Large deviation theory

min_action_method(sys::ContinuousTimeDynamicalSystem, x_i, x_f, T::Real; kwargs...)

Minimizes an action functional to obtain a minimum action path (instanton) between an initial state x_i and final state x_f in phase space.

This algorithm uses the Optimization.jl package to minimize the specified action functional (either fw_action or om_action) for the system sys over paths connecting x_i to x_f in time T.

The path is initialized as a straight line between x_i and x_f, parameterized in time via N equidistant points and total time T. Thus, the time step between discretized path points is $\Delta t = T/N$. To set an initial path different from a straight line, see the multiple dispatch method

min_action_method(sys::ContinuousTimeDynamicalSystem, init::Matrix, T::Real; kwargs...).

Returns a MinimumActionPath object containing the optimized path and the action value.

Keyword arguments

• functional = "FW": type of action functional to minimize. Defaults to fw_action, alternative: "OM" for om_action.
• N = 100: number of path points to use for the discretization of the path.
• noise_strength = nothing: noise strength for the action functional. Specify only if functional = "OM".
• method = Optimisers.Adam(): minimization algorithm (see Optimization.jl)
• ad_type = Optimization.AutoFiniteDiff(): type of automatic differentiation to use (see Optimization.jl)
• maxiter = 100: maximum number of iterations before the algorithm stops.
• abstol=1e-8: absolute tolerance of action gradient to determine convergence
• reltol=1e-8: relative tolerance of action gradient to determine convergence
• verbose = true: whether to print Optimization information during the run
• show_progress = false: whether to print a progress bar

min_action_method(sys::ContinuousTimeDynamicalSystem, init::Matrix, T::Real; kwargs...)

Minimizes the specified action functional to obtain a minimum action path (instanton) between fixed end points given a system sys and total path time T.

The initial path init must be a matrix of size (D, N), where D is the dimension of the system and N is the number of path points. The physical time of the path is specified by T, such that the time step between consecutive path points is $\Delta t = T/N$.

geometric_min_action_method(
    sys::ContinuousTimeDynamicalSystem,
    x_i,
    x_f;
    N,
    kwargs...
) -> MinimumActionPath{_A, _B, _C, Nothing, Nothing, Nothing, Nothing, Nothing} where {_A, _B<:Real, _C}

Computes the minimizer of the geometric Freidlin-Wentzell action based on the geometric minimum action method (gMAM), using optimizers of OPtimization.jl or the original formulation by Heymann and Vanden-Eijnden^[1]. Only Freidlin-Wentzell action has a geometric formulation.

To set an initial path different from a straight line, see the multiple dispatch method

• geometric_min_action_method(sys::CoupledSDEs, init::Matrix, arclength::Real; kwargs...).

Keyword arguments

• maxiter::Int=100: maximum number of optimization iterations before the algorithm stops
• abstol=1e-8: absolute tolerance of action gradient to determine convergence
• reltol=1e-8: relative tolerance of action gradient to determine convergence
• method = Adam(): minimization algorithm (see Optimization.jl)
• =0.1: step size parameter in gradient descent HeymannVandenEijnden implementation.
• verbose=false: if true, print additional output
• show_progress=true: if true, display a progress bar

Optimization algorithms

The method keyword argument takes solver methods of the Optimization.jl package; alternatively, the option solver = "HeymannVandenEijnden" uses the original gMAM algorithm^[1].

geometric_min_action_method(
    sys::ContinuousTimeDynamicalSystem,
    init::Matrix;
    maxiter,
    abstol,
    reltol,
    method,
    AD,
    ϵ,
    verbose,
    show_progress
) -> MinimumActionPath{_A, _B, _C, Nothing, Nothing, Nothing, Nothing, Nothing} where {_A, _B<:Real, _C}

Runs the geometric Minimum Action Method (gMAM) to find the minimum action path (instanton) from an initial condition init, given a system sys and total arc length arclength.

The initial path init must be a matrix of size (D, N), where D is the dimension of the system and N is the number of path points.

For more information see the main method, geometric_min_action_method(sys::CoupledSDEs, x_i, x_f, arclength::Real; kwargs...).

sgmam(
    sys::SgmamSystem,
    x_initial::Array{T, 2};
    ϵ,
    iterations,
    show_progress,
    reltol
) -> MinimumActionPath{_A, _B, _C, Nothing, Nothing, Matrix{Float64}, Matrix{Float64}, Matrix{Float64}} where {_A, _B<:Real, _C}

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 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 implementation is based on the work of Grafke et al. (2019).

A structure representing a system with Hamiltonian functions Hx and Hp.

This system operates in an extended phase space where the Hamiltonian is assumed to be quadratic in the extended momentum. The phase space coordinates x are doubled to form a 2n-dimensional extended phase space.

struct MinimumActionPath{D, T<:Real, V, Phis, Ahis, Lambda, PV, GPV}

The minimum action path between two points in a D-dimensional phase space.

Fields

• path::StateSpaceSet{D, T, V} where {D, T<:Real, V}: The path matrix.

• action::Real: The action value associated to the path.

• path_history::Any: The history of action of the paths in the optimisation algorithm (optional).

• action_history::Any: The history of action of the paths in the optimisation algorithm (optional).

• λ::Any: The Lagrange multiplier parameter for the minimum action path (optional).

• generalized_momentum::Any: The generalized momentum of the phase space variables (optional).

• path_velocity::Any: The path velocity (optional).

Constructors

MinimumActionPath(
    path,
    action;
    path_history,
    action_history,
    λ,
    generalized_momentum,
    path_velocity
)

defined at /home/runner/work/CriticalTransitions.jl/CriticalTransitions.jl/src/largedeviations/MinimumActionPath.jl:29.

fw_action(sys::CoupledSDEs, path, time) -> Any

Calculates the Freidlin-Wentzell action of a given path with time points time in a drift field specified by the deterministic dynamics f = dynamic_rule(sys) and (normalized) noise covariance matrix covariance_matrix(sys).

The path must be a (D x N) matrix, where D is the dimensionality of the system sys and N is the number of path points. The time array must have length N.

Returns a single number, which is the value of the action functional

$S_T[\phi_t] = \frac{1}{2} \int_0^T || \dot \phi_t - f(\phi_t) ||^2_Q \text{d}t$

where $\phi_t$ denotes the path in state space, $b$ the drift field, and $T$ the total time of the path. The subscript $Q$ refers to the generalized norm $||a||_Q^2 := \langle a, Q^{-1} b \rangle$ (see anorm). Here $Q$ is the noise covariance matrix normalized by $D/L_1(Q)$, with $L_1$ being the L1 matrix norm.

geometric_action(sys::CoupledSDEs, path) -> Any
geometric_action(sys::CoupledSDEs, path, arclength) -> Any

Calculates the geometric action of a given path with specified arclength for the drift field specified by the deterministic dynamics f = dynamic_rule(sys) and (normalized) noise covariance matrix covariance_matrix(sys).

For a given path $\varphi$, the geometric action $\bar S$ corresponds to the minimum of the Freidlin-Wentzell action $S_T(\varphi)$ over all travel times $T>0$, where $\varphi$ denotes the path's parameterization in physical time (see fw_action). It is given by the integral

$\bar S[\varphi] = \int_0^L \left( ||\varphi'||_Q \, ||f(\varphi)||_Q - \langle \varphi', \, f(\varphi) \rangle_Q \right) \, \text{d}s$

where $s$ is the arclength coordinate, $L$ the arclength, $f$ the drift field, and the subscript $Q$ refers to the generalized dot product $\langle a, b \rangle_Q := a^{\top} \cdot Q^{-1} b$ (see anorm). Here $Q$ is the noise covariance matrix normalized by $D/L_1(Q)$, with $L_1$ being the L1 matrix norm.

Returns the value of the geometric action $\bar S$.

om_action(sys::CoupledSDEs, path, time, noise_strength)

Calculates the Onsager-Machlup action of a given path with time points time for the drift field f = dynamic_rule(sys) at given noise_strength.

The path must be a (D x N) matrix, where D is the dimensionality of the system sys and N is the number of path points. The time array must have length N.

Returns a single number, which is the value of the action functional

$S^{\sigma}_T[\phi_t] = \frac{1}{2} \int_0^T \left( || \dot \phi_t - f(\phi_t) ||^2_Q + \sigma^2 \nabla \cdot f \right) \, \text{d} t$

where $\phi_t$ denotes the path in state space, $b$ the drift field, $T$ the total time of the path, and $\sigma$ the noise strength. The subscript $Q$ refers to the generalized norm $||a||_Q^2 := \langle a, Q^{-1} b \rangle$ (see anorm). Here $Q$ is the noise covariance matrix normalized by $D/L_1(Q)$, with $L_1$ being the L1 matrix norm.

action(
    sys::CoupledSDEs,
    path::Matrix,
    time,
    functional;
    noise_strength
) -> Any

Computes the action functional specified by functional for a given CoupledSDEs sys and path parameterized by time.

• functional = "FW": Returns the Freidlin-Wentzell action (fw_action)
• functional = "OM": Returns the Onsager-Machlup action (om_action)

string_method(
    sys::Union{Function, SgmamSystem},
    x_initial::Matrix;
    ϵ,
    iterations,
    show_progress
) -> Any

Compute the string method for a given system using E et al. (2007).

The string method is an iterative algorithm used to find minimum energy path (MEP) between two points in phase space. It works by evolving a discretized path (string) according to the system's drift while maintaining equal arc-length parametrization between points.

This implementation allows for computation between arbitrary points, not just stable fixed points.

Arguments

• sys::SgmamSystem: The doubled phase space system for which the string method is computed
• x_initial: Initial path discretized as a matrix where each column represents a point on the path
• ϵ::Real: Step size for the evolution step
• iterations::Int64: Maximum number of iterations for path convergence
• show_progress::Bool: Whether to display a progress meter during computation

Returns

• x: The final converged path representing the MEP

string_method(
    sys::ContinuousTimeDynamicalSystem,
    init;
    kwargs...
) -> Any

Compute the string method for a given system using E et al. (2007).

The string method is an iterative algorithm used to find minimum energy path (MEP) between two points in phase space. It works by evolving a discretized path (string) according to the system's drift while maintaining equal arc-length parametrization between points.

This implementation allows for computation between arbitrary points, not just stable fixed points.

Arguments

• sys::CoupledSDEs: The system for which the string method is computed
• x_initial: Initial path discretized as a matrix where each column represents a point on the path
• ϵ::Real: Step size for the evolution step
• iterations::Int64: Maximum number of iterations for path convergence
• show_progress::Bool: Whether to display a progress meter during computation

Returns

• x: The final converged path representing the MEP