22 research outputs found
Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation
We study tensor product multiscale many-particle spaces with finite-order weights and their application for the electronic Schrödinger equation. Any numerical solution of the electronic Schrödinger equation using conventional discretization schemes is impossible due to its high dimensionality. Therefore, typically Monte Carlo methods (VMC/DMC) or nonlinear model approximations like Hartree-Fock (HF), coupled cluster (CC) or density functional theory (DFT) are used. In this work we develop and implement in parallel a numerical method based on adaptive sparse grids and a particle-wise subspace splitting with respect to one-particle functions which stem from a nonlinear rank-1 approximation. Sparse grids allow to overcome the exponential complexity exhibited by conventional discretization procedures and deliver a convergent numerical approach with guaranteed convergence rates. In particular, the introduced weighted many-particle tensor product multiscale approximation spaces include the common configuration interaction (CI) spaces as a special case. To realize our new approach, we first introduce general many-particle Sobolev spaces, which particularly include the standard Sobolev spaces as well as Sobolev spaces of dominated mixed smoothness. For this novel variant of sparse grid spaces we show estimates for the approximation and complexity orders with respect to the smoothness and decay parameters. With known regularity properties of the electronic wave function it follows that, up to logarithmic terms, the convergence rate is independent of the number of electrons and almost the same as in the two-electron case. However, besides the rate, also the dependence of the complexity constants on the number of electrons plays an important role for a truly practical method. Based on a splitting of the one-particle space we construct a subspace splitting of the many-particle space, which particularly includes the known ANOVA decomposition, the HDMR decomposition and the CI decomposition as special cases. Additionally, we introduce weights for a restriction of this subspace splitting. In this way weights of finite order q lead to many-particle spaces in which the problem of an approximation of an N-particle function reduces to the problem of the approximation of q-particle functions. To obtain as small as possible constants with respect to the cost complexity, we introduce a heuristic adaptive scheme to build a sequence of finite-dimensional subspaces of a weighted tensor product multiscale many-particle approximation space. Furthermore, we construct a multiscale Gaussian frame and apply Gaussians and modulated Gaussians for the nonlinear rank-1 approximation. In this way, all matrix entries of the corresponding discrete eigenvalue problem can be computed in terms of analytic formulae for the one and two particle operator integrals. Finally, we apply our novel approach to small atomic and diatomic systems with up to 6 electrons (18 space dimensions). The numerical results demonstrate that our new method indeed allows for convergence with expected rates.Tensorprodukt-Multiskalen-Mehrteilchenräume mit Gewichten endlicher Ordnung für die elektronische Schrödingergleichung In der vorliegenden Arbeit beschäftigen wir uns mit gewichteten Tensorprodukt-Multiskalen-Mehrteilchen-Approximationsräumen und deren Anwendung zur numerischen Lösung der elektronischen Schrödinger-Gleichung. Aufgrund der hohen Problemdimension ist eine direkte numerische Lösung der elektronischen Schrödinger-Gleichung mit Standard-Diskretisierungsverfahren zur linearen Approximation unmöglich, weshalb üblicherweise Monte Carlo Methoden (VMC/DMC) oder nichtlineare Modellapproximationen wie Hartree-Fock (HF), Coupled Cluster (CC) oder Dichtefunktionaltheorie (DFT) verwendet werden. In dieser Arbeit wird eine numerische Methode auf Basis von adaptiven dünnen Gittern und einer teilchenweisen Unterraumzerlegung bezüglich Einteilchenfunktionen aus einer nichtlinearen Rang-1 Approximation entwickelt und für parallele Rechnersysteme implementiert. Dünne Gitter vermeiden die in der Dimension exponentielle Komplexität üblicher Diskretisierungsmethoden und führen zu einem konvergenten numerischen Ansatz mit garantierter Konvergenzrate. Zudem enthalten unsere zugrunde liegenden gewichteten Mehrteilchen Tensorprodukt-Multiskalen-Approximationsräume die bekannten Configuration Interaction (CI) Räume als Spezialfall. Zur Konstruktion unseres Verfahrens führen wir zunächst allgemeine Mehrteilchen-Sobolevräume ein, welche die Standard-Sobolevräume sowie Sobolevräume mit dominierender gemischter Glattheit beinhalten. Wir analysieren die Approximationseigenschaften und schätzen Konvergenzraten und Kostenkomplexitätsordnungen in Abhängigkeit der Glattheitsparameter und Abfalleigenschaften ab. Mit Hilfe bekannter Regularitätseigenschaften der elektronischen Wellenfunktion ergibt sich, dass die Konvergenzrate bis auf logarithmische Terme unabhängig von der Zahl der Elektronen und fast identisch mit der Konvergenzrate im Fall von zwei Elektronen ist. Neben der Rate spielt allerdings die Abhängigkeit der Konstanten in der Kostenkomplexität von der Teilchenzahl eine wichtige Rolle. Basierend auf Zerlegungen des Einteilchenraumes konstruieren wir eine Unterraumzerlegung des Mehrteilchenraumes, welche insbesondere die bekannte ANOVA-Zerlegung, die HDMR-Zerlegung sowie die CI-Zerlegung als Spezialfälle beinhaltet. Eine zusätzliche Gewichtung der entsprechenden Unterräume mit Gewichten von endlicher Ordnung q führt zu Mehrteilchenräumen, in denen sich das Approximationsproblem einer N-Teilchenfunktion zu Approximationsproblemen von q-Teilchenfunktionen reduziert. Mit dem Ziel, Konstanten möglichst kleiner Größe bezüglich der Kostenkomplexität zu erhalten, stellen wir ein heuristisches adaptives Verfahren zur Konstruktion einer Sequenz von endlich-dimensionalen Unterräumen eines gewichteten Mehrteilchen-Tensorprodukt-Multiskalen-Approximationsraumes vor. Außerdem konstruieren wir einen Frame aus Multiskalen-Gauss-Funktionen und verwenden Einteilchenfunktionen im Rahmen der Rang-1 Approximation in der Form von Gauss- und modulierten-Gauss-Funktionen. Somit können die zur Aufstellung der Matrizen des zugehörigen diskreten Eigenwertproblems benötigten Ein- und Zweiteilchenintegrale analytisch berechnet werden. Schließlich wenden wir unsere Methode auf kleine Atome und Moleküle mit bis zu sechs Elektronen (18 Raumdimensionen) an. Die numerischen Resultate zeigen, dass sich die aus der Theorie zu erwartenden Konvergenzraten auch praktisch ergeben
The octet rule in chemical space: Generating virtual molecules
We present a generator of virtual molecules that selects valid chemistry on
the basis of the octet rule. Also, we introduce a mesomer group key that allows
a fast detection of duplicates in the generated structures.
Compared to existing approaches, our model is simpler and faster, generates
new chemistry and avoids invalid chemistry. Its versatility is illustrated by
the correct generation of molecules containing third-row elements and a
surprisingly adept handling of complex boron chemistry.
Without any empirical parameters, our model is designed to be valid also in
unexplored regions of chemical space. One first unexpected finding is the high
prevalence of dipolar structures among generated molecules.Comment: 24 pages, 10 figure
Feature-based prediction of properties of cross-linked epoxy polymers by molecular dynamics and machine learning techniques
Epoxy polymers are used in wide range of applications. The properties and
performance of epoxy polymers depend upon various factors like the type of
constituents and their proportions used and other process parameters. The
conventional way of developing epoxy polymers is usually labor-intensive and
may not be fully efficient, which has resulted in epoxy polymers having a
limited performance range due to the use of predetermined blend combinations,
compositions and development parameters. Hence, in order to experiment with
more design parameters, robust and easy computational techniques need to be
established. To this end, we developed and analyzed in this study a new machine
learning (ML) based approach to predict the mechanical properties of epoxy
polymers based on their basic structural features. The results from molecular
dynamics (MD) simulations have been used to derive the ML model. The salient
feature of our work is that for the development of epoxy polymers based on
EPON-862, several new hardeners were explored in addition to the conventionally
used ones. The influence of additional parameters like the proportion of curing
agent used and the extent of curing on the mechanical properties of epoxy
polymers were also investigated. This method can be further extended by
providing the epoxy polymer with the desired properties through knowledge of
the structural characteristics of its constituents. The findings of our study
can thus lead toward development of efficient design methodologies for epoxy
polymeric systems
Parameterized Neural Networks for Finance
We discuss and analyze a neural network architecture, that enables learning a
model class for a set of different data samples rather than just learning a
single model for a specific data sample. In this sense, it may help to reduce
the overfitting problem, since, after learning the model class over a larger
data sample consisting of such different data sets, just a few parameters need
to be adjusted for modeling a new, specific problem. After analyzing the method
theoretically and by regression examples for different one-dimensional
problems, we finally apply the approach to one of the standard problems asset
managers and banks are facing: the calibration of spread curves. The presented
results clearly show the potential that lies within this method. Furthermore,
this application is of particular interest to financial practitioners, since
nearly all asset managers and banks which are having solutions in place may
need to adapt or even change their current methodologies when ESG ratings
additionally affect the bond spreads.Comment: 24 pages, 17 figure
Predicting Properties of Oxide Glasses Using Informed Neural Networks
Many modern-day applications require the development of new materials with
specific properties. In particular, the design of new glass compositions is of
great industrial interest. Current machine learning methods for learning the
composition-property relationship of glasses promise to save on expensive
trial-and-error approaches. Even though quite large datasets on the composition
of glasses and their properties already exist (i.e., with more than 350,000
samples), they cover only a very small fraction of the space of all possible
glass compositions. This limits the applicability of purely data-driven models
for property prediction purposes and necessitates the development of models
with high extrapolation power. In this paper, we propose a neural network model
which incorporates prior scientific and expert knowledge in its learning
pipeline. This informed learning approach leads to an improved extrapolation
power compared to blind (uninformed) neural network models. To demonstrate
this, we train our models to predict three different material properties, that
is, the glass transition temperature, the Young's modulus (at room
temperature), and the shear modulus of binary oxide glasses which do not
contain sodium. As representatives for conventional blind neural network
approaches we use five different feed-forward neural networks of varying widths
and depths. For each property, we set up model ensembles of multiple trained
models and show that, on average, our proposed informed model performs better
in extrapolating the three properties of previously unseen sodium borate glass
samples than all five conventional blind models.Comment: 25 page
ATK-ForceField: A New Generation Molecular Dynamics Software Package
ATK-ForceField is a software package for atomistic simulations using
classical interatomic potentials. It is implemented as a part of the Atomistix
ToolKit (ATK), which is a Python programming environment that makes it easy to
create and analyze both standard and highly customized simulations. This paper
will focus on the atomic interaction potentials, molecular dynamics, and
geometry optimization features of the software, however, many more advanced
modeling features are available. The implementation details of these algorithms
and their computational performance will be shown. We present three
illustrative examples of the types of calculations that are possible with
ATK-ForceField: modeling thermal transport properties in a silicon germanium
crystal, vapor deposition of selenium molecules on a selenium surface, and a
simulation of creep in a copper polycrystal.Comment: 28 pages, 9 figure
On the Interplay of Subset Selection and Informed Graph Neural Networks
Machine learning techniques paired with the availability of massive datasets
dramatically enhance our ability to explore the chemical compound space by
providing fast and accurate predictions of molecular properties. However,
learning on large datasets is strongly limited by the availability of
computational resources and can be infeasible in some scenarios. Moreover, the
instances in the datasets may not yet be labelled and generating the labels can
be costly, as in the case of quantum chemistry computations. Thus, there is a
need to select small training subsets from large pools of unlabelled data
points and to develop reliable ML methods that can effectively learn from small
training sets. This work focuses on predicting the molecules atomization energy
in the QM9 dataset. We investigate the advantages of employing domain
knowledge-based data sampling methods for an efficient training set selection
combined with informed ML techniques. In particular, we show how maximizing
molecular diversity in the training set selection process increases the
robustness of linear and nonlinear regression techniques such as kernel methods
and graph neural networks. We also check the reliability of the predictions
made by the graph neural network with a model-agnostic explainer based on the
rate distortion explanation framework