size: 49.8 KB
• Stars: 32
• Watchers: 3
• Forks: 4
• Open Issues: 8
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

Awesome Lists containing this project

• awesome-sciml - rjrosati/SymbolicTensors.jl: Manipulate tensors symbolically in Julia! Currently needs a SymPy dependency, but work is ongoing to change the backend to SymbolicUtils.jl

README

        
# SymbolicTensors

[![Build Status](https://travis-ci.com/rjrosati/SymbolicTensors.jl.svg?token=zMDX3GmCZbdBcf9JWMdp&branch=master)](https://travis-ci.com/rjrosati/SymbolicTensors.jl)

[![codecov](https://codecov.io/gh/rjrosati/SymbolicTensors.jl/branch/master/graph/badge.svg?token=JKgibtSJzc)](https://codecov.io/gh/rjrosati/SymbolicTensors.jl)

Many numerical tensor manipulation packages exist (e.g. `Einsum.jl`), but treating tensors at a purely numeric level throws away a lot of potential optimizations.

Often, it's possible to exploit the symmetries of a problem to dramatically reduce the calculation steps necessary, or perform some tensor contractions symbolically rather than numerically.

`SymbolicTensors.jl` is designed to exploit these simplifications to perform symbolic calculations and generate more efficient input into numeric tensor packages than you would write by hand. It based on `SymPy.jl`, `sympy.tensor.tensor`, `xTensor`, and `ITensors.jl`.

See the talk about this package given at JuliaCon 2020: https://www.youtube.com/watch?v=_b4JIv044GY

## Example calculations

```julia

using SymbolicTensors

using SymPy

spacetime = TensorIndexType("spacetime","f")

@indices spacetime μ ν σ ρ η

x = TensorHead("x",[spacetime])

δ = spacetime.delta

# one way to write the metric on a sphere

g = 4*δ(-μ,-ν)/(1+x(μ)*x(ν)*δ(-μ,-ν))^2

# compute the christoffel symbols

Γ = (diff(g(-μ,-ν),x(σ)) - diff(g(-ν,-σ),x(μ)) + diff(g(-σ,-μ),x(ν)))/2

Γ = factor(contract_metric(canon_bp(Γ),spacetime.metric))

```

At this point `Γ` is a symbolic tensor expression, written in Einstein notation.

It could represent 2 dimensional or 10,000 dimensional tensors in the same form -- this is one of the great advantages of working at the symbolic tensor level.

We could manipulate `Γ` further, or convert it into an array.

```julia

# convert Γ to Array{Sym}

xarr = symbols("x y",real=true)

garr = replace_with_arrays(g,Dict(x(ρ) => xarr, spacetime.delta(-ρ,-η) => [1 0; 0 1]))

Γarr = replace_with_arrays(Γ,Dict(x(ρ) => xarr, spacetime.delta(-ρ,-η) => [1 0; 0 1], spacetime => garr))

```

Now that we have an array in `Γarr`, `Quote` allows us to convert it into a native Julia function.

```julia

# Then Quote it and eval to a Julia function

quot = Quote("Christoffel",Γarr,[ x for x in xarr])

Christ = eval(quot)

Christ(0,1)

```

## Rough TODO

* Nicer errors

* testing coverage

* add conversion to ITensors or equivalent

* documentation

* symbolic derivatives?

## known bugs

* check issues