88 research outputs found
Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks
As an alternative to classical numerical solvers for partial differential
equations (PDEs) subject to boundary value constraints, there has been a surge
of interest in investigating neural networks that can solve such problems
efficiently. In this work, we design a general solution operator for two
different time-independent PDEs using graph neural networks (GNNs) and spectral
graph convolutions. We train the networks on simulated data from a finite
elements solver on a variety of shapes and inhomogeneities. In contrast to
previous works, we focus on the ability of the trained operator to generalize
to previously unseen scenarios. Specifically, we test generalization to meshes
with different shapes and superposition of solutions for a different number of
inhomogeneities. We find that training on a diverse dataset with lots of
variation in the finite element meshes is a key ingredient for achieving good
generalization results in all cases. With this, we believe that GNNs can be
used to learn solution operators that generalize over a range of properties and
produce solutions much faster than a generic solver. Our dataset, which we make
publicly available, can be used and extended to verify the robustness of these
models under varying conditions
Towards Learning Self-Organized Criticality of Rydberg Atoms using Graph Neural Networks
Self-Organized Criticality (SOC) is a ubiquitous dynamical phenomenon
believed to be responsible for the emergence of universal scale-invariant
behavior in many, seemingly unrelated systems, such as forest fires, virus
spreading or atomic excitation dynamics. SOC describes the buildup of
large-scale and long-range spatio-temporal correlations as a result of only
local interactions and dissipation. The simulation of SOC dynamics is typically
based on Monte-Carlo (MC) methods, which are however numerically expensive and
do not scale beyond certain system sizes. We investigate the use of Graph
Neural Networks (GNNs) as an effective surrogate model to learn the dynamics
operator for a paradigmatic SOC system, inspired by an experimentally
accessible physics example: driven Rydberg atoms. To this end, we generalize
existing GNN simulation approaches to predict dynamics for the internal state
of the node. We show that we can accurately reproduce the MC dynamics as well
as generalize along the two important axes of particle number and particle
density. This paves the way to model much larger systems beyond the limits of
traditional MC methods. While the exact system is inspired by the dynamics of
Rydberg atoms, the approach is quite general and can readily be applied to
other systems
Curve Your Enthusiasm: Concurvity Regularization in Differentiable Generalized Additive Models
Generalized Additive Models (GAMs) have recently experienced a resurgence in
popularity due to their interpretability, which arises from expressing the
target value as a sum of non-linear transformations of the features. Despite
the current enthusiasm for GAMs, their susceptibility to concurvity - i.e.,
(possibly non-linear) dependencies between the features - has hitherto been
largely overlooked. Here, we demonstrate how concurvity can severly impair the
interpretability of GAMs and propose a remedy: a conceptually simple, yet
effective regularizer which penalizes pairwise correlations of the non-linearly
transformed feature variables. This procedure is applicable to any
differentiable additive model, such as Neural Additive Models or NeuralProphet,
and enhances interpretability by eliminating ambiguities due to self-canceling
feature contributions. We validate the effectiveness of our regularizer in
experiments on synthetic as well as real-world datasets for time-series and
tabular data. Our experiments show that concurvity in GAMs can be reduced
without significantly compromising prediction quality, improving
interpretability and reducing variance in the feature importances
Experimental Evidence for the Incorporation of Two Metals at Equivalent Lattice Positions in Mixed-Metal MetalâOrganic Frameworks
Metalâorganic frameworks containing multiple metals distributed over crystallographically equivalent framework positions (mixedâmetal MOFs) represent an interesting class of materials, since the close vicinity of isolated metal centers often gives rise to synergistic effects. However, appropriate characterization techniques for detailed investigations of these mixedâmetal metalâorganic framework materials, particularly addressing the distribution of metals within the lattice, are rarely available. The synthesis of mixedâmetal FeCuBTC materials in direct syntheses proved to be difficult and only a thorough characterization using various techniques, like powder Xâray diffraction, Xâray absorption spectroscopy and electron paramagnetic resonance spectroscopy, unambiguously evidenced the formation of a mixedâmetal FeCuBTC material with HKUSTâ1 structure, which contained bimetallic FeâCu paddlewheels as well as monometallic CuâCu and FeâFe units under optimized synthesis conditions. The inâdepth characterization showed that other synthetic procedures led to impurities, which contained the majority of the applied iron and were impossible or difficult to identify using solely standard characterization techniques. Therefore, this study shows the necessity to characterize mixedâmetal MOFs extensively to unambiguously prove the incorporation of both metals at the desired positions. The controlled positioning of metal centers in mixedâmetal metalâorganic framework materials and the thorough characterization thereof is particularly important to derive structureâproperty or structureâactivity correlations
From Anderson to anomalous localization in cold atomic gases with effective spin-orbit coupling
We study the dynamics of a one-dimensional spin-orbit coupled Schrodinger
particle with two internal components moving in a random potential. We show
that this model can be implemented by the interaction of cold atoms with
external lasers and additional Zeeman and Stark shifts. By direct numerical
simulations a crossover from an exponential Anderson-type localization to an
anomalous power-law behavior of the intensity correlation is found when the
spin-orbit coupling becomes large. The power-law behavior is connected to a
Dyson singularity in the density of states emerging at zero energy when the
system approaches the quasi-relativistic limit of the random mass Dirac model.
We discuss conditions under which the crossover is observable in an experiment
with ultracold atoms and construct explicitly the zero-energy state, thus
proving its existence under proper conditions.Comment: 4 pages and 4 figure
A Green's function approach to transmission of massless Dirac fermions in graphene through an array of random scatterers
We consider the transmission of massless Dirac fermions through an array of
short range scatterers which are modeled as randomly positioned -
function like potentials along the x-axis. We particularly discuss the
interplay between disorder-induced localization that is the hallmark of a
non-relativistic system and two important properties of such massless Dirac
fermions, namely, complete transmission at normal incidence and periodic
dependence of transmission coefficient on the strength of the barrier that
leads to a periodic resonant transmission. This leads to two different types of
conductance behavior as a function of the system size at the resonant and the
off-resonance strengths of the delta function potential. We explain this
behavior of the conductance in terms of the transmission through a pair of such
barriers using a Green's function based approach. The method helps to
understand such disordered transport in terms of well known optical phenomena
such as Fabry Perot resonances.Comment: 22 double spaced single column pages. 15 .eps figure
Robust optical delay lines via topological protection
Phenomena associated with topological properties of physical systems are
naturally robust against perturbations. This robustness is exemplified by
quantized conductance and edge state transport in the quantum Hall and quantum
spin Hall effects. Here we show how exploiting topological properties of
optical systems can be used to implement robust photonic devices. We
demonstrate how quantum spin Hall Hamiltonians can be created with linear
optical elements using a network of coupled resonator optical waveguides (CROW)
in two dimensions. We find that key features of quantum Hall systems, including
the characteristic Hofstadter butterfly and robust edge state transport, can be
obtained in such systems. As a specific application, we show that the
topological protection can be used to dramatically improve the performance of
optical delay lines and to overcome limitations related to disorder in photonic
technologies.Comment: 9 pages, 5 figures + 12 pages of supplementary informatio
Demonstration of Universal Parametric Entangling Gates on a Multi-Qubit Lattice
We show that parametric coupling techniques can be used to generate selective
entangling interactions for multi-qubit processors. By inducing coherent
population exchange between adjacent qubits under frequency modulation, we
implement a universal gateset for a linear array of four superconducting
qubits. An average process fidelity of is estimated for
three two-qubit gates via quantum process tomography. We establish the
suitability of these techniques for computation by preparing a four-qubit
maximally entangled state and comparing the estimated state fidelity against
the expected performance of the individual entangling gates. In addition, we
prepare an eight-qubit register in all possible bitstring permutations and
monitor the fidelity of a two-qubit gate across one pair of these qubits.
Across all such permutations, an average fidelity of
is observed. These results thus offer a path to a scalable architecture with
high selectivity and low crosstalk
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