Data-driven discovery of partial differential equations

Data-driven discovery of partial differential equations Samuel Rudy Steven L. Brunton Joshua L. Proctor J. Nathan Kutz

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Data-driven discovery of partial differential equations 2017 Science Advances 10.1126/sciadv.1602614 rudy2017-data-driven-discovery-of-partial-d rudy2017-data-driven-discovery-of-partial-d.txt, the project's staged copy a2484cc436f85ee9da6dc656e95245f0edf263f1 cc-by-nc physics

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them the governing partial differential equations of a spatiotemporal system, recovered from time series measurements Quotation withheld because this record is not cleared for redistribution. whether the governing equations of a system can be discovered from data where first-principles derivation is intractable Quotation withheld because this record is not cleared for redistribution. a sparse regression method that selects the terms of a PDE from a large candidate library, with parsimony enforced by Pareto analysis and the library's sufficiency written down as an assumption Quotation withheld because this record is not cleared for redistribution. method Quotation withheld because this record is not cleared for redistribution. spatiotemporal dynamics whose governing equations have eluded analytic derivation Quotation withheld because this record is not cleared for redistribution. system-or-tool-building system-or-tool-building Quotation withheld because this record is not cleared for redistribution. sparsity-promoting regression selects the few library terms that represent the data, which is what BYPASSES a combinatorially large search over all candidate models Quotation withheld because this record is not cleared for redistribution. parsimony is enforced by Pareto analysis - accuracy against complexity - which is the same device the evolutionary and sparse symbolic-regression methods use to avoid overfitting Quotation withheld because this record is not cleared for redistribution. THE METHOD CAN ONLY DISCOVER WHAT IS ALREADY IN ITS LIBRARY, and the paper states the sufficiency of the library as an ASSUMPTION: `if we assume that Theta is an overcomplete library, meaning Theta has a sufficiently rich column space that the dynamics will be in its range` Quotation withheld because this record is not cleared for redistribution. THE NAMED FAILURE IS RIGHT TERMS AND WRONG NUMBERS: on Kuramoto-Sivashinsky the correct terms are identified and there is substantial coefficient error under noise, which the paper calls a limitation of the method rather than of the example Quotation withheld because this record is not cleared for redistribution. the validation is REDISCOVERY - a method for finding equations nobody knows, tested on Navier-Stokes, the quantum harmonic oscillator and the diffusion equation Quotation withheld because this record is not cleared for redistribution. the method is run on systems whose governing equations are already known - Navier-Stokes, the quantum harmonic oscillator, the diffusion equation - and asked to REDISCOVER them eliminates: a discovery procedure that produces a plausible equation for any data, since on these systems the right answer is known in advance and the method must return it · outcome: held Quotation withheld because this record is not cleared for redistribution. Gaussian noise is added to every example and the method is re-run, and the data is also subsampled spatially eliminates: a recovery that depends on clean, densely sampled data - noise is stated to significantly complicate both differentiation and library building · outcome: partial Quotation withheld because this record is not cleared for redistribution. multiple time series taken with DIFFERENT INITIAL DATA are used where terms are potentially nonunique eliminates: two different equations that fit one trajectory equally well being indistinguishable - a linear wave equation against Korteweg-de Vries · outcome: held Quotation withheld because this record is not cleared for redistribution. overcomplete library overcomplete library Quotation withheld because this record is not cleared for redistribution. sparse regression sparse regression Quotation withheld because this record is not cleared for redistribution. Pareto analysis Pareto analysis Quotation withheld because this record is not cleared for redistribution. parsimonious model parsimonious model Quotation withheld because this record is not cleared for redistribution. overcomplete library - AS DEFINED HERE: a candidate set whose column space is rich enough that the true dynamics lie in its range. The word `assume` is the paper's, and everything the method can find is inside it Quotation withheld because this record is not cleared for redistribution. parsimonious model - the point on the accuracy-against-complexity Pareto front, not the model with fewest terms and not the most accurate one Quotation withheld because this record is not cleared for redistribution. extends → the sparse and evolutionary symbolic-regression methods for discovering dynamical systems, carried from ODEs to partial differential equations Quotation withheld because this record is not cleared for redistribution. critiques → first-principles derivation as the only route to a governing equation, on the ground that many complex systems have eluded it Quotation withheld because this record is not cleared for redistribution. the method assumes an overcomplete library whose column space is rich enough to contain the dynamics the method assumes an overcomplete library whose column space is rich enough to contain the dynamics Quotation withheld because this record is not cleared for redistribution. on Kuramoto-Sivashinsky the CORRECT TERMS ARE IDENTIFIED and the COEFFICIENTS ARE SUBSTANTIALLY WRONG under noise - the paper separates the two failure modes and calls the second a limitation of the method on Kuramoto-Sivashinsky the CORRECT TERMS ARE IDENTIFIED and the COEFFICIENTS ARE SUBSTANTIALLY WRONG under noise - the paper separates the two failure modes and calls the second a limitation of the method Quotation withheld because this record is not cleared for redistribution. the method distinguishes a linear wave equation from Korteweg-de Vries by using multiple time series with different initial data the method distinguishes a linear wave equation from Korteweg-de Vries by using multiple time series with different initial data Quotation withheld because this record is not cleared for redistribution. every example is re-identified after Gaussian noise is added, and after significant spatial subsampling every example is re-identified after Gaussian noise is added, and after significant spatial subsampling Quotation withheld because this record is not cleared for redistribution. a candidate matrix with a sufficiently rich column space that the dynamics will be in its range - stated as an assumption, not established overcomplete library Quotation withheld because this record is not cleared for redistribution. the model selected by Pareto analysis, balancing model complexity against regression accuracy parsimonious model Quotation withheld because this record is not cleared for redistribution. canonical problems named in the abstract - Navier-Stokes, the quantum harmonic oscillator and the diffusion equation - as the validation set 3 Quotation withheld because this record is not cleared for redistribution. sparse regression sparse regression Quotation withheld because this record is not cleared for redistribution. noise noise Quotation withheld because this record is not cleared for redistribution. initial data initial data Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.1126/sciadv.1602614.