47 research outputs found
The Differential Counting Polynomial
The aim of this paper is a quantitative analysis of the solution set of a
system of polynomial nonlinear differential equations, both in the ordinary and
partial case. Therefore, we introduce the differential counting polynomial, a
common generalization of the dimension polynomial and the (algebraic) counting
polynomial. Under mild additional asumptions, the differential counting
polynomial decides whether a given set of solutions of a system of differential
equations is the complete set of solutions
The Differential Dimension Polynomial for Characterizable Differential Ideals
We generalize the differential dimension polynomial from prime differential
ideals to characterizable differential ideals. Its computation is algorithmic,
its degree and leading coefficient remain differential birational invariants,
and it decides equality of characterizable differential ideals contained in
each other
On the Ext-computability of Serre quotient categories
To develop a constructive description of in categories of
coherent sheaves over certain schemes, we establish a binatural isomorphism
between the -groups in Serre quotient categories
and a direct limit of -groups in the
ambient Abelian category . For the isomorphism
follows if the thick subcategory is
localizing. For the higher extension groups we need further assumptions on
. With these categories in mind we cannot assume
to have enough projectives or injectives and
therefore use Yoneda's description of .Comment: updated bibliography and deleted remaining occurrences of "maximally
Thomas decompositions of parametric nonlinear control systems
This paper presents an algorithmic method to study structural properties of
nonlinear control systems in dependence of parameters. The result consists of a
description of parameter configurations which cause different control-theoretic
behaviour of the system (in terms of observability, flatness, etc.). The
constructive symbolic method is based on the differential Thomas decomposition
into disjoint simple systems, in particular its elimination properties
On monads of exact reflective localizations of Abelian categories
In this paper we define Gabriel monads as the idempotent monads associated to
exact reflective localizations in Abelian categories and characterize them by a
simple set of properties. The coimage of a Gabriel monad is a Serre quotient
category. The Gabriel monad induces an equivalence between its coimage and its
image, the localizing subcategory of local objects.Comment: fixed Prop. 2.10, updated bibliograph
Algorithmic Thomas Decomposition of Algebraic and Differential Systems
In this paper, we consider systems of algebraic and non-linear partial
differential equations and inequations. We decompose these systems into
so-called simple subsystems and thereby partition the set of solutions. For
algebraic systems, simplicity means triangularity, square-freeness and
non-vanishing initials. Differential simplicity extends algebraic simplicity
with involutivity. We build upon the constructive ideas of J. M. Thomas and
develop them into a new algorithm for disjoint decomposition. The given paper
is a revised version of a previous paper and includes the proofs of correctness
and termination of our decomposition algorithm. In addition, we illustrate the
algorithm with further instructive examples and describe its Maple
implementation together with an experimental comparison to some other
triangular decomposition algorithms.Comment: arXiv admin note: substantial text overlap with arXiv:1008.376
Interpretable Anomaly Detection in Cellular Networks by Learning Concepts in Variational Autoencoders
This paper addresses the challenges of detecting anomalies in cellular
networks in an interpretable way and proposes a new approach using variational
autoencoders (VAEs) that learn interpretable representations of the latent
space for each Key Performance Indicator (KPI) in the dataset. This enables the
detection of anomalies based on reconstruction loss and Z-scores. We ensure the
interpretability of the anomalies via additional information centroids (c)
using the K-means algorithm to enhance representation learning. We evaluate the
performance of the model by analyzing patterns in the latent dimension for
specific KPIs and thereby demonstrate the interpretability and anomalies. The
proposed framework offers a faster and autonomous solution for detecting
anomalies in cellular networks and showcases the potential of deep
learning-based algorithms in handling big data
Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients
Partial differential equations (PDEs) are important tools to model physical
systems and including them into machine learning models is an important way of
incorporating physical knowledge. Given any system of linear PDEs with constant
coefficients, we propose a family of Gaussian process (GP) priors, which we
call EPGP, such that all realizations are exact solutions of this system. We
apply the Ehrenpreis-Palamodov fundamental principle, which works as a
non-linear Fourier transform, to construct GP kernels mirroring standard
spectral methods for GPs. Our approach can infer probable solutions of linear
PDE systems from any data such as noisy measurements, or pointwise defined
initial and boundary conditions. Constructing EPGP-priors is algorithmic,
generally applicable, and comes with a sparse version (S-EPGP) that learns the
relevant spectral frequencies and works better for big data sets. We
demonstrate our approach on three families of systems of PDEs, the heat
equation, wave equation, and Maxwell's equations, where we improve upon the
state of the art in computation time and precision, in some experiments by
several orders of magnitude.Comment: 26 pages, 8 figures; ICML 2023 (oral); updated with expanded
appendices and ancillary files. Code available at
https://github.com/haerski/EPGP. For animations, see
https://mathrepo.mis.mpg.de/EPGP/index.html. For a presentation see
https://icml.cc/virtual/2023/oral/25571. The paper and all ancillary files
are released under CC-B