The Trajectory-Adaptive Multilevel Sampling (TAMS) is a promising method to determine probabilities of noise-induced transition in multi-stable high-dimensional dynamical systems. In this paper, we focus on two improvements of the current algorithm related to (i) the choice of the target set and (ii) the formulation of the score function. In particular, we use confidence ellipsoids determined from linearised dynamics in the choice of the target set. Furthermore, we define a score function based on empirical transition paths computed at relatively high noise levels. The suggested new TAMS method is applied to two typical problems illustrating the benefits of the modifications.

Systems from various areas of physics exhibit multiple stable states. In such multi-stable systems, transitions between states can occur as a result of small-scale processes, usually referred to as noise-induced transitions

A central issue in models of these multi-stable systems is the computation of transition probabilities between different states. If we exclude very special classes of systems, analytical results are generally not available. The Eyring–Kramers formula

In order to sample tails of distributions more effectively, various methods have been developed, generally referred to as rare-event algorithms. One of the promising methods to compute transition probabilities is the Trajectory-Adaptive Multilevel Sampling (TAMS) method

The aim of this work is to propose improvements to the use of the TAMS algorithm to be able to compute transitions in multi-stable systems more efficiently. The first type of improvement is the choice of the target set, which is often determined from rather arbitrary thresholds. This choice also raises more broadly the question of a precise definition of what we consider a noise-induced transition between two (stable) states. In the second type of improvement, we propose a more systematic method of defining a score function, i.e. based on empirical transition paths. The modified TAMS method is first applied to an idealised gradient system and then to a system representing a box model of the AMOC

In Sect. 2, we describe the methods developed to improve the TAMS algorithm. In Sect. 3 we show how to incorporate these techniques into the definition of the score function and present the results for idealised dynamical systems and the AMOC model. A discussion follows in Sect. 4, assessing the strengths and the limitations of the new TAMS method.

We consider finite-dimensional dynamical systems described by stochastic differential equations (SDEs) of the following form:

A prominent example of such a system is a model of a free particle moving in a two-dimensional double-well potential (with

The transition probability that a trajectory starting in

Illustration of the TAMS algorithm. First, simulate

Consider a general SDE (

Once the score function has been chosen, a threshold needs to be defined for the TAMS algorithm to converge, so that the occurrence of a transition can be detected. In other words, we do not expect each trajectory that undergoes a transition to reach exactly the destination equilibrium

For this purpose, we use the concept of a confidence ellipsoid. This is an ellipsoidal neighbourhood around a stable equilibrium state, inside which a trajectory subject to the locally linearised dynamics stays, within a certain confidence level

Confidence ellipsoids for the two-dimensional double-well potential system (Eq.

Consider a general SDE system given by Eq. (

The

The second line of improvement of the score function concerns the estimation of typical transition paths of the dynamical system. In the zero noise limit, the Freidlin–Wentzell theory of large deviations predicts that transition paths will cluster around the most probable transition path, called the instanton

The idea is to first accumulate transition paths at a noise level where transitions are frequent enough (typically

The main steps of the path-finding algorithm are listed below:

the trajectory of the typical transition path starts in the box of the histogram containing the initial state

the next box in this trajectory corresponds to the neighbour which has the highest nonzero histogram value but which has not already been visited by the typical transition path;

the algorithm stops if it reaches the box containing the target state

In a general system, transitions induced at high noise do not necessarily follow similar transition paths at lower noise levels. However, high-noise estimates are robust as long as there is no drastic change in the behaviour of the system at an intermediate noise level, which would signal a physical phase transition in the system

In this section, we apply both modifications to TAMS (ellipsoids in the score function and typical path estimation) to different problems.

First, we apply the modified TAMS method to the two-dimensional double-well system defined by Fig.

We first compute the level

In order to show how to design a score function based on a typical transition path, we consider a two-dimensional system slightly less trivial than the double-well system, i.e. a two-dimensional system with the following potential,

The dynamics of the system is quite interesting, as it exhibits two distinct regimes for transition paths, depending on the noise level

Figure

Given a typical transition path

The score function increases from 0 to 1 along the trajectory. Thus, it encodes the preferred direction that the system has to follow. This contrasts with the generic score functions

Next, we applied TAMS with the path-based score function

We show in Fig.

The performances of the numerical methods are next measured using the work-normalised relative error

The results are shown in Fig.

Finally, as a main application of the modification of the target set

For a differential-algebraic system of equations (DAEs), such as the system in Eq. (

For the system in Eq. (

Level sets of the score function

When constructing an improved score function, in order to evaluate whether a state belongs to the neighbourhood of

To be able to assess the relevance of a proper definition of the target set in the TAMS algorithm and hence the importance of using the improved version of the score function, we computed transition probabilities of the AMOC from the present-climate state to the collapsed state for reasonable values of the atmospheric forcing and noise. In particular, we chose

Results of the transition probabilities for the AMOC model using different score functions.

When running TAMS to compute transition probabilities between two states of the Atlantic circulation in this model, with different versions of the score function, we found a considerable discrepancy between the obtained values. In particular, it appears that setting a very high threshold in the Gaussian score function makes the algorithm detect no transitions (we set up the algorithm so that it stops when the probabilities involved are smaller than

We presented and applied several improvements to the TAMS rare-event algorithm, when used to compute transitions in multistable systems. The first improvement was based on a more rigorous criterion to define noise-induced transitions involving confidence ellipsoids

This method, while quite general, has several limitations. While the modified score function

Next, we proposed a systematic method of defining a score function, designed to approximate the static committor, based on empirical transition paths. We proposed an algorithm to estimate the typical transition path under a high noise level, which is then used to define a family of score functions with a single decay parameter

One key limitation of our approach of constructing the path-based score function

Another way to define the reduced space

Another potential issue of our modified TAMS method is the fact that the score function

Further testing of the ideas presented in this work in high-dimensional systems such as discretised PDEs would give more insight as to the effectiveness of our approach compared to the more generic score functions used up to now. Moreover, incorporating some form of time dependence into the score function

The equations determining the evolution of the AMOC in this model are the salinity budgets of the different boxes, together with the variation of the volume of the pycnocline, and the salt and volume conservation equations

Reference parameters used in Eqs. (

The software is available at

Datasets for Figs. 6 and 8 are available from Wang and Castellana (

PW and DC designed the algorithms, ran the simulations and prepared the figures. All the authors discussed the results and contributed to the writing of the manuscript.

The authors declare that they have no conflict of interest.

The authors thank Sven Baars for insightful discussions. Pascal Wang acknowledges the hospitality of the Institute for Marine and Atmospheric research Utrecht where the scientific work was conducted.

This research has been supported by the European Commission, H2020 Research Infrastructures (CRITICS (grant no. 643073)).

This paper was edited by Balasubramanya Nadiga and reviewed by two anonymous referees.