Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

• URL: http://arxiv.org/abs/2012.06681v2
• Date: Thu, 10 Jun 2021 18:47:05 GMT
• Title: Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms
• Authors: Alexander Engel, Graeme Smith, Scott E. Parker
• Abstract summary: We describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) We then explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation)
• Score: 74.17312533172291
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. We describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously-studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.

Related papers

• Learning Controlled Stochastic Differential Equations [61.82896036131116]
  This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled differential equations with non-uniform diffusion. We provide strong theoretical guarantees, including finite-sample bounds for (L2), (Linfty), and risk metrics, with learning rates adaptive to coefficients' regularity. Our method is available as an open-source Python library.
  arXiv  Detail & Related papers  (2024-11-04T11:09:58Z)
• Metric-Entropy Limits on Nonlinear Dynamical System Learning [4.069144210024563]
  We show that recurrent neural networks (RNNs) are capable of learning nonlinear systems that satisfy a Lipschitz property and forget past inputs fast enough in a metric-entropy optimal manner. As the sets of sequence-to-sequence maps we consider are significantly more massive than function classes generally considered in deep neural network approximation theory, a refined metric-entropy characterization is needed.
  arXiv  Detail & Related papers  (2024-07-01T12:57:03Z)
• Koopman-based Deep Learning for Nonlinear System Estimation [1.3791394805787949]
  We present a novel data-driven linear estimator based on Koopman operator theory to extract meaningful finite-dimensional representations of complex non-linear systems. Our estimator is also adaptive to a diffeomorphic transformation of the estimated nonlinear system, which enables it to compute optimal state estimates without re-learning.
  arXiv  Detail & Related papers  (2024-05-01T16:49:54Z)
• A Polynomial Time Quantum Algorithm for Exponentially Large Scale Nonlinear Differential Equations via Hamiltonian Simulation [1.6003521378074745]
  We introduce a class of systems of nonlinear ODEs that can be efficiently solved on quantum computers. Specifically, we employ the Koopman-von Neumann linearization to map the system of nonlinear ODEs to Hamiltonian dynamics. This allows us to use the optimal Hamiltonian simulation technique for solving the nonlinear ODEs with $O(rm log(N))$ overhead.
  arXiv  Detail & Related papers  (2023-05-01T04:22:56Z)
• Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learning [11.339982217541822]
  We present a data-driven methodology for modeling the dynamics of nonlinear systems. In this work, we propose using quadratic systems as the common structure, inspired by the lifting principle.
  arXiv  Detail & Related papers  (2022-11-01T10:03:34Z)
• Challenges for quantum computation of nonlinear dynamical systems using linear representations [2.2000635322691378]
  We show that a necessary projection into a feasible finite-dimensional space will in practice induce numerical artifacts which can be hard to eliminate or even control. As a result, a practical, reliable and accurate way to use quantum computation for solving general nonlinear dynamical systems is still an open problem.
  arXiv  Detail & Related papers  (2022-02-04T15:29:53Z)
• Exact solutions of interacting dissipative systems via weak symmetries [77.34726150561087]
  We analytically diagonalize the Liouvillian of a class Markovian dissipative systems with arbitrary strong interactions or nonlinearity. This enables an exact description of the full dynamics and dissipative spectrum. Our method is applicable to a variety of other systems, and could provide a powerful new tool for the study of complex driven-dissipative quantum systems.
  arXiv  Detail & Related papers  (2021-09-27T17:45:42Z)
• StarNet: Gradient-free Training of Deep Generative Models using Determined System of Linear Equations [47.72653430712088]
  We present an approach for training deep generative models based on solving determined systems of linear equations. A network that uses this approach, called a StarNet, has the following desirable properties.
  arXiv  Detail & Related papers  (2021-01-03T08:06:42Z)
• Active Learning for Nonlinear System Identification with Guarantees [102.43355665393067]
  We study a class of nonlinear dynamical systems whose state transitions depend linearly on a known feature embedding of state-action pairs. We propose an active learning approach that achieves this by repeating three steps: trajectory planning, trajectory tracking, and re-estimation of the system from all available data. We show that our method estimates nonlinear dynamical systems at a parametric rate, similar to the statistical rate of standard linear regression.
  arXiv  Detail & Related papers  (2020-06-18T04:54:11Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.