Computing large deviation prefactors of stochastic dynamical systems based on machine learning

We study the large deviation function of the displacement of a Brownian particle confined on a ring. In the zero noise limit this large deviation function has a cusp at zero velocity given by the Freidlin–Wentzell theory. We develop a WKB approach to analyse how this cusp is rounded in the weak noise limit.

The currents generated by noise-induced activation processes in a periodic potential are investigated analytically, by digital simulation and by performing analog experiments. The noise is taken to be quasimonochromatic and the potential to be a smoothed sawtooth. Two analytic approaches are studied. The first involves a perturbative expansion in inverse powers of the frequency characterizing quasimonochromatic noise and the second is a direct numerical integration of the deterministic differential equations obtained in the limit of weak noise. These results, together with the digital and analog experiments, show that the system does indeed give rise, in general, to a net transport of particles. All techniques also show that a current reversal exists for a particular value of the noise parameters.

The emergence of transition phenomena between metastable states induced by noise plays a fundamental role in a broad range of nonlinear systems. The computation of the most probable paths is a key issue to understanding the mechanism of transition behaviors. The shooting method is a common technique for this purpose to solve the Euler-Lagrange equation for the associated action functional, while losing its efficacy in high-dimensional systems. In the present work, we develop a machine learning framework to compute the most probable paths in the sense of Onsager-Machlup action functional theory. Specifically, we reformulate the boundary value problem of a Hamiltonian system and design a neural network to remedy the shortcomings of the shooting method. The successful applications of our algorithms to several prototypical examples demonstrate its efficacy and accuracy for stochastic systems with both (Gaussian) Brownian noise and (non-Gaussian) Lévy noise. This approach is effective in exploring the internal mechanisms of rare events triggered by random fluctuations in various scientific fields.

We present several numerical approaches to investigate rare events in stochastic systems, with a specific focus on application to biological models. We first present several aspects concerning variance reduction of Monte-Carlo methods, with a focus on importance sampling. We show how these methods can be applied to basic biological models. We show that these techniques can be useful in dealing with multiscale continuous-time Markov chains that are important in the context of biochemical reaction networks. We treat with more detail the problem of first passage time for a linear diffusion process, arising from the integrate-and-fire neuron model, to show the kind of mathematical problems that may arise when trying to design an optimal importance sampling scheme. This leads to the observation that large deviation theory can be helpful to solve these questions. We also review a numerical method designed to estimate large deviations quantities such as the quasipotential and the optimal path and apply this method to estimate numerically the quasipotential of the Morris–Lecar neuron model perturbed by small noise.

Here we study large deviations in networks that are more likely to result from the accumulation of many slightly unusual events. We are particularly interested in analyzing large deviations where the most probable path changes direction. These deviations arise when a large deviation in one part of the system cascades across the network to produce a large deviation in another part of the network.

We present a new framework for finding the optimal transition paths of metastable stochastic chemical kinetic systems with large system sizes. The optimal transition paths are identified, in terms of reaction advancement coordinates, to be the most probable paths according to large deviation theory for the limiting dynamics governed by stochastic differential equations. Dynamical equations for the optimal transition paths are obtained using the variational principle. A multiscale minimum action method is proposed as a numerical scheme to solve the optimal transition paths. Applications to the toggle switch model are presented.

In this paper we propose a simple and efficient method for computing accurate estimates (in closed form) of the first passage time density of the Ornstein-Uhlenbeck neuronal model through a fixed boundary (i.e. the interspike statistics of the stochastic leaky integrate-and-fire neuron model). This new approach can also provide very tight upper and lower bounds (in closed form) for the exact first passage time density in a systematic manner. Unlike previous approximate analytical attempts, this novel approximation scheme not only goes beyond the linear response and weak noise limit, but it can also be systematically improved to yield the exact results. Furthermore, it is straightforward to extend our approach to study the more general case of a deterministically modulated boundary.KeywordsStochastic ResonanceBarrier OptionRegular Spike CellGauss Quadrature MethodPassage Time ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Marginal steady state belongs to a special class of states in nonlinear dynamics. To realize this state we consider a cubic autocatalytic reaction A + 2B → 3B in a continuous-stirred-tank-reactor, where the flow rate of the reactant A can be controlled to manipulate the dynamical behavior of the open system. We demonstrate that when the flow rate is weakly noisy the autocatalytic reaction admits of a steady state which is marginal in nature and is surrounded by infinite number of periodic trajectories. When the uncatalyzed reaction A → B is included in the reaction scheme, there exists a marginal steady state which is a critical state corresponding to the point of transition between the flow branch and the equilibrium branch, similar to gas-liquid critical point of transition. This state loses its stability in the weak noise limit.

The position of propagating population fronts fluctuates because of the discreteness of the individuals and stochastic character of processes of birth, death, and migration. Here we consider a Markov model of a population front propagating into a metastable state, and focus on the weak noise limit. For typical, small fluctuations the front motion is diffusive, and we calculate the front diffusion coefficient. We also determine the probability distribution of rare, large fluctuations of the front position and, for a given average front velocity, find the most likely population density profile of the front. Implications of the theory for population extinction risk are briefly considered.

The theory of LΩEGF and the scaling theory on multidimensional Fokker-Planck equations with gradient potential are applied to derive the probability transition rate for Brownian motion in a symmetric two-dimensional bistable potential. Based on the rate method in adiabatic elimination, the phenomenon of stochastic resonance is studied in this system. Compared with the known results, the analytic expression for signal-to-noise ratio derived in this study is independent of the intensity of friction,i.e. not only applicable to an overdamped system, but also applicable to a lightly damped system. Numerical simulations fruther verify the significance of the analytically approximate result.

Learning, large deviations and rare events

Robust weighted kernel logistic regression in imbalanced and rare events data

AutoScore-Imbalance: An interpretable machine learning tool for development of clinical scores with rare events data

As a primary branch of Operations Research, Queueing Theory models and analyzes engineering systems with random fluctuations. With the development of internet and computation techniques, the engineering systems today are much bigger in scale and more complicated in structure than 20 years ago, which raises numerous new problems to researchers in the field of queueing theory. The aim of this thesis is to explore new methods and tools, from both algorithmic and analytical perspectives, that are useful to solve such problems. In Chapter 1 and 2, we introduce some techniques of asymptotic analysis that are relatively new to queueing applications in order to give more accurate probabilistic characterization of queueing models with large scale and complicated structure. In particular, Chapter 1 gives the first functional large deviation result for infinite-server system with general inter-arrival and service times. The functional approach we use enables a nice description of the whole system over the entire time horizon of interest, which is important in real problems. In Chapter 2, we construct a queueing model for the so-called limit order book that is used in main financial markets worldwide. We use an asymptotic approach called multi-scale modeling to disentangle the complicated dependence among the elements in the trading system and to reduce the model dimensionality. The asymptotic regime we use is inspired by empirical observations and the resulting limit process explains and reproduces stylized features of real market data. Chapter 2 also provides a nice example of novel applications of queueing models in systems, such as the electronic trading system, that are traditionally outside the scope of queueing theory. Chapter 3 and 4 focus on stochastic simulation methods for performance evaluation of queueing models where analytic approaches fail. In Chapter 3, we develop a perfect sampling algorithm to generate exact samples from the stationary distribution of stochastic fluid networks in polynomial time. Our approach can be used for time-varying networks with general inter-arrival and service times, whose stationary distributions have no analytic expression. In Chapter 4, we focus on the stochastic systems with continuous random fluctuations, for instance, the workload arrives to the system in continuous flow like a Levy process. We develop a general framework of simulation algorithms featuring a deterministic error bound and an almost square root convergence rate. As an application, we apply this framework to estimate the stationary distributions of reflected Brownian motions and the performance of our algorithm is better than existing prevalent numeric methods.

A hybrid modelling framework of machine learning and extreme value theory for crash risk estimation using traffic conflicts