arXiv subjects

Andrej Junginger

Publications and source records attributed to Andrej Junginger.

At least 19 recordsLinked to original sources

arXiv2021

Informed Equation Learning

Distilling data into compact and interpretable analytic equations is one of the goals of science. Instead, contemporary supervised machine learning methods mostly produce unstructured and dense maps from input to output. Particularly in deep learning, this property is owed to the generic nature of simple standard link functions. To learn equations rather than maps, standard non-linearities can be replaced with structured building blocks of atomic functions. However, without strong priors on sparsity and structure, representational complexity and numerical conditioning limit this direct approach. To scale to realistic settings in science and engineering, we propose an informed equation learning system. It provides a way to incorporate expert knowledge about what are permitted or prohibited equation components, as well as a domain-dependent structured sparsity prior. Our system then utilizes a robust method to learn equations with atomic functions exhibiting singularities, as e.g. logarithm and division. We demonstrate several artificial and real-world experiments from the engineering domain, in which our system learns interpretable models of high predictive power.

cs.LG↗

arXiv2019

Invariant Manifolds and Rate Constants in Driven Chemical Reactions

Reaction rates of chemical reactions under nonequilibrium conditions can be determined through the construction of the normally hyperbolic invariant manifold (NHIM) [and moving dividing surface (DS)] associated with the transition state trajectory. Here, we extend our recent methods by constructing points on the NHIM accurately even for multidimensional cases. We also advance the implementation of machine learning approaches to construct smooth versions of the NHIM from a known high-accuracy set of its points. That is, we expand on our earlier use of neural nets, and introduce the use of Gaussian process regression for the determination of the NHIM. Finally, we compare and contrast all of these methods for a challenging two-dimensional model barrier case so as to illustrate their accuracy and general applicability.

physics.chem-ph↗

arXiv2018

Unpaired High-Resolution and Scalable Style Transfer Using Generative Adversarial Networks

Neural networks have proven their capabilities by outperforming many other approaches on regression or classification tasks on various kinds of data. Other astonishing results have been achieved using neural nets as data generators, especially in settings of generative adversarial networks (GANs). One special application is the field of image domain translations. Here, the goal is to take an image with a certain style (e.g. a photography) and transform it into another one (e.g. a painting). If such a task is performed for unpaired training examples, the corresponding GAN setting is complex, the neural networks are large, and this leads to a high peak memory consumption during, both, training and evaluation phase. This sets a limit to the highest processable image size. We address this issue by the idea of not processing the whole image at once, but to train and evaluate the domain translation on the level of overlapping image subsamples. This new approach not only enables us to translate high-resolution images that otherwise cannot be processed by the neural network at once, but also allows us to work with comparably small neural networks and with limited hardware resources. Additionally, the number of images required for the training process is significantly reduced. We present high-quality results on images with a total resolution of up to over 50 megapixels and emonstrate that our method helps to preserve local image details while it also keeps global consistency.

cs.CV↗

arXiv2018

Binary contraction method for the construction of time-dependent dividing surfaces in driven chemical reactions

Transition state theory formally provides a simplifying approach for determining chemical reaction rates and pathways. Given an underlying potential energy surface for a reactive system, one can determine the dividing surface in phase space which separates reactant and product regions, and thereby also these regions. This is often a difficult task, and it is especially demanding for high-dimensional time-dependent systems or when a non-local dividing surface is required. Recently, approaches relying on Lagrangian descriptors have been successful at resolving the dividing surface in some of these challenging cases, but this method can also be computationally expensive due to the necessity of integrating the corresponding phase space function. In this paper, we present an alternative method by which time-dependent, locally recrossing-free dividing surfaces can be constructed without the calculation of any auxiliary phase space function, but only from simple dynamical properties close to the energy barrier.

physics.chem-ph↗

arXiv2018

Neural network approach to time-dependent dividing surfaces in classical reaction dynamics

In a dynamical system, the transition between reactants and products is typically mediated by an energy barrier whose properties determine the corresponding pathways and rates. The latter is the flux through a dividing surface (DS) between the two corresponding regions and it is exact only if it is free of recrossings. For time-independent barriers, the DS can be attached to the top of the corresponding saddle point of the potential energy surface, and in time-dependent systems, the DS is a moving object. The precise determination of reaction rates, eg using transition state theory, requires the actual construction of a DS for a given saddle geometry which is in general a demanding methodical and computational task, especially in high-dimensional systems. In this paper, we demonstrate how such time-dependent, global, and recrossing-free DSs can be constructed using neural networks. In our approach, the neural network uses the bath coordinates and time as input and it is trained in a way that its output provides the position of the DS along the reaction coordinate. An advantage of this procedure is that, once the neural network is trained, the complete information about the dynamical phase space separation is stored in the network's parameters, and a precise distinction between reactants and products can be made for all possible system configurations, all times, and with little computational effort. We demonstrate this general method for two- and three-dimensional systems, and explain its straightforward extension to even more degrees of freedom.

physics.chem-ph↗

arXiv2018

Ensemble Methods as a Defense to Adversarial Perturbations Against Deep Neural Networks

Deep learning has become the state of the art approach in many machine learning problems such as classification. It has recently been shown that deep learning is highly vulnerable to adversarial perturbations. Taking the camera systems of self-driving cars as an example, small adversarial perturbations can cause the system to make errors in important tasks, such as classifying traffic signs or detecting pedestrians. Hence, in order to use deep learning without safety concerns a proper defense strategy is required. We propose to use ensemble methods as a defense strategy against adversarial perturbations. We find that an attack leading one model to misclassify does not imply the same for other networks performing the same task. This makes ensemble methods an attractive defense strategy against adversarial attacks. We empirically show for the MNIST and the CIFAR-10 data sets that ensemble methods not only improve the accuracy of neural networks on test data but also increase their robustness against adversarial perturbations.

stat.ML↗

arXiv2017

Obtaining time-dependent multi-dimensional dividing surfaces using Lagrangian descriptors

Dynamics between reactants and products are often mediated by a rate-determining barrier and an associated dividing surface leading to the transition state theory rate. This framework is challenged when the barrier is time-dependent because its motion can give rise to recrossings across the fixed dividing surface. A non-recrossing time-dependent dividing surface can nevertheless be attached to the TS trajectory resulting in recrossing-free dynamics. We extend the formalism---constructed using Lagrangian Descriptors---to systems with additional bath degrees of freedom. The propagation of reactant ensembles provides a numerical demonstration that our dividing surface is recrossing-free and leads to exact TST rates.

physics.chem-ph↗

arXiv2017

Chemical dynamics between wells across a time-dependent barrier: Self-similarity in the Lagrangian descriptor and reactive basins

In chemical or physical reaction dynamics, it is essential to distinguish precisely between reactants and products for all time. This task is especially demanding in time-dependent or driven systems because therein the dividing surface (DS) between these states often exhibits a nontrivial time-dependence. The so-called transition state (TS) trajectory has been seen to define a DS which is free of recrossings in a large number of one-dimensional reactions across time-dependent barriers, and, thus, allows one to determine exact reaction rates. A fundamental challenge to applying this method is the construction of the TS trajectory itself. The minimization of Lagrangian descriptors (LDs) provides a general and powerful scheme to obtain that trajectory even when perturbation theory fails. Both approaches encounter possible breakdowns when the overall potential is bounded, admitting the possibility of returns to the barrier long after trajectories have reached the product or reactant wells. Such global dynamics cannot be captured by perturbation theory. Meanwhile, in the LD-DS approach, it leads to the emergence of additional local minima which make it difficult to extract the optimal branch associated with the desired TS trajectory. In this work, we illustrate this behavior for a time-dependent double-well potential revealing a self-similar structure of the LD, and we demonstrate how the reflections and side-minima can be addressed by an appropriate modification of the LD associated with the direct rate across the barrier.

physics.chem-ph↗

arXiv2017

Variational principle for the determination of unstable periodic orbits and instanton trajectories at saddle points

The complexity of arbitrary dynamical systems and chemical reactions, in particular, can often be resolved if only the appropriate periodic orbit - in the form of a limit cycle, dividing surface, instanton trajectories or some other related structure - can be uncovered. Determining such a periodic orbit, no matter how beguilingly simple it appears, is often very challenging. We present a method for the direct construction of unstable periodic orbits and instanton trajectories at saddle points by means of Lagrangian descriptors. Such structures result from the minimization of a scalar-valued phase space function without need for any additional constraints or knowledge. We illustrate the approach for two-degree of freedom systems at a rank-1 saddle point of the underlying potential energy surface by constructing both periodic orbits at energies above the saddle point as well as instanton trajectories below the saddle point energy.

physics.chem-ph↗

arXiv2017

Construction of Darboux coordinates in noncanonical Hamiltonian systems via normal form expansions

Darboux's theorem guarantees the existence of local canonical coordinates on symplectic manifolds under certain conditions. We demonstrate a general method to construct such Darboux coordinates in the vicinity of a fixed point of a noncanonical Hamiltonian system via normal form expansions. The procedure serves as a tool to naturally extract canonical coordinates and at the same time to transform the Hamiltonian into its Poincare-Birkhoff normal form. The method is general in the sense that it is applicable for arbitrary degrees of freedom, in arbitrary orders of the local expansion, and it is independent of the precise form of the Hamiltonian. The method presented allows for the general and systematic investigation of noncanonical Hamiltonian systems in the vicinity of fixed points, which e.g. correspond to ground, excited or transition states. As an exemplary field of application, we discuss a variational approach to quantum systems which defines a noncanonical Hamiltonian system for the variational parameters and which directly allows to apply transition state theory to quantum mechanical wave packets.

nlin.CD↗

arXiv2016

Dynamics of Solitons in the One-Dimensional Nonlinear Schrödinger Equation

We investigate bright solitons in the one-dimensional Schrödinger equation in the framework of an extended variational approach. We apply the latter to the stationary ground state of the system as well as to coherent collisions between two or more solitons. Using coupled Gaussian trial wave functions, we demonstrate that the variational approach is a powerful method to calculate the soliton dynamics. This method has the advantage that it is computationally faster compared to numerically exact grid calculations. In addition, it goes far beyond the capability of analytical ground state solutions, because the variational approach provides the ability to treat excited solitons as well as dynamical interactions between different wave packets. To demonstrate the power of the variational approach, we calculate the stationary ground state of the soliton and compare it with the analytical solution showing the convergence to the exact solution. Furthermore, we extend our calculations to nonstationary solitons by investigating the breathing oscillations of excited solitons and the coherent collisions of several wave packets in both the low- and high-energy regime. Comparisons of the variational approach with numerically exact simulations on grids reveal excellent agreement in the high-energy regime while deviations can be observed for low energies.

quant-ph↗

arXiv2015

Normal form expansions and thermal decay rates of Bose-Einstein condensates with short- and long-range interaction

The thermally induced coherent collapse of Bose-Einstein condensates at finite temperature is the dominant decay mechanism near the critical scattering length in condensates with at least partially attractive interaction. The collapse dynamics out of the ground state is mediated by a transition state whose properties determine the corresponding decay rate or lifetime of the condensate. In this paper, we perform normal form expansions of the ground and the transition state of condensates with short-range scattering interaction as well as with anisotropic and long-range dipolar interaction in a variational framework. This method allows one to determine the local properties of these states, i.e. their mean-field energy, their normal modes, the coupling between different modes, and the structure of the reaction channel to any desired order. We discuss the physical interpretation of the transition state as a certain density distribution of the atomic cloud and the behavior of the single normal form contributions in dependence on the s-wave scattering length. Moreover, we investigate the convergence of the local normal form when using extended Gaussian variational approaches, and present the condensate's decay rate.

cond-mat.quant-gas↗

arXiv2013

Transition states and thermal collapse of dipolar Bose-Einstein condensates

We investigate thermally excited, dipolar Bose-Einstein condensates. Quasi-particle excitations of the atomic cloud cause density fluctuations which can induce the collapse of the condensate if the inter-particle interaction is attractive. Within a variational approach, we identify the collectively excited stationary states of the gas which form transition states on the way to the BEC's collapse. We analyze transition states with different $m$-fold rotational symmetry and identify the one which mediates the collapse. The latter's symmetry depends on the trap aspect ratio of the external trapping potential which determines the shape of the BEC. Moreover, we present the collapse dynamics of the BEC and calculate the corresponding decay rate using transition state theory. We observe that the thermally induced collapse mechanism is important near the critical scattering length, where the lifetime of the condensate can be significantly reduced. Our results are valid for an arbitrary strength of the dipole-dipole interaction. Specific applications are discussed for the elements $^{52}$Cr, $^{164}$Dy and $^{168}$Er with which dipolar BECs have been experimentally realized.

cond-mat.quant-gas↗

arXiv2012

Quantum-classical lifetimes of Rydberg molecules

A remarkable property of Rydberg atoms is the possibility to create molecules formed by one highly excited atom and another atom in the ground state. The first realisation of such a Rydberg molecule has opened an active field of physical investigations, and showed that its basic properties can be described within a simple model regarding the ground state atom as a small perturber that is bound by a low-energy scattering process with the Rydberg electron [C. H. Greene et al, Phys. Rev. Lett., 85: 2458 (2000)]. Besides the good agreement between theory and the experiment concerning the vibrational states of the molecule, the experimental observations yield the astonishing feature that the lifetime of the molecule is clearly reduced as compared to the bare Rydberg atom [B. Butscher et al, J. Phys. B: At. Mol. Opt. Phys. 44, 184004 (2011)]. With focus on this yet unexplained observation, we investigate in this paper the vibrational ground state of the molecule in a quantum-classical framework. We show that the Rydberg wave function is continuously detuned by the presence of the moving ground state atom and that the timescale on which the detuning significantly exceeds the natural linewidth is in good agreement with the observed reduced lifetimes of the Rydberg molecule.

physics.atom-ph↗

arXiv2012

Symmetry-breaking thermally induced collapse of dipolar Bose-Einstein condensates

We investigate a Bose-Einstein condensate with additional long-range dipolar interaction in a cylindrically symmetric trap within a variational framework. Compared to the ground state of this system, little attention has as yet been payed to its unstable excited states. For thermal excitations, however, the latter is of great interest, because it forms the "activated complex" that mediates the collapse of the condensate. For a certain value of the s-wave scatting length our investigations reveal a bifurcation in the transition state, leading to the emergence of two additional and symmetry-breaking excited states. Because these are of lower energy than their symmetric counterpart, we predict the occurrence of a symmetry-breaking thermally induced collapse of dipolar condensates. We show that its occurrence crucially depends on the trap geometry and calculate the thermal decay rates of the system within leading order transition state theory with the help of a uniform rate formula near the rank-2 saddle which allows to smoothly pass the bifurcation.

cond-mat.quant-gas↗

arXiv2012

Quantum-classical model for the formation of Rydberg molecules

A fascinating aspect of Rydberg atoms is their ability to form huge but very weakly bound molecules with a ground state atom, only held together by a scattering process between the latter and the Rydberg electron. Beyond the usual way of creating such molecules by laser excitation from two ground state atoms with a distance of less than the Rydberg radius, we demonstrate that Rydberg molecules can also be formed by capturing a ground state atom which is initially located outside the range of the Rydberg atom when it comes in contact with it. To demonstrate this effect, we investigate the scattering process between the Rydberg electron and the ground state atom within a quantum-classical framework. In this picture capturing results from a dissipative finite-mass correction term in the classical equations of motion. We show that and under which conditions the capturing takes place.

physics.atom-ph↗

arXiv2012

Transition state theory for wave packet dynamics. I. Thermal decay in metastable Schrödinger systems

We demonstrate the application of transition state theory to wave packet dynamics in metastable Schrödinger systems which are approached by means of a variational ansatz for the wave function and whose dynamics is described within the framework of a time-dependent variational principle. The application of classical transition state theory, which requires knowledge of a classical Hamilton function, is made possible by mapping the variational parameters to classical phase space coordinates and constructing an appropriate Hamiltonian in action variables. This mapping, which is performed by a normal form expansion of the equations of motion and an additional adaptation to the energy functional, as well as the requirements to the variational ansatz are discussed in detail. The applicability of the procedure is demonstrated for a cubic model potential for which we calculate thermal decay rates of a frozen Gaussian wave function. The decay rate obtained with a narrow trial wave function agrees perfectly with the results using the classical normal form of the corresponding point particle. The results with a broader trial wave function go even beyond the classical approach, i.e., they agree with those using the quantum normal form. The method presented here will be applied to Bose-Einstein condensates in the following paper [A. Junginger, M. Dorwarth, J. Main, and G. Wunner, submitted to J. Phys. A].

nlin.CD↗

arXiv2012

Transition state theory for wave packet dynamics. II. Thermal decay of Bose-Einstein condensates with long-range interaction

We apply transition state theory to coupled Gaussian wave packets and calculate thermal decay rates of Bose-Einstein condensates with additional long-range interaction. The ground state of such a condensate is metastable if the contact interaction is attractive and a sufficient thermal excitation may lead to its collapse. The use of transition state theory is made possible by describing the condensate within a variational framework and locally mapping the variational parameters to classical phase space as has been demonstrated in the preceding paper [A. Junginger, J. Main, and G. Wunner, submitted to J. Phys. A]. We apply this procedure to Gaussian wave packets and present results for condensates with monopolar 1/r-interaction comparing decay rates obtained by using different numbers of coupled Gaussian trial wave functions as well as different normal form orders.

cond-mat.quant-gas↗