524 research outputs found
On concentration in vortex sheets
The question of energy concentration in approximate solution sequences
, as , of the two-dimensional incompressible Euler
equations with vortex-sheet initial data is revisited. Building on a novel
identity for the structure function in terms of vorticity, the vorticity
maximal function is proposed as a quantitative tool to detect concentration
effects in approximate solution sequences. This tool is applied to numerical
experiments based on the vortex-blob method, where vortex sheet initial data
without distinguished sign are considered, as introduced in \emph{[R.~Krasny,
J. Fluid Mech. \textbf{167}:65-93 (1986)]}. Numerical evidence suggests that no
energy concentration appears in the limit of zero blob-regularization , for the considered initial data
Statistical solutions of hyperbolic conservation laws I: Foundations
We seek to define statistical solutions of hyperbolic systems of conservation
laws as time-parametrized probability measures on -integrable functions. To
do so, we prove the equivalence between probability measures on spaces
and infinite families of \textit{correlation measures}. Each member of this
family, termed a \textit{correlation marginal}, is a Young measure on a
finite-dimensional tensor product domain and provides information about
multi-point correlations of the underlying integrable functions. We also prove
that any probability measure on a space is uniquely determined by certain
moments (correlation functions) of the equivalent correlation measure.
We utilize this equivalence to define statistical solutions of
multi-dimensional conservation laws in terms of an infinite set of equations,
each evolving a moment of the correlation marginal. These evolution equations
can be interpreted as augmenting entropy measure-valued solutions, with
additional information about the evolution of all possible multi-point
correlation functions. Our concept of statistical solutions can accommodate
uncertain initial data as well as possibly non-atomic solutions even for atomic
initial data.
For multi-dimensional scalar conservation laws we impose additional entropy
conditions and prove that the resulting \textit{entropy statistical solutions}
exist, are unique and are stable with respect to the -Wasserstein metric on
probability measures on
Operator learning with PCA-Net: upper and lower complexity bounds
PCA-Net is a recently proposed neural operator architecture which combines
principal component analysis (PCA) with neural networks to approximate
operators between infinite-dimensional function spaces. The present work
develops approximation theory for this approach, improving and significantly
extending previous work in this direction: First, a novel universal
approximation result is derived, under minimal assumptions on the underlying
operator and the data-generating distribution. Then, two potential obstacles to
efficient operator learning with PCA-Net are identified, and made precise
through lower complexity bounds; the first relates to the complexity of the
output distribution, measured by a slow decay of the PCA eigenvalues. The other
obstacle relates to the inherent complexity of the space of operators between
infinite-dimensional input and output spaces, resulting in a rigorous and
quantifiable statement of a "curse of parametric complexity", an
infinite-dimensional analogue of the well-known curse of dimensionality
encountered in high-dimensional approximation problems. In addition to these
lower bounds, upper complexity bounds are finally derived. A suitable
smoothness criterion is shown to ensure an algebraic decay of the PCA
eigenvalues. Furthermore, it is shown that PCA-Net can overcome the general
curse for specific operators of interest, arising from the Darcy flow and the
Navier-Stokes equations
Computation of measure-valued solutions for the incompressible Euler equations
We combine the spectral (viscosity) method and ensemble averaging to propose
an algorithm that computes admissible measure valued solutions of the
incompressible Euler equations. The resulting approximate young measures are
proved to converge (with increasing numerical resolution) to a measure valued
solution. We present numerical experiments demonstrating the robustness and
efficiency of the proposed algorithm, as well as the appropriateness of measure
valued solutions as a solution framework for the Euler equations. Furthermore,
we report an extensive computational study of the two dimensional vortex sheet,
which indicates that the computed measure valued solution is non-atomic and
implies possible non-uniqueness of weak solutions constructed by Delort
Seamless Integration of RESTful Services into the Web of Data
We live in an era of ever-increasing abundance of data. To cope with the information overload we suffer from every single day, more sophisticated methods are required to access, manipulate, and analyze these humongous amounts of data. By embracing the heterogeneity, which is unavoidable at such a scale, and accepting the fact that the data quality and meaning are fuzzy, more adaptable, flexible, and extensible systems can be built. RESTful services combined with Semantic Web technologies could prove to be a viable path to achieve that. Their combination a1lows data integration on an unprecedented sca1e and solves some of the problems Web developers are continuously struggling with. This paper introduces a novel approach to create machine-readable descriptions for RESTful services as a first step towards this ambitious goal. It also shows how these descriptions along with analgorithm to translate SPARQL queries to HTTP requests can be used to integrate RESTful services into a global read-write Web of Data
The curse of dimensionality in operator learning
Neural operator architectures employ neural networks to approximate operators
mapping between Banach spaces of functions; they may be used to accelerate
model evaluations via emulation, or to discover models from data. Consequently,
the methodology has received increasing attention over recent years, giving
rise to the rapidly growing field of operator learning. The first contribution
of this paper is to prove that for general classes of operators which are
characterized only by their - or Lipschitz-regularity, operator learning
suffers from a curse of dimensionality, defined precisely here in terms of
representations of the infinite-dimensional input and output function spaces.
The result is applicable to a wide variety of existing neural operators,
including PCA-Net, DeepONet and the FNO. The second contribution of the paper
is to prove that the general curse of dimensionality can be overcome for
solution operators defined by the Hamilton-Jacobi equation; this is achieved by
leveraging additional structure in the underlying solution operator, going
beyond regularity. To this end, a novel neural operator architecture is
introduced, termed HJ-Net, which explicitly takes into account characteristic
information of the underlying Hamiltonian system. Error and complexity
estimates are derived for HJ-Net which show that this architecture can provably
beat the curse of dimensionality related to the infinite-dimensional input and
output function spaces
Error Bounds for Learning with Vector-Valued Random Features
This paper provides a comprehensive error analysis of learning with
vector-valued random features (RF). The theory is developed for RF ridge
regression in a fully general infinite-dimensional input-output setting, but
nonetheless applies to and improves existing finite-dimensional analyses. In
contrast to comparable work in the literature, the approach proposed here
relies on a direct analysis of the underlying risk functional and completely
avoids the explicit RF ridge regression solution formula in terms of random
matrices. This removes the need for concentration results in random matrix
theory or their generalizations to random operators. The main results
established in this paper include strong consistency of vector-valued RF
estimators under model misspecification and minimax optimal convergence rates
in the well-specified setting. The parameter complexity (number of random
features) and sample complexity (number of labeled data) required to achieve
such rates are comparable with Monte Carlo intuition and free from logarithmic
factors.Comment: 25 pages, 1 tabl
The Nonlocal Neural Operator: Universal Approximation
Neural operator architectures approximate operators between
infinite-dimensional Banach spaces of functions. They are gaining increased
attention in computational science and engineering, due to their potential both
to accelerate traditional numerical methods and to enable data-driven
discovery. A popular variant of neural operators is the Fourier neural operator
(FNO). Previous analysis proving universal operator approximation theorems for
FNOs resorts to use of an unbounded number of Fourier modes and limits the
basic form of the method to problems with periodic geometry. Prior work relies
on intuition from traditional numerical methods, and interprets the FNO as a
nonstandard and highly nonlinear spectral method. The present work challenges
this point of view in two ways: (i) the work introduces a new broad class of
operator approximators, termed nonlocal neural operators (NNOs), which allow
for operator approximation between functions defined on arbitrary geometries,
and includes the FNO as a special case; and (ii) analysis of the NNOs shows
that, provided this architecture includes computation of a spatial average
(corresponding to retaining only a single Fourier mode in the special case of
the FNO) it benefits from universal approximation. It is demonstrated that this
theoretical result unifies the analysis of a wide range of neural operator
architectures. Furthermore, it sheds new light on the role of nonlocality, and
its interaction with nonlinearity, thereby paving the way for a more systematic
exploration of nonlocality, both through the development of new operator
learning architectures and the analysis of existing and new architectures
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