Related papers: Sparse Dynamics for Partial Differential Equations
In this paper, we propose a general sparse decomposition of dynamical systems provided that the vector field and constraint set possess certain sparse structures, which we call subsystems. This notion is based on causal dependence in the…
With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
In applications of nonlinear and complex dynamical systems, a common situation is that the system can be measured but its structure and the detailed rules of dynamical evolution are unknown. The inverse problem is to determine the system…
Stiff forces, which bind objects together or otherwise confine motion, are found widely in soft-matter systems - colloids with short range attractions, ligand-receptor contacts, particles in optical traps, fibres that resist stretching,…
We study the performance of sparse regression methods and propose new techniques to distill the governing equations of dynamical systems from data. We first look at the generic methodology of learning interpretable equation forms from data,…
Learning governing equations allows for deeper understanding of the structure and dynamics of data. We present a random sampling method for learning structured dynamical systems from under-sampled and possibly noisy state-space…
We produce approximation bounds on a semidefinite programming relaxation for sparse principal component analysis. These bounds control approximation ratios for tractable statistics in hypothesis testing problems where data points are…
Sticky diffusion models a Markovian particle experiencing reflection and temporary adhesion phenomena at the boundary. Numerous numerical schemes exist for approximating stopped or reflected stochastic differential equations (SDEs), but…
Nonlinear dynamics are ubiquitous in science and engineering applications, but the physics of most complex systems is far from being fully understood. Discovering interpretable governing equations from measurement data can help us…
Automatic machine learning of empirical models from experimental data has recently become possible as a result of increased availability of computational power and dedicated algorithms. Despite the successes of non-parametric inference and…
The sparse representation of signals defined on Euclidean domains has been successfully applied in signal processing. Bringing the power of sparse representations to non-regular domains is still a challenge, but promising approaches have…
In the first part of the series papers, we set out to answer the following question: given specific restrictions on a set of samplers, what kind of signal can be uniquely represented by the corresponding samples attained, as the foundation…
In sparse coding, we attempt to extract features of input vectors, assuming that the data is inherently structured as a sparse superposition of basic building blocks. Similarly, neural networks perform a given task by learning features of…
This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and…
We develop a sparse spectral method for a class of fractional differential equations, posed on $\mathbb{R}$, in one dimension. These equations can include sqrt-Laplacian, Hilbert, derivative and identity terms. The numerical method utilizes…
We advocate an optimization procedure for variable density sampling in the context of compressed sensing. In this perspective, we introduce a minimization problem for the coherence between the sparsity and sensing bases, whose solution…
Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a…
In many problems of data-driven modeling for dynamical systems, the governing equations are not known a priori and must be selected phenomenologically from a large set of candidate interactions and basis functions. In such situations, point…
Robust physics discovery is of great interest for many scientific and engineering fields. Inspired by the principle that a representative model is the one simplest possible, a new model selection criteria considering both model's Parsimony…