Tobias Grafke

Jiayao Shao, Tobias Grafke, and Robert S MacKay

Abstract

Stochastic dynamical systems allow modelling of transitions induced by disturbances, in particular from an attracting equilibrium and crossing the stable manifold of a saddle. In the small-noise limit, the probability of such transitions is governed by a large deviation principle. We illustrate a computational approach-the Method of Division-for approximating rare transition events, including their most likely paths, transition rates, and associated probabilities. To cater for realistic applications, we allow unbounded time, coloured and degenerate forcing. Its effectiveness is demonstrated on two examples: an inverted double-well potential and a simplified roll-heave ship capsize model.

arXiv

Paolo Bernuzzi, Tobias Grafke, SIAM Multiscale Model. Simul. 23(3) (2025) 1274

Abstract

Noise-induced transitions between multistable states happen in a multitude of systems, such as species extinction in biology, protein folding, or tipping points in climate science. Large deviation theory is the rigorous language to describe such transitions for non-equilibrium systems in the small noise limit. At its core, it requires the computation of the most likely transition pathway, solution to a PDE constrained optimization problem. Standard methods struggle to compute the minimiser in the particular coexistence of (1) multistability, i.e. coexistence of multiple long-lived states, and (2) degenerate noise, i.e. stochastic forcing acting only on a small subset of the system's degrees of freedom. In this paper, we demonstrate how to adapt existing methods to compute the large deviation minimiser in this setting by combining ideas from optimal control, large deviation theory, and numerical optimisation. We show the efficiency of the introduced method in various applications in biology, medicine, and fluid dynamics, including the transition to turbulence in subcritical pipe flow.

doi:10.1137/24M169713X

arXiv

Summary: In stochastic systems, multiple long-lived states might coexists, in which the system spends a large amount of time before spontaneously and often catastrophically switching to a different long-lived state. For example, in molecular dynamics, the reactant and the product state both correspond to local minima in an energy landscape, but thermal activation pushed the system across the separating energy barrier, corresponding to the chemical reaction. Similarly, Earth's climate might spend a long time in a certain long-lived state, but climate change might push it across a tipping point, such that the state of the climate abruptly changes. Often, these changes are hard to predict and rare on the intrinsic time scale of the system. In this project, algorithms are developed to simulate only those trajectories of the stochastic system that exhibit the transition. This transition path sampling or bridge sampling allows to generate for example conditional statistics, analyze physical causes responsible for the transition, or even develop early warning signs for an anticipated transition in progress.

Relevant publications

1. T. Grafke, "Sampling conditioned diffusions via Pathspace Projected Monte Carlo", ArXiv (2025) (link)

2. T. Grafke, A. Laio, "Metadynamics for transition paths in irreversible dynamics", Multiscale Modeling & Simulation, Vol. 22, Iss. 1 (2024) (link)

Tobias Grafke

Abstract

We present an algorithm to sample stochastic differential equations conditioned on rather general constraints, including integral constraints, endpoint constraints, and stochastic integral constraints. The algorithm is a pathspace Metropolis-adjusted manifold sampling scheme, which samples stochastic paths on the submanifold of realizations that adhere to the conditioning constraint. We demonstrate the effectiveness of the algorithm by sampling a dynamical condensation phase transition, conditioning a random walk on a fixed Levy stochastic area, conditioning a stochastic nonlinear wave equation on high amplitude waves, and sampling a stochastic partial differential equation model of turbulent pipe flow conditioned on relaminarization events.

arXiv

A. Svirsky, T. Grafke, and A. Frishman

Abstract

Pipe flow is a canonical example of a subcritical flow, the transition to turbulence requiring a finite perturbation. The Reynolds number (Re) serves as the control parameter for this transition, going from the ordered (laminar) to the chaotic (turbulent) phase with increasing Re. Just above the critical Re, where turbulence can be sustained indefinitely, turbulence spreads via the self-replication of localized turbulent structures called puffs. To reveal the workings behind this process, we consider transitions between one and two-puff states, which dynamically are transitions between two distinct chaotic saddles. We use direct numerical simulations to explore the phase space boundary between these saddles, adapting a bisection algorithm to identify an attracting state on the boundary, termed an edge state. At Re=2200, we also examine spontaneous transitions between the two saddles, demonstrating the relevance of the found edge state to puff self-replication. Our analysis reveals that the process of self-replication follows a previously proposed splitting mechanism, with the found edge state as its tipping point. Additionally, we report results for lower values of Re, where the bisection algorithm yields a different type of edge state. As we cannot directly observe splits at this Re, the self-replication mechanism here remains an open question. Our analysis suggests how this question could be addressed in future studies, and paves the way to probing the turbulence proliferation mechanism in other subcritical flows.

arXiv

Jelle Soons, Tobias Grafke, and Henk A. Dijkstra, J. Fluid Mech. 1009 (2025), A53

Abstract

There is a reasonable possibility that the present-day Atlantic Meridional Overturning Circulation is in a bi-stable regime and hence it is relevant to compute probabilities and pathways of noise-induced transitions between the stable equilibrium states. Here, the most probable transition pathway of a noise-induced collapse of the northern overturning circulation in a spatially-continuous two-dimensional model with surface temperature and stochastic salinity forcings is directly computed using Large Deviation Theory (LDT). This pathway reveals the fluid dynamical mechanisms of such a collapse. Paradoxically it starts off with a strengthening of the northern overturning circulation before a short but strong salinity pulse induces a second overturning cell. The increased atmospheric energy input of this two-cell configuration cannot be mixed away quickly enough, leading to the collapse of the northern overturning cell and finally resulting in a southern overturning circulation. Additionally, the approach allows us to compare the probability of this collapse under different parameters in the deterministic part of the salinity surface forcing, which quantifies the increase in collapse probability as the bifurcation point of the system is approached.

doi:10.1017/jfm.2025.248

arXiv

Jelle Soons, Tobias Grafke, René M. van Westen, and Henk A. Dijkstra

Abstract

Recently the global average temperature has temporarily exceeded the 1.5°C goal of the Paris Agreement, and so an overshoot of various climate tipping elements becomes increasingly likely. In this study we analyze the physical processes of an overshoot of the Atlantic Meridional Overturning Circulation (AMOC), one of the major tipping elements, using a conceptual box model. Here either the atmospheric temperature above the North Atlantic, or the freshwater forcing into the North Atlantic overshoot their respective critical boundaries. In both cases a higher forcing rate can prevent a collapse of the AMOC, since a higher rate of forcing causes initially a fresher North Atlantic, which in turn results in a higher northward transport by the subtropical gyre supplementing the salinity loss in time. For small exceedance amplitudes the AMOC is still resilient as the forcing rates can be low and so other state variables outside of the North Atlantic can adjust. Contrarily, for larger overshoots the trajectories are dynamically similar and we find a lower limit in volume and exceedance time for respectively freshwater and temperature forcing in order to prevent a collapse. Moreover, for a large overshoot an increased air-sea temperature coupling has a destabilizing effect, while the reverse holds for an overshoot close to the tipping point. The understanding of the physics of the AMOC overshoot behavior is important for interpreting results of Earth System Models and for evaluating the effects of mitigation and intervention strategies.

arXiv

T. Schorlepp, and T. Grafke

Abstract

We show how to efficiently compute asymptotically sharp estimates of extreme event probabilities in stochastic differential equations (SDEs) with small multiplicative Brownian noise. The underlying approximation is known as sharp large deviation theory or precise Laplace asymptotics in mathematics, the second-order reliability method (SORM) in reliability engineering, and the instanton or optimal fluctuation method with 1-loop corrections in physics. It is based on approximating the tail probability in question with the most probable realization of the stochastic process, and local perturbations around this realization. We first recall and contextualize the relevant classical theoretical result on precise Laplace asymptotics of diffusion processes [Ben Arous (1988), Stochastics, 25(3), 125-153], and then show how to compute the involved infinite-dimensional quantities — operator traces and Carleman-Fredholm determinants — numerically in a way that is scalable with respect to the time discretization and remains feasible in high spatial dimensions. Using tools from automatic differentiation, we achieve a straightforward black-box numerical computation of the SORM estimates in JAX. The method is illustrated in examples of SDEs and stochastic partial differential equations, including a two-dimensional random advection-diffusion model of a passive scalar. We thereby demonstrate that it is possible to obtain efficient and accurate SORM estimates for very high-dimensional problems, as long as the infinite-dimensional structure of the problem is correctly taken into account. Our JAX implementation of the method is made publicly available.

arXiv