Terms Of Stochastic Differential Equations Research Articles

Deep Physics Corrector: A physics enhanced deep learning architecture for solving stochastic differential equations

We propose a novel gray-box modeling algorithm for physical systems governed by stochastic differential equations (SDE). The proposed approach, referred to as the Deep Physics Corrector (DPC), blends approximate physics represented in terms of SDE with deep neural network (DNN). The primary idea here is to exploit DNN to model the missing physics. We hypothesize that combining incomplete physics with data will make the model interpretable and allow better generalization. The primary bottleneck associated with training surrogate models for stochastic simulators is often associated with selecting the suitable loss function. Among the different loss functions available in the literature, we use the conditional maximum mean discrepancy (CMMD) loss function in DPC because of its proven performance. Overall, physics-data fusion and CMMD allow DPC to learn from sparse data. We illustrate the performance of the proposed DPC on four benchmark examples from the literature. The results obtained are highly accurate, indicating its possible application as a surrogate model for stochastic simulators.

Spatial parameterization of attachment processes in molecular motor-cargo systems

Intracellular transport is conducted largely by molecular motor proteins which process along cytoskeletal filaments, from which they can attach or detach. We describe an analytical framework to characterize motor attachment or reattachment rates to microtubules as a function of the physical and geometric properties of the motor, the cargo to which it is attached, and possibly a second motor attached to the same cargo and a microtubule. The biophysical model is coarse-grained at the level of the macromolecular motors and formulated in terms of stochastic differential equations, allowing for rotation of the cargo and nonlinear force laws for the motor-cargo tether.

Reconstructing complex system dynamics from time series: a method comparison

Modeling complex systems with large numbers of degrees of freedom has become a grand challenge over the past decades. In many situations, only a few variables are actually observed in terms of measured time series, while the majority of variables—which potentially interact with the observed ones—remain hidden. A typical approach is then to focus on the comparably few observed, macroscopic variables, assuming that they determine the key dynamics of the system, while the remaining ones are represented by noise. This naturally leads to an approximate, inverse modeling of such systems in terms of stochastic differential equations (SDEs), with great potential for applications from biology to finance and Earth system dynamics. A well-known approach to retrieve such SDEs from small sets of observed time series is to reconstruct the drift and diffusion terms of a Langevin equation from the data-derived Kramers–Moyal (KM) coefficients. For systems where interactions between the observed and the unobserved variables are crucial, the Mori–Zwanzig formalism (MZ) allows to derive generalized Langevin equations that contain non-Markovian terms representing these interactions. In a similar spirit, the empirical model reduction (EMR) approach has more recently been introduced. In this work we attempt to reconstruct the dynamical equations of motion of both synthetical and real-world processes, by comparing these three approaches in terms of their capability to reconstruct the dynamics and statistics of the underlying systems. Through rigorous investigation of several synthetical and real-world systems, we confirm that the performance of the three methods strongly depends on the intrinsic dynamics of the system at hand. For instance, statistical properties of systems exhibiting weak history-dependence but strong state-dependence of the noise forcing, can be approximated better by the KM method than by the MZ and EMR approaches. In such situations, the KM method is of a considerable advantage since it can directly approximate the state-dependent noise. However, limitations of the KM approximation arise in cases where non-Markovian effects are crucial in the dynamics of the system. In these situations, our numerical results indicate that methods that take into account interactions between observed and unobserved variables in terms of non-Markovian closure terms (i.e., the MZ and EMR approaches), perform comparatively better.

Probabilistic Interpretation of the Cauchy Problem Solution for Systems of Nonlinear Parabolic Equations

We reduce solution of the Cauchy problem for forward and backward systems of nonlinear parabolic equations to solution of certain stochastic problem formulated in terms of stochastic differential equations. Then we state existence and uniqueness theorems for solutions of the resulting stochastic systems and deduce existence and uniqueness of the Cauchy problem solutions for the original parabolic systems. As a by-product we derive stochastic representations of solutions to the Cauchy problem for some classes of nonlinear parabolic equation systems.

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.

P.408 Protective effect of morinda citrifolia l. Juice on cognition and memory in diazepam-induced amnesia in rats

Understanding if and how mutants reach fixation in populations is an important question in evolutionary biology. We study the impact of population growth has on the success of mutants. To systematically understand the effects of growth we decouple competition from reproduction; competition follows a birth–death process and is governed by an evolutionary game, while growth is determined by an externally controlled branching rate. In stochastic simulations we find non-monotonic behaviour of the fixation probability of mutants as the speed of growth is varied; the right amount of growth can lead to a higher success rate. These results are observed in both coordination and coexistence game scenarios, and we find that the ‘one-third law’ for coordination games can break down in the presence of growth. We also propose a simplified description in terms of stochastic differential equations to approximate the individual-based model.

Stochastic modeling of direct radiation transmission in particle-laden turbulent flow

Direct radiation transmission in turbulent flows laden with heavy particles plays a fundamental role in systems such as clouds and particle solar receivers. Owing to their inertia, particles are preferentially concentrated by the turbulence, and the resulting voids and clusters lead to deviations in mean transmission from the classical Beer-Lambert law for exponential extinction. Additionally, the transmission fluctuations can exceed those of Poissonian media by an order of magnitude, which implies a gross misprediction in transmission statistics if the correlations in particle positions are neglected. On the other hand, tracking millions of particles in a turbulence simulation can be prohibitively expensive. We propose stochastic processes as computationally inexpensive reduced order models for the instantaneous particle number density field and radiation transmission therein. Results from the stochastic processes are compared to Monte Carlo Ray Tracing (MCRT) simulations using the particle positions obtained from the point-particle direct numerical simulation (DNS) of isotropic turbulence at a Taylor Reynolds number of 150. Accurate transmission statistics are predicted with respect to MCRT by matching the mean, variance, and correlation length of the DNS number density fields. Formulation of the stochastic processes in terms of stochastic differential equations allows an exact solution for arbitrary moments of radiation transmission to be derived from the Kolmogorov backwards equations. Higher order statistics such as the transmission variance, and correlations between particle number density, transmission, and absorption are compared to MCRT and discussed.

Data Driven Approach to the Dynamics of Import and Export of G7 Countries.

The dynamics of imports plus exports of 226 product classes by the G7 countries between 1962 and 2000 is described in terms of stochastic differential equations. The model allows interesting comparisons among the different economies related to the compositions of the national baskets. Synthetic solutions can also be used to estimate hidden and unexploited growth potentials. These prerogatives are strictly connected with the fact that a network structure is at the basis of the model. Such a network expresses the mutual influences of different products through resource transfers, and is a key ingredient producing cooperative growth effects which can be quantified and distinguished from those generated by deterministic drifts and representing direct resource inputs. An analysis of this network, which differs substantially from those previously considered within the economic complexity approach, allows to estimate the centrality of different products in each national basket, highlighting the most essential commodities for each economy. Solutions of the model give the possibility of performing counterfactual analyses aimed at estimating how much the growth of each country could have profited from a general strengthening, or weakening, of the links in the same products network.

Probabilistic Approach to Finite State Mean Field Games

We study mean field games and corresponding N-player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite state mean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric $$\varepsilon _N$$-Nash equilibria for the N-player game, both in open-loop and in feedback strategies (not relaxed), with $$\varepsilon _N\le \frac{\text {constant}}{\sqrt{N}}.$$ Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity.

Intrinsic stochastic differential equations as jets

We explain how Itô stochastic differential equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We also show how jets can be used to derive graphical representations of Itô SDEs, and we show how jets can be used to derive the differential operators associated with SDEs in a coordinate-free manner. We relate jets to vector flows, giving a geometric interpretation of the Itô–Stratonovich transformation. We show how percentiles can be used to give an alternative coordinate-free interpretation of the coefficients of one-dimensional SDEs. We relate this to the jet approach. This allows us to interpret the coefficients of SDEs in terms of ‘fan diagrams’. In particular, the median of an SDE solution is associated with the drift of the SDE in Stratonovich form for small times.

Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales M. We first apply the 2D-Fock transform for operators from the testing functional space S(M) to the generalized functional space S⁎(M) and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales M. And also we prove the continuity and continuous dependence on initial values of the solution.

Stochastic load generated by the beam irregularities

The problem investigated in this paper comes from railway engineering. It is known that geometrical irregularities of the rail head rolling surface produce additional force when the train runs on track. This force can be quite significant and should not be neglected in the analysis, especially when one deals with high-speed railways. In this paper, an analytical method of modelling of such irregularities is presented. The detailed description of this method is associated with its practical application to the analysis of the rail track dynamic response to moving train. However, stochastic analysis of the presented model is omitted in this paper and left for further work. This should include a number of realisations along with statistical analysis of results, or description of the rail track subjected to moving train in terms of stochastic differential equations, which is the main direction of the authors future investigations.

Effects of population growth on the success of invading mutants

Understanding if and how mutants reach fixation in populations is an important question in evolutionary biology. We study the impact of population growth has on the success of mutants. To systematically understand the effects of growth we decouple competition from reproduction; competition follows a birth–death process and is governed by an evolutionary game, while growth is determined by an externally controlled branching rate. In stochastic simulations we find non-monotonic behaviour of the fixation probability of mutants as the speed of growth is varied; the right amount of growth can lead to a higher success rate. These results are observed in both coordination and coexistence game scenarios, and we find that the ‘one-third law’ for coordination games can break down in the presence of growth. We also propose a simplified description in terms of stochastic differential equations to approximate the individual-based model.

Collective dynamics of multimode bosonic systems induced by weak quantum measurement

In contrast to the fully projective limit of strong quantum measurement, where the evolution is locked to a small subspace (quantum Zeno dynamics), or even frozen completely (quantum Zeno effect), the weak non-projective measurement can effectively compete with standard unitary dynamics leading to nontrivial effects. Here we consider global weak measurement addressing collective variables, thus preserving quantum superpositions due to the lack of which path information. While for certainty we focus on ultracold atoms, the idea can be generalized to other multimode quantum systems, including various quantum emitters, optomechanical arrays, and purely photonic systems with multiple-path interferometers (photonic circuits). We show that light scattering from ultracold bosons in optical lattices can be used for defining macroscopically occupied spatial modes that exhibit long-range coherent dynamics. Even if the measurement strength remains constant, the quantum measurement backaction acts on the atomic ensemble quasi-periodically and induces collective oscillatory dynamics of all the atoms. We introduce an effective model for the evolution of the spatial modes and present an analytic solution showing that the quantum jumps drive the system away from its stable point. We confirm our finding describing the atomic observables in terms of stochastic differential equations.

Transient evolution of eigenmodes in dynamic cavities and time-varying media

In this paper, we investigate the perturbation of natural eigenmodes of dynamic cavities with boundaries moving at quasi-static speeds relative to the wave velocity. For an arbitrarily shaped source-free cavity, the amplitude of the irrotational mode is modeled as a damped harmonic oscillator with time-varying eigenfrequency, i.e., a parametric oscillator. It is found that the effect of the pure Doppler shift of the resonance frequencies of the eigenmodes is small at nonrelativistic speeds. However, it is known that any spectrum of eigenenergies that is perturbed by a space- and/or time-fluctuating medium can develop frequency shifts of arbitrary magnitude. By using a linear dynamic (time-dependent) shift for the cavity broad resonances, we find that Doppler-like large shifts result in a mere frequency modulation of the total (resultant) field amplitude, while nonuniform red or blue shift can create a hybrid amplitude and frequency modulation. Interestingly, the combined action of red and blue shifts of uniform magnitude can also create a hybrid modulation. If the angle between modal wave vector and stirrer speed is accounted for in the static (time-independent) shift, the resulting red and blue shifts lead to irregular hybrid modulations. This can occur even for regular perturbations in regular cavities. In addition, owing to the stochastic nature of mode-stirred cavities, the effect of random Doppler-like shifts is also investigated, leading to a Fokker-Planck equation whose diffusion coefficient shows quadratic dependence on the mode amplitude. Thus, the analysis of random perturbations offers an effective framework for observed instantaneous Doppler effects in closed electromagnetic environments. The mathematical framework obtained in terms of stochastic differential equations is useful to predict the nonstationary response of dynamic cavities with complicated or unknown boundary geometry.

A stochastic model for examining along-wind loading uncertainty and intervention costs due to wind-induced damage on tall buildings

This study describes some recent results of a numerical study on the wind-induced response of a tall building, contaminated by along-wind loading uncertainty. The study also proposes the use of “intervention costs” in the form of a dimensionless random variable, which nonlinearly depends on the dynamic response of the structure, for the examination of the structural performance during high-wind events.The CAARC building is employed as the benchmark structure. Three nonlinear reduced-order models are used to describe the dynamic response. In these models various hypotheses are introduced to systematically and efficiently simulate both loading uncertainty and cost variability. The generalized models are formulated in terms of stochastic differential equations and are approximately solved by equivalent stochastic linearization. In the case of loading uncertainty analysis, the joint probability density function of the roof-top lateral dynamic displacements in the two primary bending planes of the building is determined as a function of the mean wind speed and the standard deviation of a parametric error term. Dependence of the cross-correlation among the two response components on the input standard deviation of the parametric error term is observed. In the case of the intervention cost analysis, it is noted that the main contributing factor to the distribution of the intervention cost random variable is the mean wind speed.The combination of wind loading uncertainty and projected intervention costs on the dynamic response of the structure is the first step towards the implementation of alternative but rational analysis methods for performance-based wind engineering, applied to tall buildings.

Optimal controllability of manpower system with linear quadratic performance index

In classical manpower systems analysis, control of the system usually results in a set of admissible controls. This forms the basis for the use of the concepts of optimal control to distinguish this set of admissible controls for optimality. In this paper, the concepts of classical deterministic optimal control are extended to examine the optimal controllability of manpower system modeled by stochastic differential equations in terms of the differential flow matrices for both time varying and time invariant manpower systems. Necessary and sufficient conditions for controllability are given. The Hamilton–Jacobi–Bellman (HJB) equation is used to obtain an algebraic Riccati equation for an optimal tracking linear quadratic problem in a finite time horizon. A 2-norm optimality criterion which is equivalent to a minimum effort criterion is used to obtain a 2-norm optimal control for the system. An optimal time control is also obtained.

Constructing a stochastic model of bumblebee flights from experimental data.

The movement of organisms is subject to a multitude of influences of widely varying character: from the bio-mechanics of the individual, over the interaction with the complex environment many animals live in, to evolutionary pressure and energy constraints. As the number of factors is large, it is very hard to build comprehensive movement models. Even when movement patterns in simple environments are analysed, the organisms can display very complex behaviours. While for largely undirected motion or long observation times the dynamics can sometimes be described by isotropic random walks, usually the directional persistence due to a preference to move forward has to be accounted for, e.g., by a correlated random walk. In this paper we generalise these descriptions to a model in terms of stochastic differential equations of Langevin type, which we use to analyse experimental search flight data of foraging bumblebees. Using parameter estimates we discuss the differences and similarities to correlated random walks. From simulations we generate artificial bumblebee trajectories which we use as a validation by comparing the generated ones to the experimental data.

Active Brownian particles with velocity-alignment and active fluctuations

We consider a model of active Brownian particles (ABPs) with velocity alignment in two spatial dimensions with passive and active fluctuations. Here, active fluctuations refers to purely non-equilibrium stochastic forces correlated with the heading of an individual active particle. In the simplest case studied here, they are assumed to be independent stochastic forces parallel (speed noise) and perpendicular (angular noise) to the velocity of the particle. On the other hand, passive fluctuations are defined by a noise vector independent of the direction of motion of a particle, and may account, for example, for thermal fluctuations. We derive a macroscopic description of the ABP gas with velocity-alignment interaction. Here, we start from the individual-based description in terms of stochastic differential equations (Langevin equations) and derive equations of motion for the coarse-grained kinetic variables (density, velocity and temperature) via a moment expansion of the corresponding probability density function. We focus here on the different impact of active and passive fluctuations on onset of collective motion and show how active fluctuations in the active Brownian dynamics can change the phase-transition behaviour of the system. In particular, we show that active angular fluctuations lead to an earlier breakdown of collective motion and to the emergence of a new bistable regime in the mean-field case.

Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

In this article we study the fractal Navier-Stokes equations by using stochastic Lagrangian particle path approach in Constantin and Iyer \cite{Co-Iy}. More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by L\'evy processes. Basing on this representation, a self-contained proof for the existence of local unique solution for the fractal Navier-Stokes equation with initial data in $\mW^{1,p}$ is provided, and in the case of two dimensions or large viscosity, the existence of global solution is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for L\'evy processes with time dependent and discontinuous drifts is proved.