453 research outputs found
Neural signature kernels as infinite-width-depth-limits of controlled ResNets
Motivated by the paradigm of reservoir computing, we consider randomly
initialized controlled ResNets defined as Euler-discretizations of neural
controlled differential equations (Neural CDEs). We show that in the
infinite-width-then-depth limit and under proper scaling, these architectures
converge weakly to Gaussian processes indexed on some spaces of continuous
paths and with kernels satisfying certain partial differential equations (PDEs)
varying according to the choice of activation function. In the special case
where the activation is the identity, we show that the equation reduces to a
linear PDE and the limiting kernel agrees with the signature kernel of Salvi et
al. (2021). In this setting, we also show that the width-depth limits commute.
We name this new family of limiting kernels neural signature kernels. Finally,
we show that in the infinite-depth regime, finite-width controlled ResNets
converge in distribution to Neural CDEs with random vector fields which,
depending on whether the weights are shared across layers, are either
time-independent and Gaussian or behave like a matrix-valued Brownian motion
Electron Thermal Runaway in Atmospheric Electrified Gases: a microscopic approach
Thesis elaborated from 2018 to 2023 at the Instituto de AstrofĂsica de AndalucĂa under the supervision of Alejandro Luque (Granada, Spain) and Nikolai Lehtinen (Bergen, Norway). This thesis presents a new database of atmospheric electron-molecule collision cross sections which was published separately under the DOI :
With this new database and a new super-electron management algorithm which significantly enhances high-energy electron statistics at previously unresolved ratios, the thesis explores general facets of the electron thermal runaway process relevant to atmospheric discharges under various conditions of the temperature and gas composition as can be encountered in the wake and formation of discharge channels
Optimal control of bioproduction in the presence of population heterogeneity
International audienceCell-to-cell variability, born of stochastic chemical kinetics, persists even in large isogenic populations. In the study of single-cell dynamics this is typically accounted for. However, on the population level this source of heterogeneity is often sidelined to avoid the inevitable complexity it introduces. The homogeneous models used instead are more tractable but risk disagreeing with their heterogeneous counterparts and may thus lead to severely suboptimal control of bioproduction. In this work, we introduce a comprehensive mathematical framework for solving bioproduction optimal control problems in the presence of heterogeneity. We study population-level models in which such heterogeneity is retained, and propose order-reduction approximation techniques. The reduced-order models take forms typical of homogeneous bioproduction models, making them a useful benchmark by which to study the importance of heterogeneity. Moreover, the derivation from the heterogeneous setting sheds light on parameter selection in ways a direct homogeneous outlook cannot, and reveals the source of approximation error. With view to optimally controlling bioproduction in microbial communities, we ask the question: when does optimising the reduced-order models produce strategies that work well in the presence of population heterogeneity? We show that, in some cases, homogeneous approximations provide remarkably accurate surrogate models. Nevertheless, we also demonstrate that this is not uniformly true: overlooking the heterogeneity can lead to significantly suboptimal control strategies. In these cases, the heterogeneous tools and perspective are crucial to optimise bioproduction
Action for classical, quantum, closed and open systems
The action functional can be used to define classical, quantum, closed, and
open dynamics in a generalization of the variational principle and in the path
integral formalism in classical and quantum dynamics, respectively. These
schemes are based on an unusual feature, a formal redoubling of the degrees of
freedom. Five arguments to motivate such a redoubling are put forward to
demonstrate that such a formalism is natural. The common elements of the
different arguments is the causal time arrow. Some lessons concerning
decoherence, dissipation and the classical limits are mentioned, too.Comment: 39 pages 4 figure
Demographic effects of aggregation in the presence of a component Allee effect
Intraspecific interactions are key drivers of population dynamics because
they establish relations between individual fitness and population density. The
component Allee effect is defined as a positive correlation between any fitness
component of a focal organism and population density, and it can lead to
positive density dependence in the population per capita growth rate. The
spatial structure is key to determining whether and to which extent a component
Allee effect will manifest at the demographic level because it determines how
individuals interact with one another. However, existing spatial models to
study the Allee effect impose a fixed spatial structure, which limits our
understanding of how a component Allee effect and the spatial dynamics jointly
determine the existence of demographic Allee effects. To fill this gap, we
introduce a spatially-explicit theoretical framework where spatial structure
and population dynamics are emergent properties of the individual-level
demographic rates. Depending on the intensity of the individual processes the
population exhibits a variety of spatial patterns that determine the
demographic-level by-products of an existing individual-level component Allee
effect. We find that aggregation increases population abundance and allows
populations to survive in harsher environments and at lower global population
densities when compared with uniformly distributed organisms. Moreover,
aggregation can prevent the component Allee effect from manifesting at the
population level or restrict it to the level of each independent group. These
results provide a mechanistic understanding of how component Allee effects
operate for different spatial population structures and show at the population
level. Our results contribute to better understanding population dynamics in
the presence of Allee effects and can potentially inform population management
strategies
Comparing real and synthetic observations of protostellar disks
Nascent envelope disk structures around protostars play a crucial role in the process of star and planet formation. As ALMA reveals unprecedented details of the envelope, disk, and outflow structures in nearby protostellar systems, a consistent interpretation for these observations remains absent, instead, highly simplified models are often adopted to partially fit the observed features. In this project, we aim to generate more realistic synthetic observations of the nascent protostellar disk and envelope system, using existing radiation and non-ideal magnetohydrodynamic simulations of protostellar collapse and disk formation. The main goal of the project is to provide multi-facet interpretation of the current continuum and polarization observations of protostellar sources at their earliest stages, and offer more realistic constraints on the dust growth in the early protoplanetary disks
Mixed Chebyshev and Legendre polynomials differentiation matrices for solving initial-boundary value problems
A new form of basis functions structures has been constructed. These basis functions constitute a mix of Chebyshev polynomials and Legendre polynomials. The main purpose of these structures is to present several forms of differentiation matrices. These matrices were built from the perspective of pseudospectral approximation. Also, an investigation of the error analysis for the proposed expansion has been done. Then, we showed the presented matrices' efficiency and accuracy with several test functions. Consequently, the correctness of our matrices is demonstrated by solving ordinary differential equations and some initial boundary value problems. Finally, some comparisons between the presented approximations, exact solutions, and other methods ensured the efficiency and accuracy of the proposed matrices
Selected Topics in Gravity, Field Theory and Quantum Mechanics
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories
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Multiscale Modeling with Meshfree Methods
Multiscale modeling has become an important tool in material mechanics because material behavior can exhibit varied properties across different length scales. The use of multiscale modeling is essential for accurately capturing these characteristics and predicting material behavior. Mesh-free methods have also been gaining attention in recent years due to their innate ability to handle complex geometries and large deformations. These methods provide greater flexibility and efficiency in modeling complex material behavior, especially for problems involving discontinuities, such as fractures and cracks. Moreover, mesh-free methods can be easily extended to multiple lengths and time scales, making them particularly suitable for multiscale modeling.
The thesis focuses on two specific problems of multiscale modeling with mesh-free methods. The first problem is the atomistically informed constitutive model for the study of high-pressure induced densification of silica glass. Molecular Dynamics (MD) simulations are carried out to study the atomistic level responses of fused silica under different pressure and strain-rate levels, Based on the data obtained from the MD simulations, a novel continuum-based multiplicative hyper-elasto-plasticity model that accounts for the anomalous densification behavior is developed and then parameterized using polynomial regression and deep learning techniques. To incorporate dynamic damage evolution, a plasticity-damage variable that controls the shrinkage of the yield surface is introduced and integrated into the elasto-plasticity model. The resulting coupled elasto-plasticity-damage model is reformulated to a non-ordinary state-based peridynamics (NOSB-PD) model for the computational efficiency of impact simulations. The developed peridynamics (PD) model reproduces coarse-scale quantities of interest found in MD simulations and can simulate at a component level. Finally, the proposed atomistically-informed multiplicative hyper-elasto-plasticity-damage model has been validated against limited available experimental results for the simulation of hyper-velocity impact simulation of projectiles on silica glass targets.
The second problem addressed in the thesis involves the upscaling approach for multi-porosity media, analyzed using the so-called MultiSPH method, which is a sequential SPH (Smoothed Particle Hydrodynamics) solver across multiple scales. Multi-porosity media is commonly found in natural and industrial materials, and their behavior is not easily captured with traditional numerical methods. The upscaling approach presented in the thesis is demonstrated on a porous medium consisting of three scales, it involves using SPH methods to characterize the behavior of individual pores at the microscopic scale and then using a homogenization technique to upscale to the meso and macroscopic level. The accuracy of the MultiSPH approach is confirmed by comparing the results with analytical solutions for simple microstructures, as well as detailed single-scale SPH simulations and experimental data for more complex microstructures
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