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A stochastic variant of replicator dynamics in zero-sum games and its invariant measures
No full textWe study the behavior of a stochastic variant of replicator dynamics in two-agent zero-sum games. We characterize the statistics of such systems by their invariant measures which can be shown to be entirely supported on the boundary of the space of mixed strategies. Depending on the noise strength we can furthermore characterize these invariant measures by finding accumulation of mass at specific parts of the boundary. In particular, regardless of the magnitude of noise, we show that any invariant probability measure is a convex combination of Dirac measures on pure strategy profiles, which correspond to vertices/corners of the agents' simplices. Thus, in the presence of stochastic perturbations, even in the most classic zero-sum settings, such as Matching Pennies, we observe a stark disagreement between the axiomatic prediction of Nash equilibrium and the evolutionary emergent behavior derived by an assumption of stochastically adaptive, learning agents
A multi-view latent variable model reveals cellular heterogeneity in complex tissues for paired multimodal single-cell data
No full textMotivation:
Single-cell multimodal assays allow us to simultaneously measure two different molecular features of the same cell, enabling new insights into cellular heterogeneity, cell development and diseases. However, most existing methods suffer from inaccurate dimensionality reduction for the joint-modality data, hindering their discovery of novel or rare cell subpopulations.
Results:
Here, we present VIMCCA, a computational framework based on variational-assisted multi-view canonical correlation analysis to integrate paired multimodal single-cell data. Our statistical model uses a common latent variable to interpret the common source of variances in two different data modalities. Our approach jointly learns an inference model and two modality-specific non-linear models by leveraging variational inference and deep learning. We perform VIMCCA and compare it with 10 existing state-of-the-art algorithms on four paired multi-modal datasets sequenced by different protocols. Results demonstrate that VIMCCA facilitates integrating various types of joint-modality data, thus leading to more reliable and accurate downstream analysis. VIMCCA improves our ability to identify novel or rare cell subtypes compared to existing widely used methods. Besides, it can also facilitate inferring cell lineage based on joint-modality profiles.
Availability and implementation:
The VIMCCA algorithm has been implemented in our toolkit package scbean (≥0.5.0), and its code has been archived at https://github.com/jhu99/scbean under MIT license
Nonuniqueness of generalised weak solutions to the primitive and Prandtl equations
Get PDFWe develop a convex integration scheme for constructing nonunique weak solutions
to the hydrostatic Euler equations (also known as the inviscid primitive equations of
oceanic and atmospheric dynamics) in both two and three dimensions. We also develop
such a scheme for the construction of nonunique weak solutions to the three-dimensional
viscous primitive equations, as well as the two-dimensional Prandtl equations.
While in [D.W. Boutros, S. Markfelder and E.S. Titi, arXiv:2208.08334 (2022)] the
classical notion of weak solution to the hydrostatic Euler equations was generalised,
we introduce here a further generalisation. For such generalised weak solutions we
show the existence and nonuniqueness for a large class of initial data. Moreover, we
construct infinitely many examples of generalised weak solutions which do not conserve
energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler
equations (which are the average and the fluctuation of the horizontal velocity in the
z-coordinate, respectively) that are constructed have different regularities
Spatially resolved protein map of intact human cytomegalovirus virions
No full textHerpesviruses assemble large enveloped particles that are difficult to characterize structurally due to their size, fragility and complex multilayered proteome with partially amorphous nature. Here we used crosslinking mass spectrometry and quantitative proteomics to derive a spatially resolved interactome map of intact human cytomegalovirus virions. This enabled the de novo allocation of 32 viral proteins into four spatially resolved virion layers, each organized by a dominant viral scaffold protein. The viral protein UL32 engages with all layers in an N-to-C-terminal radial orientation, bridging nucleocapsid to viral envelope. We observed the layer-specific incorporation of 82 host proteins, of which 39 are selectively recruited. We uncovered how UL32, by recruitment of PP-1 phosphatase, antagonizes binding to 14-3-3 proteins. This mechanism assures effective viral biogenesis, suggesting a perturbing role of UL32-14-3-3 interaction. Finally, we integrated these data into a coarse-grained model to provide global insights into the native configuration of virus and host protein interactions inside herpesvirions
On the pitchfork bifurcation for the Chafee-Infante equation with additive noise
No full textWe investigate pitchfork bifurcations for a stochastic reaction diffusion equation perturbed by an infinite-dimensional Wiener process. It is well-known that the random attractor is a singleton, independently of the value of the bifurcation parameter; this phenomenon is often referred to as the “destruction” of the bifurcation by the noise. Analogous to the results of Callaway et al. (AIHP Prob Stat 53:1548–1574, 2017) for a 1D stochastic ODE, we show that some remnant of the bifurcation persists for this SPDE model in the form of a positive finite-time Lyapunov exponent. Additionally, we prove finite-time expansion of volume with increasing dimension as the bifurcation parameter crosses further eigenvalues of the Laplacian
A porous-media model for reactive fluid-rock interaction in a dehydrating rock
Get PDFWe study the GENERIC structure of models for reactive two-phase flows and their connection to
a porous-media model for reactive fluid-rock interaction used in Geosciences. For this we discuss the
equilibration of fast dissipative processes in the GENERIC framework. Mathematical properties of
the porous-media model and first results on its mathematical analysis are provided. The mathematical
assumptions imposed for the analysis are critically validated with the thermodynamical rock data
sets
Interpolating Between BSDEs and PINNs: Deep Learning for Elliptic and Parabolic Boundary Value Problems
No full textSolving high-dimensional partial differential equations is a recurrent challenge in economics, science and en-
gineering. In recent years, a great number of computational approaches have been developed, most of them
relying on a combination of Monte Carlo sampling and deep learning based approximation. For elliptic and
parabolic problems, existing methods can broadly be classified into those resting on reformulations in terms
of backward stochastic differential equations (BSDEs) and those aiming to minimize a regression-type L2-error
(physics-informed neural networks, PINNs). In this paper, we review the literature and suggest a methodology
based on the novel diffusion loss that interpolates between BSDEs and PINNs. Our contribution opens the door
towards a unified understanding of numerical approaches for high-dimensional PDEs, as well as for implementa-
tions that combine the strengths of BSDEs and PINNs. We also provide generalizations to eigenvalue problems
and perform extensive numerical studies, including calculations of the ground state for nonlinear Schrödinger
operators and committor functions relevant in molecular dynamics
Enhancing ECG Analysis of Implantable Cardiac Monitor Data: An Efficient Pipeline for Multi-Label Classification
No full textImplantable Cardiac Monitor (ICM) devices are demonstrating as of today, the fastest-growing market for implantable cardiac devices. As such, they are becoming increasingly common in patients for measuring heart electrical activity. ICMs constantly monitor and record a patient's heart rhythm and when triggered - send it to a secure server where health care professionals (denote HCPs from here on) can review it. These devices employ a relatively simplistic rule-based algorithm (due to energy consumption constraints) to alert for abnormal heart rhythms. This algorithm is usually parameterized to an over-sensitive mode in order to not miss a case (resulting in a relatively high false-positive rate) and this, combined with the device's nature of constantly monitoring the heart rhythm and its growing popularity, results in HCPs having to analyze and diagnose an increasingly growing amount of data. In order to reduce the load on the latter, automated methods for ECG analysis are nowadays becoming a great tool to assist HCPs in their analysis. While state-of-the-art algorithms are data-driven rather than rule-based, training data for ICMs often consist of specific characteristics that make its analysis unique and particularly challenging. This study presents the challenges and solutions in automatically analyzing ICM data and introduces a method for its classification that outperforms existing methods on such data. It does so by combining high-frequency noise detection (which often occurs in ICM data) with a semi-supervised learning pipeline that allows for re-labeling of training episodes, and by using segmentation and dimension reduction techniques that are robust to morphology variations of the sECG signal (which are typical to ICM data). As a result, it performs better than state-of-the-art techniques on such data with e.g. F1 score of 0.51 vs. 0.38 of our baseline state-of-the-art technique in correctly calling Atrial Fibrilation in ICM data. As such, it could be used in numerous ways such as aiding HCPs in the analysis of ECGs originating from ICMs by, e.g., suggesting a rhythm type
Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation
Get PDFWe study the vanishing viscosity/diffusivity limit for the transport equation of a passive scalar f(x,t)∈R along a divergence-free vector field u(x,t)∈R2, given by ∂f∂t+∇⋅(uf)=0; and the associated advection-diffusion equation of f along u for positive viscosity/diffusivity parameter ν>0, expressed by ∂f∂t+∇⋅(uf)−νΔf=0. We demonstrate failure of the vanishing viscosity limit of the advection-diffusion equation to select unique solutions, or to select entropy-admissible solutions, to transport along u.
First, we construct a bounded divergence-free vector field u which admits, for each (non-constant) initial datum, two weak solutions to the initial value problem for the transport equation. Moreover, we show that both these solutions are renormalised weak solutions, and are strong limits along different subsequences of vanishing viscosity of solutions to the corresponding advection-diffusion equation.
Second, we construct a second bounded divergence-free vector field u admitting, for any initial datum, a weak solution to the transport equation which is perfectly mixed to its spatial average, and after some delay in time, it unmixes to its initial state. Moreover, we show that this entropy-inadmissible unmixing is the unique weak vanishing viscosity limit of the corresponding advection-diffusion equation
WaveTrain: A Python Package for Numerical Quantum Mechanics of Chain-Like Systems Based on Tensor Trains
No full textWaveTrain is an open-source software for numerical simulations of chain-like quantum systems with nearest-neighbor (NN) interactions only. The Python package is centered around tensor train (TT, or matrix product) format representations of Hamiltonian operators and (stationary or time-evolving) state vectors. It builds on the Python tensor train toolbox Scikit_tt, which provides efficient construction methods and storage schemes for the TT format. Its solvers for eigenvalue problems and linear differential equations are used in WaveTrain for the time-independent and time-dependent Schrödinger equations, respectively. Employing efficient decompositions to construct low-rank representations, the tensor-train ranks of state vectors are often found to depend only marginally on the chain length N. This results in the computational effort growing only slightly more than linearly with N, thus mitigating the curse of dimensionality. As a complement to the classes for full quantum mechanics, WaveTrain also contains classes for fully classical and mixed quantum-classical (Ehrenfest or mean field) dynamics of bipartite systems. The graphical capabilities allow visualization of quantum dynamics ‘on the fly’, with a choice of several different representations based on reduced density matrices. Even though developed for treating quasi one-dimensional excitonic energy transport in molecular solids or conjugated organic polymers, including coupling to phonons, WaveTrain can be used for any kind of chain-like quantum systems, with or without periodic boundary conditions, and with NN interactions only.
The present work describes version 1.0 of our WaveTrain software, based on version 1.2 of Scikit_tt, both of which are freely available from the GitHub platform where they will also be further developed. Moreover, WaveTrain is mirrored at SourceForge, within the framework of the WavePacket project for numerical quantum dynamics. Worked-out demonstration examples with complete input and output, including animated graphics, are available