اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدRoeNets: Predicting Discontinuity of Hyperbolic Systems from Continuous Data
54
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We introduce Roe Neural Networks (RoeNets) that can predict the discontinuity of the hyperbolic conservation laws (HCLs) based on short-term discontinuous and even continuous training data. Our methodology is inspired by Roe approximate Riemann solver (P. L. Roe, J. Comput. Phys., vol. 43, 1981, pp. 357--372), which is one of the most fundamental HCLs numerical solvers. In order to accurately solve the HCLs, Roe argues the need to construct a Roe matrix that fulfills Property U, including diagonalizable with real eigenvalues, consistent with the exact Jacobian, and preserving conserved quantities. However, the construction of such matrix cannot be achieved by any general numerical method. Our model made a breakthrough improvement in solving the HCLs by applying Roe solver under a neural network perspective. To enhance the expressiveness of our model, we incorporate pseudoinverses into a novel context to enable a hidden dimension so that we are flexible with the number of parameters. The ability of our model to predict long-term discontinuity from a short window of continuous training data is in general considered impossible using traditional machine learning approaches. We demonstrate that our model can generate highly accurate predictions of evolution of convection without dissipation and the discontinuity of hyperbolic systems from smooth training data.
قيم البحث
اقرأ أيضاً
Concise, accurate descriptions of physical systems through their conserved quantities abound in the natural sciences. In data science, however, current research often focuses on regression problems, without routinely incorporating additional assumpti
ons about the system that generated the data. Here, we propose to explore a particular type of underlying structure in the data: Hamiltonian systems, where an energy is conserved. Given a collection of observations of such a Hamiltonian system over time, we extract phase space coordinates and a Hamiltonian function of them that acts as the generator of the system dynamics. The approach employs an autoencoder neural network component to estimate the transformation from observations to the phase space of a Hamiltonian system. An additional neural network component is used to approximate the Hamiltonian function on this constructed space, and the two components are trained jointly. As an alternative approach, we also demonstrate the use of Gaussian processes for the estimation of such a Hamiltonian. After two illustrative examples, we extract an underlying phase space as well as the generating Hamiltonian from a collection of movies of a pendulum. The approach is fully data-driven, and does not assume a particular form of the Hamiltonian function.
We study the problem of predicting rare critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster process drives its evolution and induces cri
tical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as the limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional. We discuss the mathematical and broader implications of our results.
In an abstract sense, quantum data hiding is the manifestation of the fact that two classes of quantum measurements can perform very differently in the task of binary quantum state discrimination. We investigate this phenomenon in the context of cont
inuous variable quantum systems. First, we look at the celebrated case of data hiding against the set of local operations and classical communication. While previous studies have placed upper bounds on its maximum efficiency in terms of the local dimension and are thus not applicable to continuous variable systems, we tackle this latter case by establishing more general bounds that rely solely on the local mean photon number of the states employed. Along the way, we perform a quantitative analysis of the error introduced by the non-ideal Braunstein--Kimble quantum teleportation protocol, determining how much two-mode squeezing and local detection efficiency is needed in order to teleport an arbitrary local state of known mean energy with a prescribed accuracy. Finally, following a seminal proposal by Winter, we look at data hiding against the set of Gaussian operations and classical computation, providing the first example of a relatively simple scheme that works with a single mode only. The states employed can be generated from a two-mode squeezed vacuum by local photon counting; the larger the squeezing, the higher the efficiency of the scheme.
Stochastic dynamical systems with continuous symmetries arise commonly in nature and often give rise to coherent spatio-temporal patterns. However, because of their random locations, these patterns are not well captured by current order reduction tec
hniques and a large number of modes is typically necessary for an accurate solution. In this work, we introduce a new methodology for efficient order reduction of such systems by combining (i) the method of slices, a symmetry reduction tool, with (ii) any standard order reduction technique, resulting in efficient mixed symmetry-dimensionality reduction schemes. In particular, using the Dynamically Orthogonal (DO) equations in the second step, we obtain a novel nonlinear Symmetry-reduced Dynamically Orthogonal (SDO) scheme. We demonstrate the performance of the SDO scheme on stochastic solutions of the 1D Korteweg-de Vries and 2D Navier-Stokes equations.
Predicting features of complex, large-scale quantum systems is essential to the characterization and engineering of quantum architectures. We present an efficient approach for constructing an approximate classical description, called the classical sh
adow, of a quantum system from very few quantum measurements that can later be used to predict a large collection of features. This approach is guaranteed to accurately predict M linear functions with bounded Hilbert-Schmidt norm from only order of log(M) measurements. This is completely independent of the system size and saturates fundamental lower bounds from information theory. We support our theoretical findings with numerical experiments over a wide range of problem sizes (2 to 162 qubits). These highlight advantages compared to existing machine learning approaches.
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
