15 research outputs found
Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning
The mean exit time escaping basin of attraction in the presence of white
noise is of practical importance in various scientific fields. In this work, we
propose a strategy to control mean exit time of general stochastic dynamical
systems to achieve a desired value based on the quasipotential concept and
machine learning. Specifically, we develop a neural network architecture to
compute the global quasipotential function. Then we design a systematic
iterated numerical algorithm to calculate the controller for a given mean exit
time. Moreover, we identify the most probable path between metastable
attractors with help of the effective Hamilton-Jacobi scheme and the trained
neural network. Numerical experiments demonstrate that our control strategy is
effective and sufficiently accurate
Computing non-equilibrium trajectories by a deep learning approach
Predicting the occurence of rare and extreme events in complex systems is a
well-known problem in non-equilibrium physics. These events can have huge
impacts on human societies. New approaches have emerged in the last ten years,
which better estimate tail distributions. They often use large deviation
concepts without the need to perform heavy direct ensemble simulations. In
particular, a well-known approach is to derive a minimum action principle and
to find its minimizers.
The analysis of rare reactive events in non-equilibrium systems without
detailed balance is notoriously difficult either theoretically and
computationally. They are described in the limit of small noise by the
Freidlin-Wentzell action. We propose here a new method which minimizes the
geometrical action instead using neural networks: it is called deep gMAM. It
relies on a natural and simple machine-learning formulation of the classical
gMAM approach. We give a detailed description of the method as well as many
examples. These include bimodal switches in complex stochastic (partial)
differential equations, quasi-potential estimates, and extreme events in
Burgers turbulence
A Dynamical Systems Approach for Most Probable Escape Paths over Periodic Boundaries
Analyzing when noisy trajectories, in the two dimensional plane, of a
stochastic dynamical system exit the basin of attraction of a fixed point is
specifically challenging when a periodic orbit forms the boundary of the basin
of attraction. Our contention is that there is a distinguished Most Probable
Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy
escaping paths in the case of small noise slightly away from the limit of
vanishing noise. It is well known that, before exiting, noisy trajectories will
tend to cycle around the periodic orbit as the noise vanishes, but we observe
that the escaping paths are stubbornly resistant to cycling as soon as the
noise becomes at all significant. Using a geometric dynamical systems approach,
we isolate a subset of the unstable manifold of the fixed point in the
Euler-Lagrange system, which we call the River. Using the Maslov index we
identify a subset of the River which is comprised of local minimizers. The
Onsager-Machlup (OM) functional, which is treated as a perturbation of the
Friedlin-Wentzell functional, provides a selection mechanism to pick out a
specific MPEP. Much of the paper is focused on the system obtained by reversing
the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo
simulations, we show that the prediction provided by OM-selected MPEP matches
closely the escape hatch chosen by noisy trajectories at a certain level of
small noise.Comment: 28 pages, 15 figure
Numerical Geometric Acoustics
Sound propagation in air is accurately described by a small perturbation of the ambient pressure away from a quiescent state. This is the realm of linear acoustics, where the propagation of a time-harmonic wave can be modeled using the Helmholtz equation. When the wavelength is small relative to the size of a scattering obstacle, techniques from geometric optics are applicable. Geometric methods such as raytracing are often used for computational room acoustics simulations in situations where the geometry of the built environment is sufficiently complicated. At the same time, the high-frequency approximation of the Helmholtz equation is described by two partial differential equations: the eikonal equation, whose solution gives the first arrival time of a geometric acoustics/optics wavefront as a field; and a transport equation, the solution of which describes the amplitude of that wavefield. Phenomena related to high-frequency acoustic diffraction are frequently omitted from these models because of their complexity. These phenomena can be modeled using a high-frequency diffraction theory, such as the uniform theory of diffraction. Despite their shortcomings, geometric methods for room acoustics provide a useful trade-off between realism and computational efficiency.
Motivated by the limitations of geometric methods, we approach the problem of geometric acoustics using numerical methods for solving partial differential equations. Our focus is offline sound propagation in a high-frequency regime where directly solving the wave or Helmholtz equations is infeasible. To this end, we conduct a broad-based survey of semi-Lagrangian solvers for the eikonal equation, which make the local ray information of the solution explicit. We develop efficient, first-order solvers for the eikonal equation in 3D, called ordered line integral methods (OLIMs). The OLIMs provide intuition about how to design work-efficient semi-Lagrangian eikonal solvers, but their first order accuracy is not sufficient to compute the amplitude consistently. Motivated by the requirements of sound propagation simulations, we develop higher-order semi-Lagrangian eikonal solvers which we term jet marching methods (JMMs). JMMs augment the efficiency of OLIMs by additionally transporting higher-order derivative information of the eikonal in a causal fashion, which allows for high-order solution of the eikonal equation using compact stencils. We use the information made available locally by our JMMs to use paraxial raytracing to simultaneously solve the transport equation yielding the amplitude. We initially develop a JMM which handles a smoothly varying speed of sound on a regular grid in 2D. Motivated by the requirements of room acoustics applications, we develop a second-order JMM for solving the eikonal equation on a tetrahedron mesh for a constant speed of sound as a special case. As before, we use paraxial raytracing to compute the amplitude. Additionally, we compute multiple arrivals by reinitializing the eikonal equation on reflecting walls and diffracting edges. To compute these scattered fields, we devise algorithms which allow us to apply reflection and diffraction boundary conditions for the eikonal and amplitude. For the amplitude, we construct algorithms that allow us to apply the uniform theory of diffraction in a semi-Lagrangian setting efficiently
Emergent Structure and Dynamics from Stochastic Pairwise Crosslinking in Chromosomal Polymer Models
The spatio-temporal organization of the genome is critical to the ability of the cell to store huge amounts of information in highly compacted DNA while also performing vital cellular functions. Experimental methods provide a window into the geometry of the chromatin but cannot provide a full picture in space and time.Polymer models have been shown to reproduce properties of chromatin and can be used to make simulated observations, informing biological experimentation. We apply a previously-studied model of the full yeast genome with dynamic protein crosslinking in the nucleolus which showed the emergence of clustering when the crosslinking timescale was sufficiently fast. We investigate the the crosslinking timescale at finer resolution and newly identify the presence of a \textit{flexible clustering} regime for intermediate timescales, which maximizes mixing of nucleolar beads, of significant interest due to the role mixing plays in nuclear processes. In order to robustly identify spatio-temporal clustering structure, we map our problem to a multi-layer network and then apply the multi-layer modularity community detection algorithm, showing the presence of spatio-temporal community structure in the fast and intermediate clustering regimes. We perform analysis of the relationship between cluster size and the ensuing stability of clusters,revealing a heterogeneous collection of clusters in which cluster size correlates with stability. We view the stochastic switching as producing an effective thermal equilibrium byextending the WKB approach for deriving quasipotentials in switching systems to the case of an overdamped Langevin equation with switching force term, and derive the associated Hamilton-Jacobi equation. We apply the string method for finding most-probable transition paths, revealing previously unreported numerical challenges; we present modifications to the algorithms to overcome them. We show that our methods can correctly compute asymptotic escape times by comparison to Monte Carlo simulations, and verified an important principle: the effective force is often significantly weaker than a naive average of the switching suggests. Through this multifaceted approach, we have shown how stochastic crosslinking leads to complex emergent structure, with different timescales optimizing different properties, and shown how the structure can be analyzed using both network data based tools and through stochastic averaging principles.Doctor of Philosoph