GiggleLiu

### A Pluto.jl notebook ###
# v0.11.14

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local el = $(esc(element))

GiggleLiu

### A Pluto.jl notebook ###
# v0.11.14

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local el = $(esc(element))

GiggleLiu

using NiLang, NiLang.AD

function f!(du, u, p, t)
    y₁, y₂, y₃ = u
    k₁, k₂, k₃ = p
    du[1] = -k₁*y₁ + k₃*y₂*y₃
    du[2] =  k₁*y₁ - k₃*y₂*y₃ - k₂*y₂^2
    du[3] =  y₁ + y₂ + y₃ - 1
    return nothing
end

GiggleLiu

### A Pluto.jl notebook ###
# v0.11.14

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local el = $(esc(element))

GiggleLiu

# References
# * Yao: https://arxiv.org/abs/1912.10877
# * ForwardDiff: https://arxiv.org/abs/1607.07892
# * Tropical Numbers: https://en.wikipedia.org/wiki/Tropical_geometry
using ForwardDiff, Yao, LinearAlgebra

# define tropical numbers with the following property.
# x ⊕ y := max(x ,y)
# x ⊗ y := x + y
struct Tropical{T} <: Number

GiggleLiu

# `]add OMEinsum#master`
# `]add TupleTools`
using OMEinsum
using CuArrays
using CUDAdrv

using CUDAnative, TupleTools
using OMEinsum: index_map, map_prod
using Base.Cartesian
using GPUArrays

GiggleLiu

using OMEinsum, LinearAlgebra
using TupleTools
import OMEinsum: einsum!

"""The rule type used for dispatch."""
struct EinRule{S} end
EinRule(S) = EinRule{S}()

global einsum_rules = NTuple{2, Any}[]

GiggleLiu

using TupleTools
using Base.Cartesian
using CuArrays, CUDAnative

"""
A naive implementation of `einsum!`
    * `ixs`: input tensor indices,
    * `xs`: input tensors,
    * `iy`: output tensor indices,
    * `y`: accumulated tensor, notice it is initialized to 0 as output!

GiggleLiu

#=
To use this script, please install Flux.jl first by typing `]add Flux` in a Julia REPL.
=#
import LinearAlgebra
using Flux.Tracker: @grad, data, track, TrackedTuple, TrackedArray
using Flux
import Flux.Tracker: _forward

"""
backward function for svd.

GiggleLiu

using KrylovKit, Yao

"""
    J1J2{D}
    J1J2(size::Int...; J2::Real, periodic::Bool) -> J1J2

D ∈ {1, 2} is the dimension target system.
"""
struct J1J2{D}
    size::NTuple{D, Int}