Min working example.ipynb

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   "source": [
    "<h1>Neural CLE Minimum Working Example<h1>"
   ]
  },
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   "execution_count": 319,
   "id": "8515499f",
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   "source": [
    "using DiffEqFlux\n",
    "using SciMLSensitivity\n",
    "using Optimization\n",
    "using StochasticDiffEq, DiffEqCallbacks, DiffEqNoiseProcess\n",
    "using Statistics, LinearAlgebra, Random\n",
    "using Plots\n",
    "using RecursiveArrayTools\n",
    "using DiffEqBase.EnsembleAnalysis\n",
    "using BenchmarkTools"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "88e7fe4e",
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   "source": [
    "We are interested in the Birth-Death system, which could be used to describe the population dynamics of a system of cells that has both a growth and death term proportional to its current population. The mathematics of the single ODE system are as follows:\n",
    "\n",
    "$\\frac{dX}{dt} = \\alpha_1 X - \\alpha_2 X$\n",
    "\n",
    "For our purposes, what we are trying to show is that we can observe parts of the ODE from viewing the final output. So we rearrage this system for convenience (and generalizability)\n",
    "\n",
    "$\\frac{dX}{dt} = R_1 - R_2$\n",
    "\n",
    "$\\frac{dT_1}{dt} = R_1$\n",
    "\n",
    "$\\frac{dT_2}{dt} = R_2$\n",
    "\n",
    "$R_1 = \\alpha_1 X$\n",
    "\n",
    "$R_2 = \\alpha_2 X$\n",
    "\n",
    "Where $T_1$ and $T_2$ are cumulative reaction rate curves. If we can predict these from the observable data (X), it is equivalent to being able to predict the reaction rates. We can generate training data for $X_0$ = 50, $\\alpha_1$ = 0.2, $\\alpha_2$ = 0.1. \n",
    "\n",
    "The CLE of this system would look like:\n",
    "$\\frac{dS}{dt} = \\begin{bmatrix}\n",
    "R_1 - R_2\\\\\n",
    "R_1\\\\\n",
    "R_2\\\\\n",
    "\\end{bmatrix} dt + \n",
    "\\begin{bmatrix}\n",
    "\\sqrt{R_1} & -\\sqrt{R_2}\\\\\n",
    "\\sqrt{R_1} & 0\\\\\n",
    "0 & \\sqrt{R_2}\\\\\n",
    "\\end{bmatrix}dW$\n",
    "\n",
    "After we construct training data, the goal is to fit the CLE to the data using a neural network to predict $R_1$ and $R_2$."
   ]
  },
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   "execution_count": 258,
   "id": "ec772d8e",
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    {
     "data": {
      "text/plain": [
       "0.0f0:0.05f0:10.0f0"
      ]
     },
     "execution_count": 258,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "u0 = Float64[50.; 0.;0]\n",
    "datasize = 201\n",
    "tspan = (0.0f0, 10.0f0)\n",
    "tsteps = range(tspan[1], tspan[2], length = datasize)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 379,
   "id": "a4f3b0cd",
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      ]
     },
     "execution_count": 379,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function trueSDEfunc(du, u, p, t)\n",
    "    R1 = 0.2*u[1]\n",
    "    R2 = 0.1*u[1]\n",
    "    #Safety valve to prevent divergence to unstable values\n",
    "    if u[1]>1000\n",
    "        du[1] = 0\n",
    "    else\n",
    "        du[1] = R1-R2\n",
    "    end\n",
    "    du[2] = R1\n",
    "    du[3] = R2\n",
    "end\n",
    "\n",
    "function true_noise_func(du, u, p, t)\n",
    "    R1 = 0.2*u[1]\n",
    "    R2 = 0.1*u[1]\n",
    "    du[1, 1] = sqrt.(R1)[1]\n",
    "    du[1, 2] = -sqrt.(R2)[1]\n",
    "    du[2, 1] = sqrt.(R1)[1]\n",
    "    du[3, 2] = 0\n",
    "    du[3, 1] = 0\n",
    "    du[3, 2] = sqrt.(R2)[1]\n",
    "end\n",
    "prob_truesde = SDEProblem(trueSDEfunc, true_noise_func, [50,0,0], tspan, noise_rate_prototype = zeros(3, 2))\n",
    "nothing\n",
    "\n",
    "sol = solve(prob_truesde, EM(), dt=0.05)\n",
    "plot(sol)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5370f4ee",
   "metadata": {},
   "source": [
    "We can generate a consistant testing set of means and variance by averaging an ensemble of function evaluations. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 674,
   "id": "9cfcf850",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  7.724124 seconds (28.97 M allocations: 3.018 GiB, 13.34% gc time, 14.55% compilation time: 98% of which was recompilation)\n"
     ]
    }
   ],
   "source": [
    "ensembleprob = EnsembleProblem(prob_truesde)\n",
    "@time sol = solve(ensembleprob, EM(), dt=0.05, saveat = 0.05, trajectories = 100000)\n",
    "truemean = mean(sol, dims = 3)[:, :]\n",
    "truevar = var(sol, dims = 3)[:, :];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 675,
   "id": "d02338c4",
   "metadata": {},
   "outputs": [
    {
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    "plot(LinRange(0,10,201),truemean')"
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   "execution_count": 676,
   "id": "a1ea1ba6",
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     "execution_count": 676,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(LinRange(0,10,201),truevar')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c22172b7",
   "metadata": {},
   "source": [
    "Now we generate a neural SDE that can be trained on this data. We want to generate an SDEProblem that takes in two functions describing the drift and diffusion of our system. These functions, in turn, rely on the same neural network (with the same parameters) to predict the rates for them. Note that ratesmodel() only gets to access X."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 402,
   "id": "d9180917",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ratestodiff (generic function with 2 methods)"
      ]
     },
     "execution_count": 402,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ratestodrift(u, p, t)\n",
    "    R1,R2 = ratesmodel(u[1],p)\n",
    "    #Safetly valve to keep bad parameter sets from crashing\n",
    "    if abs(u[1])<1000\n",
    "        reacprogress = R1-R2\n",
    "    else\n",
    "        reacprogress = 0\n",
    "    end\n",
    "    return [reacprogress,R1,R2]\n",
    "end\n",
    "\n",
    "function ratestodiff(u, p, t)\n",
    "    R1,R2 = ratesmodel(u[1],p)\n",
    "    return [[sqrt.(R1),sqrt.(R1),0] [-sqrt.(R2),0,sqrt.(R2)]]\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "313906ff",
   "metadata": {},
   "source": [
    "We define the neural network using FastChains, which explicitly requires you to pass in a vector of parameters. p_nn is a randomly generated set of parameters that can be used as a starting point. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 564,
   "id": "3f66b09a",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[33m\u001b[1m┌ \u001b[22m\u001b[39m\u001b[33m\u001b[1mWarning: \u001b[22m\u001b[39mFastChain is being deprecated in favor of Lux.jl. Lux.jl uses functions with explicit parameters f(u,p) like FastChain, but is fully featured and documented machine learning library. See the Lux.jl documentation for more details.\n",
      "\u001b[33m\u001b[1m└ \u001b[22m\u001b[39m\u001b[90m@ DiffEqFlux C:\\Users\\ChemeGrad2021\\.julia\\packages\\DiffEqFlux\\jHIee\\src\\fast_layers.jl:9\u001b[39m\n"
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      ]
     },
     "execution_count": 564,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratesmodel = FastChain(FastDense(1 , 16, relu), FastDense(16 , 2, relu));\n",
    "p_nn = initial_params(ratesmodel);\n",
    "\n",
    "prob = SDEProblem(ratestodrift, ratestodiff, u0, tspan, p_nn, noise_rate_prototype=zeros(3,2))\n",
    "sol = solve(prob,EM(),dt = 0.05;sensealg=TrackerAdjoint())\n",
    "plot(sol.t,Array(sol)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 616,
   "id": "4430297c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "loss_neuralsde (generic function with 1 method)"
      ]
     },
     "execution_count": 616,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function getprediction(p,u=u0)\n",
    "    prob = SDEProblem(ratestodrift, ratestodiff, u, tspan, p, noise_rate_prototype=zeros(3,2))\n",
    "    sol = solve(prob,EM(),dt = 0.05;sensealg=TrackerAdjoint())\n",
    "    if (sol.retcode != ReturnCode.Success)\n",
    "        print(p)\n",
    "    end\n",
    "    return Array(sol)\n",
    "end\n",
    "\n",
    "function predict_batch_neuralsde(p, n)\n",
    "    res = map((x->getprediction(p,x)),[u0 for i in 1:n])\n",
    "    VA = VectorOfArray(res)\n",
    "    arr = convert(Array,VA)\n",
    "end\n",
    "\n",
    "function predictmeanvar(p,n)\n",
    "    samples = predict_batch_neuralsde(p,n)\n",
    "    means = mean(samples, dims = 3)\n",
    "    vars = var(samples, dims = 3, mean = means)\n",
    "    return means,vars\n",
    "end\n",
    "\n",
    "function loss_neuralsde(p; n = 100)\n",
    "  means,vars = predictmeanvar(p,n)\n",
    "  testmeans = means[1,:,:]\n",
    "  testvars = vars[1,:,:]\n",
    "  loss = sum(abs2, truemean[1,:] - testmeans) + sum(abs2, p) + sum(abs2, truevar[1,:] - testvars)\n",
    "  return loss#, means, vars\n",
    "end\n",
    "#test = Zygote.gradient(loss_neuralsde,p_nn)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2ccd240",
   "metadata": {},
   "source": [
    "Now that we have a loss function described, we can train our neural network. I had to set the training rate hyperparameter very low to avoid weird divergence in parameter updates. As such it takes a long time. The plots below shows the results - specifically (hopefully), that as the model converges to the testing data (X means and vars) it also converges to showing the hidden values correctly (R1 and R2 (cumulative))."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 687,
   "id": "5460beb6",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "opt = ADAM(0.0001)\n",
    "\n",
    "# First round of training with n = 10\n",
    "adtype = Optimization.AutoZygote()\n",
    "optf = Optimization.OptimizationFunction((x,p) -> loss_neuralsde(x), adtype)\n",
    "optprob = Optimization.OptimizationProblem(optf, result1.u)#Change to p_nn if first time running\n",
    "result1 = Optimization.solve(optprob, opt, maxiters = 1000);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 688,
   "id": "1f24c302",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.450435625903767e6"
      ]
     },
     "execution_count": 688,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loss_neuralsde(p_nn) #Inital parameter guess"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 689,
   "id": "8910c8f4",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41499.16386260459"
      ]
     },
     "execution_count": 689,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loss_neuralsde(result1.u) #Parameters after training"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 690,
   "id": "0248ffd0",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 690,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predres = predictmeanvar(p_nn,1000)#Inital parameter guess\n",
    "plt1 = plot(tsteps,[truemean[1,:],predres[1][1,:,:]],title=\"X mean\")\n",
    "plt2 = plot(tsteps,[truevar[1,:],predres[2][1,:,:]],title=\"X var\")\n",
    "plt3 = plot(tsteps,[truemean[2,:],predres[1][2,:,:]],title=\"R1 mean\")\n",
    "plt4 = plot(tsteps,[truevar[2,:],predres[2][2,:,:]],title=\"R1 var\")\n",
    "plt5 = plot(tsteps,[truemean[3,:],predres[1][3,:,:]],title=\"R2 mean\")\n",
    "plt6 = plot(tsteps,[truevar[3,:],predres[2][3,:,:]],title=\"R2 var\")\n",
    "plot(plt1,plt2,plt3,plt4,plt5,plt6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 691,
   "id": "093bc34d",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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      ]
     },
     "execution_count": 691,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predres = predictmeanvar(result1.u,1000)#Parameters after training\n",
    "plt1 = plot(tsteps,[truemean[1,:],predres[1][1,:,:]],title=\"X mean\")\n",
    "plt2 = plot(tsteps,[truevar[1,:],predres[2][1,:,:]],title=\"X var\")\n",
    "plt3 = plot(tsteps,[truemean[2,:],predres[1][2,:,:]],title=\"R1 mean\")\n",
    "plt4 = plot(tsteps,[truevar[2,:],predres[2][2,:,:]],title=\"R1 var\")\n",
    "plt5 = plot(tsteps,[truemean[3,:],predres[1][3,:,:]],title=\"R2 mean\")\n",
    "plt6 = plot(tsteps,[truevar[3,:],predres[2][3,:,:]],title=\"R2 var\")\n",
    "plot(plt1,plt2,plt3,plt4,plt5,plt6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 710,
   "id": "a16b00fa",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 710,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res = [ratesmodel(x,result1.u) for x in testX]\n",
    "VA = VectorOfArray(res)\n",
    "arr = convert(Array,VA)\n",
    "plot(testX, arr[:,1,:]',xlabel = \"X\",ylabel = \"R rate\",label = [\"NN R1\" \"NN R2\"])\n",
    "plot!(testX, [0.2*testX,0.1*testX],label = [\"True R1\" \"True R2\"])\n"
   ]
  }
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