phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | spatiotemporal dynamics whose governing equations have eluded analytic derivation— “Less well understood is how to uncover underlying physical laws and/or governing equations from time series data that exhibit spatiotemporal activity.” | |
| claim-type | method— “We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately… | |
| method — evidence | — “We demonstrate the success of the method by rediscovering a broad range of physical laws solely from time series data.” | |
| object of study — evidence | — “We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately… | |
| research question — evidence | — “Less well understood is how to uncover underlying physical laws and/or governing equations from time series data that exhibit spatiotemporal activity.” | |
| contribution — evidence | — “If we assume that **Θ** is an overcomplete library, meaning **Θ** has a sufficiently rich column space that the dynamics will be in its range, then the PDE should be well represented by Eq. 2 with a sparse vector of coefficients ξ. This is equivalent to picking enough candidate functions that the full PDE” | |
| mechanism | sparsity-promoting regression selects the few library terms that represent the data, which is what BYPASSES a combinatorially large search over all candidate models— “We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately… | mechanisms[0] |
| mechanism | 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— “Both the evolutionary ( _14_) and sparse ( _16_) symbolic regression methods avoid overfitting by selecting parsimonious models that balance model accuracy with complexity via Pareto analysis.” | mechanisms[1] |
| extends | the sparse and evolutionary symbolic-regression methods for discovering dynamical systems, carried from ODEs to partial differential equations— “Both the evolutionary ( _14_) and sparse ( _16_) symbolic regression methods avoid overfitting by selecting parsimonious models that balance model accuracy with complexity via Pareto analysis.” | claim_relations[0] |
| critiques | first-principles derivation as the only route to a governing equation, on the ground that many complex systems have eluded it— “Less well understood is how to uncover underlying physical laws and/or governing equations from time series data that exhibit spatiotemporal activity.” | claim_relations[1] |
| sense | 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— “If we assume that **Θ** is an overcomplete library, meaning **Θ** has a sufficiently rich column space that the dynamics will be in its range, then the PDE should be well represented by Eq. 2 with a sparse vector of coefficients ξ. This is equivalent to picking enough candidate… | senses[0] |
| sense | parsimonious model - the point on the accuracy-against-complexity Pareto front, not the model with fewest terms and not the most accurate one | senses[1] |
| control | 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 — held Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. — rules out 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 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[0] |
| control | Gaussian noise is added to every example and the method is re-run, and the data is also subsampled spatially — partial Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. — rules out a recovery that depends on clean, densely sampled data - noise is stated to significantly complicate both differentiation and library building Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[1] |
| control | multiple time series taken with DIFFERENT INITIAL DATA are used where terms are potentially nonunique — held Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. — rules out two different equations that fit one trajectory equally well being indistinguishable - a linear wave equation against Korteweg-de Vries Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[2] |
| weight | the sparse and evolutionary symbolic-regression methods for discovering dynamical systems, carried from ODEs to partial differential equations — 0.9 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | claim_relations[0] |
| weight | first-principles derivation as the only route to a governing equation, on the ground that many complex systems have eluded it — 0.85 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | claim_relations[1] |
| outcome | control — held Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[0] |
| outcome | control — partial Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[1] |
| outcome | control — held Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | controls[2] |
| limitation | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[0] |
| limitation | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[1] |
| limitation | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[2] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[0] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[1] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[2] |
| finding | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[3] |
| quantity | canonical problems named in the abstract - Navier-Stokes, the quantum harmonic oscillator and the diffusion equation - as the validation set Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.quantities[0] |
| variable | noise Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.variables[0] |
| variable | initial data Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.variables[1] |
| definition | overcomplete library Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[0] |
| definition | parsimonious model Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[1] |
| concept | overcomplete library — not defined in this paper's own words; the candidate function set the regression selects from - assumed rich enough to contain the answer Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[0] |
| concept | sparse regression — not defined in this paper's own words; the selection procedure - it picks few terms from many, which is what makes the search tractable Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[1] |
| concept | Pareto analysis — not defined in this paper's own words; the device that trades accuracy against complexity, and the paper's stated defence against overfitting Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[2] |
| concept | parsimonious model — not defined in this paper's own words; the model the Pareto trade selects - balanced rather than smallest Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[3] |
| sparse regression | Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.statistics[0] |
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