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docs: add sgMAM example (#131)
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oameye authored Dec 24, 2024

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3 changes: 2 additions & 1 deletion docs/pages.jl
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@@ -4,7 +4,8 @@ pages = [
"Quickstart" => "quickstart.md",
"Tutorial" => "examples/tutorial.md",
"Examples" => Any[
"Instanton for the Maier-Stein system" => "examples/gMAM_Maierstein.md"
"Anlyses for the Maier-Stein system" => "examples/gMAM_Maierstein.md",
"sgMAM for the Kerr Parametric Oscillator" => "examples/sgMAM_KPO.md",
],
"Manual" => Any[
"Define a CoupledSDEs system" => "man/CoupledSDEs.md",
93 changes: 93 additions & 0 deletions docs/src/examples/sgMAM_KPO.md
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We demonstrate the simple geometric minimum action method (sgMAM) on the Kerr parametric oscillator (KPO). The method computes the optimal path between two attractors in the phase space that minimizes the action of the system. It is a simplification of the geometric minimum action method (gMAM) by avoiding the computation of the second order derivatives of the extended Hamiltonian of the optimisation problem.

```@example GMAM
using CriticalTransitions, CairoMakie
```

The KPO equation is a nonlinear ordinary differential equation that describes the response of the nonlinear parametrically driven resonator at its dominant resonant condition. The equation of motion are of the form:
```math
\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) + \sigma\mathbf{ξ}(t)
```
where `f` is an autonemous drift function and and we have brownian noise `ξ` with intensity `σ`.

Here we define the define the drift of each seperable variable `u` and `v`. In addition, we hard-code the Jacobian of the drift function.
```@example GMAM
const λ = 3 / 1.21 * 2 / 295
const ω0 = 1.000
const ω = 1.000
const γ = 1 / 295
const η = 0
const α = -1
function fu(u, v)
return (-4 * γ * ω * u - 2 * λ * v - 4 * (ω0 - ω^2) * v - 3 * α * v * (u^2 + v^2)) /
(8 * ω)
end
function fv(u, v)
return (-4 * γ * ω * v - 2 * λ * u + 4 * (ω0 - ω^2) * u + 3 * α * u * (u^2 + v^2)) /
(8 * ω)
end
stream(u,v) = Point2f(fu(u, v), fv(u, v))
dfvdv(u, v) = (-4 * γ * ω + 6 * α * u * v) / (8 * ω)
dfudu(u, v) = (-4 * γ * ω - 6 * α * u * v) / (8 * ω)
dfvdu(u, v) = (-2 * λ + 4 * (ω0 - ω^2) + 9 * α * u^2 + 3 * α * v^2) / (8 * ω)
dfudv(u, v) = (-2 * λ - 4 * (ω0 - ω^2) - 3 * α * u^2 - 9 * α * v^2) / (8 * ω)
```

The optimisation is performed in a doubled phase space, i.e., every variable of the SDE system is considered as a generelised coordinate $\mathbf{x}$ and gets a corresponding generalised momentum $\mathbf{p}$. The makes it that also systems with dissipative flow can be solved. As such, we extend the phase space by defining the hamiltionian
```math
H = \sum_i \frac{p_i^2}{2} + f_i(\mathbf{x})p_i
```
Hence, to use the sgMAM method, we need to define the derivatives of the Hamiltonian with respect to the phase space coordinates and the generalised momentum:
```@example GMAM
function H_x(x, p) # ℜ² → ℜ²
u, v = eachrow(x)
pu, pv = eachrow(p)
H_u = @. pu * dfudu(u, v) + pv * dfvdu(u, v)
H_v = @. pu * dfudv(u, v) + pv * dfvdv(u, v)
return Matrix([H_u H_v]')
end
function H_p(x, p) # ℜ² → ℜ²
u, v = eachrow(x)
pu, pv = eachrow(p)
H_pu = @. pu + fu(u, v)
H_pv = @. pv + fv(u, v)
return Matrix([H_pu H_pv]')
end
sys = SgmamSystem(H_x, H_p)
```
We saved this function in the `SgmamSystem` struct. We want to find the optimal path between two attractors in the phase space. We define the initial trajectory as `wiggle` between the two attractors.
```@example GMAM
# setup
Nt = 500 # number of discrete time steps
s = collect(range(0; stop=1, length=Nt))
xa = [-0.0208, 0.0991]
xb = -xa
xsaddle = [0.0, 0.0]
# Initial trajectory
xx = @. (xb[1] - xa[1]) * s + xa[1] + 4 * s * (1 - s) * xsaddle[1]
yy = @. (xb[2] - xa[2]) * s + xa[2] + 4 * s * (1 - s) * xsaddle[2] + 0.01 * sin(2π * s)
x_initial = Matrix([xx yy]')
```
The optimisation is the performed by the `sgmam` function:
```@example GMAM
x_min, S_min, lambda, p, xdot = sgmam(
sys, x_initial; iterations=1_000, ϵ=10e2, show_progress=false
)
```
The function returns the optimal path `x_min`, the minmal action `S_min`, the Lagrange multipliers `lambda` associated with the optimal path, the optimal generalised momentum `p`, and the time derivative of the optimal path `xdot`. We can plot the initial trajectory and the optimal path:
```@example GMAM
fig, ax, _ = lines(x_initial[1, :], x_initial[2, :]; label="init", linewidth=3, color=:black)
lines!(x_min[1, :], x_min[2, :]; label="MLP", linewidth=3, color=:red)
streamplot!(ax, stream, (-0.08, 0.08), (-0.15, 0.15);
gridsize=(20, 20), arrow_size=10, stepsize=0.001,
colormap=[:gray, :gray]
)
axislegend(ax)
fig
```
16 changes: 10 additions & 6 deletions src/largedeviations/sgMAM.jl
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@@ -27,16 +27,20 @@ end
$(TYPEDSIGNATURES)
Performs the simplified geometric Minimal Action Method (sgMAM) on the given system `sys`.
Our implementation is only valid for additive noise.
This method computes the optimal path in the phase space of a Hamiltonian system that
minimizes the action. The Hamiltonian functions `H_x` and `H_p` define the system's dynamics
in a doubled phase. The initial state `x_initial` is evolved iteratively using constrained
gradient descent with step size parameter `ϵ` over a specified number of iterations.
The method can display a progress meter and will stop early if the relative tolerance
`reltol` is achieved.
minimizes the Freidlin–Wentzell action. The Hamiltonian functions `H_x` and `H_p` define
the system's dynamics in a doubled phase. The initial state `x_initial` is evolved
iteratively using constrained gradient descent with step size parameter `ϵ` over a specified
number of iterations. The method can display a progress meter and will stop early if the
relative tolerance `reltol` is achieved.
The function returns a tuple containing the final state, the action value,
the Lagrange multipliers, the momentum, and the state derivatives.
the Lagrange multipliers, the momentum, and the state derivatives. The implementation is
based on the work of [Grafke et al. (2019)](
https://homepages.warwick.ac.uk/staff/T.Grafke/simplified-geometric-minimum-action-method-for-the-computation-of-instantons.html.
).
"""
function sgmam(
sys::SgmamSystem,

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