Sparse dynamics for partial differential equations

By Hayden Schaeffer, Russel E. Caflisch, Cory D. Hauck, Stanley J. Osher. 2012

190 citations recorded by the source corpus.

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.

Subfields: Computational Mechanics, Geophysics, Statistical and Nonlinear Physics

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Citations in the corpus

Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations (2018, methodology): “More recently, sparsity [23] has been used to determine the governing dynamical system [24, 25, 26, 27, 28, 29, 30, 31, 32, 33].”
Discovering governing equations from data by sparse identification of nonlinear dynamical systems (2015, background): “Proc Natl Acad Sci USA 104(24):9943–9948.”
On the Compressive Spectral Method (2014, methodology): “in [13] to solve an (1)-regularized least squares problem to obtain a sparse approximation of û (the Fourier transform of u).”
Chaos as an intermittently forced linear system (2016, background): “This work is part of a growing literature that is exploring the use of sparsity in dynamics [72, 84, 64] and dynamical systems [8, 78, 14].”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, methodology): “Recently, compressed sensing, sparsity-promoting algorithms such as the lasso regression [35], [36], [37], and machine learning have been increasingly applied to characterize and control dynamical systems [38], [39], [40], [41], [42], [43], [44], [45], [46], [47].”
Data-driven discovery of coordinates and governing equations (2019, unknown): “Parsimonious models strike this balance, having the fewest terms required to capture essential interactions (1, 3, 8, 10, 15), thus promoting interpretability and generalizability.”
Sparse Identification of Nonlinear Dynamics with Control (SINDYc) (2016, methodology): “There are many other interesting methods recently developed to incorporate sparsity and nonlinear dynamics (Schaeffer et al., 2013; Ozoliņš et al., 2013; Mackey et al., 2014).”
Compressed modes for variational problems in mathematics and physics (2013, unknown): “[6] as an approximation to a truncated diffusion of Dirac delta function via the Schrödinger operator − 1 2 ∂(2) x, which is a compactly supported function.”
Dynamic Mode Decomposition with Control (2014, methodology): “Sparse measurements have recently been leveraged in a variety of complex systems, some for control [3], [32], [25], [12].”