Multidimensional approximation of nonlinear dynamical systems

Abstract

A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method MANDy which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems.

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• Physics-constrained, low-dimensional models for MHD: First-principles and data-driven approaches
• Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)
• Sparse nonlinear models of chaotic electroconvection
• SINDy-BVP: Sparse Identification of Nonlinear Dynamics for Boundary Value Problems
• Tensor-based algorithms for image classification
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• Deeptime: a Python library for machine learning dynamical models from time series data
• Scalably learning quantum many-body Hamiltonians from dynamical data
• tgEDMD: Approximation of the Kolmogorov Operator in Tensor Train Format
• Feature space approximation for kernel-based supervised learning
• On Koopman Mode Decomposition and Tensor Component Analysis
• Adaptive parameters identification for nonlinear dynamics using deep permutation invariant networks
• A quantum inspired approach to learning dynamical laws from data -- block-sparsity and gauge-mediated weight sharing
• Extraction of nonlinearity in neural networks with Koopman operator
• WaveTrain: A Python Package for Numerical Quantum Mechanics of Chain-Like Systems Based on Tensor Trains
• Machine learning and serving of discrete field theories -- when artificial intelligence meets the discrete universe
• Koopman operator-based discussion on partial observation in stochastic systems
• Solving the time-independent Schrödinger equation for chains of coupled excitons and phonons using tensor trains