Merge branch 'master' into docs_update

.buildkite/pipeline.yml

@@ -40,3 +40,23 @@ steps:
     timeout_in_minutes: 680
     # Don't run Buildkite if the commit message includes the text [skip tests]
     if: build.message !~ /\[skip tests\]/
+
+steps:
+  - label: "Documentation"
+    plugins:
+      - JuliaCI/julia#v1:
+          version: "1"
+    command: |
+      julia --project=docs -e '
+        println("--- :julia: Instantiating project")
+        using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()
+        println("+++ :julia: Building documentation")
+        include("docs/make.jl")'
+    agents:
+      queue: "juliagpu"
+      cuda: "*"
+    if: build.message !~ /\[skip docs\]/ && !build.pull_request.draft
+    timeout_in_minutes: 1000
+
+env:
+  SECRET_DOCUMENTER_KEY: "yBJHrf5zUPu3VFotb0W4TRdauMTGgNUei3ax0xrVTmcrrP3EX8zSGaj9MNZji9H6JqyNEhCqZMGcHhxR5XK0f97YjhAp5rDNpwV6FbIXY8FXgFyIOLAenYUklta1W6fNM7KTz3Dq3UnKBAprKhQBwUw3SldTuvtm+IhiVT2rFmADqxMSQcv+5LivYEgAFrrd6LX+PHhonj38VN46z5Bi3TOIGAnByVlZssX7cOwaRg/8TloLPsWAUlQiPr89Vdlow9k6SQV8W9mf00/Rq4LFd1Eq1EYTCSmReVawMxVpXh1mj7MSabf9ppVWYOmjP5Rzatk8YFWlJf80HVvzY7tXRQ==;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"

docs/pages.jl

@@ -1,6 +1,6 @@
 pages = ["index.md",
     "ODE PINN Tutorials" => Any["Introduction to NeuralPDE for ODEs" => "tutorials/ode.md",
-                                "Baysian PINNs - Lotka-Volterra" => "examples/Lotka_Volterra_BPINNs.md"
+                                "Bayesian PINNs for Coupled ODEs - Lotka-Volterra" => "examples/Lotka_Volterra_BPINNs.md"
                                 #"examples/nnrode_example.md", # currently incorrect
                                 ],
     "PDE PINN Tutorials" => Any["Introduction to NeuralPDE for PDEs" => "tutorials/pdesystem.md",

docs/src/examples/Lotka_Volterra_BPINNs.md

@@ -8,12 +8,12 @@ The Lotka–Volterra equations, also known as the predator–prey equations, are
 These differential equations are frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey.
 The populations change through time according to the pair of equations
 
-$$
+```math
 \begin{aligned}
 \frac{\mathrm{d}x}{\mathrm{d}t} &= (\alpha - \beta y(t))x(t), \\
 \frac{\mathrm{d}y}{\mathrm{d}t} &= (\delta x(t) - \gamma)y(t)
 \end{aligned}
-$$
+``` 
 
 where $x(t)$ and $y(t)$ denote the populations of prey and predator at time $t$, respectively, and $\alpha, \beta, \gamma, \delta$ are positive parameters.
 
@@ -23,7 +23,9 @@ We then solve the equations and estimate the parameters of the model with priors
 
 And also solve the equations for the constructed ODEProblem's provided ideal `p` values using a Lux.jl Neural Network, chain_lux.
 
-```julia 
+```julia
+using NeuralPDE, Flux, Lux, Plots, StatsPlots, OrdinaryDiffEq, Distributions 
+
 function lotka_volterra(u, p, t)
     # Model parameters.
     α, β, γ, δ = p
@@ -43,11 +45,11 @@ p = [1.5, 1.0, 3.0, 1.0]
 tspan = (0.0, 6.0)
 prob = ODEProblem(lotka_volterra, u0, tspan, p)
 
-# Plot simulation.
 ```
 With the [`saveat` argument](https://docs.sciml.ai/latest/basics/common_solver_opts/) we can specify that the solution is stored only at `saveat` time units(default saveat=1 / 50.0).
 
 ```julia
+# Plot solution got by Standard DifferentialEquations.jl ODE solver
 solution = solve(prob, Tsit5(); saveat = 0.05)
 plot(solve(prob, Tsit5()))
 
@@ -58,67 +60,49 @@ To make the example more realistic we add random normally distributed noise to t
 
 
 ```julia
-# Dataset creation for parameter estimation
+# Dataset creation for parameter estimation (30% noise)
 time = solution.t
-u = hcat(solution.u...) 
-x = u[1, :] + 0.5 * randn(length(u[1, :]))
-y = u[2, :] + 0.5 * randn(length(u[1, :]))
+u = hcat(solution.u...)
+x = u[1, :] + (0.3 .*u[1, :]).*randn(length(u[1, :]))
+y = u[2, :] + (0.3 .*u[2, :]).*randn(length(u[2, :]))
 dataset = [x, y, time]
 
 # Neural Networks must have 2 outputs as u -> [dx,dy] in function lotka_volterra()
-chainflux = Flux.Chain(Flux.Dense(1, 6, tanh), Flux.Dense(6, 6, tanh), Flux.Dense(6, 2)) |> Flux.f64
+chainflux = Flux.Chain(Flux.Dense(1, 6, tanh), Flux.Dense(6, 6, tanh), 
+                       Flux.Dense(6, 2)) |> Flux.f64
+chainlux = Lux.Chain(Lux.Dense(1, 7, Lux.tanh), Lux.Dense(7, 7, Lux.tanh),
+                    Lux.Dense(7, 2))
 
-chainlux = Lux.Chain(Lux.Dense(1, 6, Lux.tanh), Lux.Dense(6, 6, Lux.tanh), Lux.Dense(6, 2))
 ```
 A Dataset is required as parameter estimation is being done using provided priors in `param` keyword argument for BNNODE.
 
 ```julia
 alg1 = NeuralPDE.BNNODE(chainflux,
     dataset = dataset,
     draw_samples = 1000,
-    l2std = [
-        0.05,
-        0.05,
-    ],
-    phystd = [
-        0.05,
-        0.05,
-    ],
-    priorsNNw = (0.0,
-        3.0),
+    l2std = [0.1, 0.1],
+    phystd = [0.1, 0.1],
+    priorsNNw = (0.0, 3.0),
     param = [
-        Normal(1,
-            2),
-        Normal(2,
-            2),
-        Normal(2,
-            2),
-        Normal(0,
-            2),
-    ],
-    n_leapfrog = 30, progress = true)
+        Normal(1, 2),
+        Normal(2, 2),
+        Normal(2, 2),
+        Normal(0, 2),
+    ], progress = true)
 
 sol_flux_pestim = solve(prob, alg1)
 
 # Dataset not needed as we are solving the equation with ideal parameters
 alg2 = NeuralPDE.BNNODE(chainlux,
     draw_samples = 1000,
-    l2std = [
-        0.05,
-        0.05,
-    ],
-    phystd = [
-        0.05,
-        0.05,
-    ],
-    priorsNNw = (0.0,
-        3.0),
-    n_leapfrog = 30, progress = true)
+    phystd = [0.05, 0.05],
+    priorsNNw = (0.0, 10.0),
+    progress = true)
 
 sol_lux = solve(prob, alg2)
 
 #testing timepoints must match keyword arg `saveat`` timepoints of solve() call
-t=collect(Float64,prob.tspan[1]:1/50.0:prob.tspan[2])
+t = collect(Float64, prob.tspan[1]:(1 / 50.0):prob.tspan[2]) 
 
 ```
 
@@ -133,13 +117,14 @@ plot(t,sol_flux_pestim.ensemblesol[1])
 plot!(t,sol_flux_pestim.ensemblesol[2])
 
 # estimated ODE parameters by .estimated_ode_params, weights and biases by .estimated_nn_params
+println(sol_flux_pestim.estimated_ode_params)
 sol_flux_pestim.estimated_nn_params
-sol_flux_pestim.estimated_ode_params
-
+
 # plotting solution for x,y for chain_lux
-plot(t,sol_lux_pestim.ensemblesol[1])
-plot!(t,sol_lux_pestim.ensemblesol[2])
+plot(t,sol_lux.ensemblesol[1])
+plot!(t,sol_lux.ensemblesol[2])
 
-# estimated weights and biases by .estimated_nn_params for chain_lux
-sol_lux_pestim.estimated_nn_params
-```
+# estimated weights and biases by .estimated_nn_params for chain_lux 
+sol_lux.estimated_nn_params
+
+```

docs/src/examples/ks.md

@@ -6,13 +6,13 @@ Let's consider the Kuramoto–Sivashinsky equation, which contains a 4th-order d
 ∂_t u(x, t) + u(x, t) ∂_x u(x, t) + \alpha ∂^2_x u(x, t) + \beta ∂^3_x u(x, t) + \gamma ∂^4_x u(x, t) =  0 \, ,
 ```
 
-where `\alpha = \gamma = 1` and `\beta = 4`. The exact solution is:
+where $\alpha = \gamma = 1$ and $\beta = 4$. The exact solution is:
 
 ```math
 u_e(x, t) = 11 + 15 \tanh \theta - 15 \tanh^2 \theta - 15 \tanh^3 \theta \, ,
 ```
 
-where `\theta = t - x/2` and with initial and boundary conditions:
+where $\theta = t - x/2$ and with initial and boundary conditions:
 
 ```math
 \begin{align*}

docs/src/tutorials/param_estim.md

@@ -10,7 +10,7 @@ Consider a Lorenz System,
 \end{align*}
 ```
 
-with Physics-Informed Neural Networks. Now we would consider the case where we want to optimize the parameters `\sigma`, `\beta`, and `\rho`.
+with Physics-Informed Neural Networks. Now we would consider the case where we want to optimize the parameters $\sigma$, $\beta$, and $\rho$.
 
 We start by defining the problem,