12,286 research outputs found
Separable Hamiltonian Neural Networks
The modelling of dynamical systems from discrete observations is a challenge
faced by modern scientific and engineering data systems. Hamiltonian systems
are one such fundamental and ubiquitous class of dynamical systems. Hamiltonian
neural networks are state-of-the-art models that unsupervised-ly regress the
Hamiltonian of a dynamical system from discrete observations of its vector
field under the learning bias of Hamilton's equations. Yet Hamiltonian dynamics
are often complicated, especially in higher dimensions where the state space of
the Hamiltonian system is large relative to the number of samples. A recently
discovered remedy to alleviate the complexity between state variables in the
state space is to leverage the additive separability of the Hamiltonian system
and embed that additive separability into the Hamiltonian neural network.
Following the nomenclature of physics-informed machine learning, we propose
three separable Hamiltonian neural networks. These models embed additive
separability within Hamiltonian neural networks. The first model uses additive
separability to quadratically scale the amount of data for training Hamiltonian
neural networks. The second model embeds additive separability within the loss
function of the Hamiltonian neural network. The third model embeds additive
separability through the architecture of the Hamiltonian neural network using
conjoined multilayer perceptions. We empirically compare the three models
against state-of-the-art Hamiltonian neural networks, and demonstrate that the
separable Hamiltonian neural networks, which alleviate complexity between the
state variables, are more effective at regressing the Hamiltonian and its
vector field.Comment: 11 page
Applications of Machine Learning to Modelling and Analysing Dynamical Systems
We explore the use of Physics Informed Neural Networks to analyse nonlinear
Hamiltonian Dynamical Systems with a first integral of motion. In this work, we
propose an architecture which combines existing Hamiltonian Neural Network
structures into Adaptable Symplectic Recurrent Neural Networks which preserve
Hamilton's equations as well as the symplectic structure of phase space while
predicting dynamics for the entire parameter space. This architecture is found
to significantly outperform previously proposed neural networks when predicting
Hamiltonian dynamics especially in potentials which contain multiple
parameters. We demonstrate its robustness using the nonlinear Henon-Heiles
potential under chaotic, quasiperiodic and periodic conditions.
The second problem we tackle is whether we can use the high dimensional
nonlinear capabilities of neural networks to predict the dynamics of a
Hamiltonian system given only partial information of the same. Hence we attempt
to take advantage of Long Short Term Memory networks to implement Takens'
embedding theorem and construct a delay embedding of the system followed by
mapping the topologically invariant attractor to the true form. This
architecture is then layered with Adaptable Symplectic nets to allow for
predictions which preserve the structure of Hamilton's equations. We show that
this method works efficiently for single parameter potentials and provides
accurate predictions even over long periods of time.Comment: This is a dissertation submitted in partial fulfilment of the
requirements for the degree of Bachelor of Science (Honours) Physics at St.
Stephens College University of Delhi in 2023. The dissertation was guided by
Dr. Abhinav Gupta, Associate Professor, Department of Physics, St. Stephens
College Delh
IST Austria Thesis
In this Thesis, I study composite quantum impurities with variational techniques, both inspired by machine learning as well as fully analytic. I supplement this with exploration of other applications of machine learning, in particular artificial neural networks, in many-body physics. In Chapters 3 and 4, I study quasiparticle systems with variational approach. I derive a Hamiltonian describing the angulon quasiparticle in the presence of a magnetic field. I apply analytic variational treatment to this Hamiltonian. Then, I introduce a variational approach for non-additive systems, based on artificial neural networks. I exemplify this approach on the example of the polaron quasiparticle (Fröhlich Hamiltonian). In Chapter 5, I continue using artificial neural networks, albeit in a different setting. I apply artificial neural networks to detect phases from snapshots of two types physical systems. Namely, I study Monte Carlo snapshots of multilayer classical spin models as well as molecular dynamics maps of colloidal systems. The main type of networks that I use here are convolutional neural networks, known for their applicability to image data
A spherical Hopfield model
We introduce a spherical Hopfield-type neural network involving neurons and
patterns that are continuous variables. We study both the thermodynamics and
dynamics of this model. In order to have a retrieval phase a quartic term is
added to the Hamiltonian. The thermodynamics of the model is exactly solvable
and the results are replica symmetric. A Langevin dynamics leads to a closed
set of equations for the order parameters and effective correlation and
response function typical for neural networks. The stationary limit corresponds
to the thermodynamic results. Numerical calculations illustrate our findings.Comment: 9 pages Latex including 3 eps figures, Addition of an author in the
HTML-abstract unintentionally forgotten, no changes to the manuscrip
Thermostat-assisted continuously-tempered Hamiltonian Monte Carlo for Bayesian learning
We propose a new sampling method, the thermostat-assisted
continuously-tempered Hamiltonian Monte Carlo, for Bayesian learning on large
datasets and multimodal distributions. It simulates the Nos\'e-Hoover dynamics
of a continuously-tempered Hamiltonian system built on the distribution of
interest. A significant advantage of this method is that it is not only able to
efficiently draw representative i.i.d. samples when the distribution contains
multiple isolated modes, but capable of adaptively neutralising the noise
arising from mini-batches and maintaining accurate sampling. While the
properties of this method have been studied using synthetic distributions,
experiments on three real datasets also demonstrated the gain of performance
over several strong baselines with various types of neural networks plunged in
General Neural Networks Dynamics are a Superposition of Gradient-like and Hamiltonian-like Systems
This report presents a formalism that enables the dynamics of a broad class of neural networks to be understood. A number of previous works have analyzed the Lyapunov stability of neural network models. This type of analysis shows that the excursion of the solutions from a stable point is bounded. The purpose of this work is to present a model of the dynamics that also describes the phase space behavior as well as the structural stability of the system. This is achieved by writing the general equations of the neural network dynamics as the sum of gradient-like and Hamiltonian-like systems. In this paper some important properties of both gradient-like and Hamiltonian-like systems are developed and then it is demonstrated that a broad class of neural network models are expressible in this form
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