Data-driven discovery of dimensionless numbers and governing laws from scarce measurements

Xie, Xiaoyu; Samaei, Arash; Guo, Jiachen; Liu, Wing Kam; Gan, Zhengtao

doi:10.1038/s41467-022-35084-w

Cited by 36 publications

(11 citation statements)

References 49 publications

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“…These governing equations may not have the function of analyzing the system, but it can approach the solution of the original system numerically well. In the next step of research, we will focus on dimensional analysis theory and attempt to combine deep learning with dimensional invariance theory [53] to discover governing equations that can explain and analyze the system.…”

Section: Discussionmentioning

confidence: 99%

Governing equation discovery based on causal graph for nonlinear dynamic systems

Jia,

Zhou,

et al. 2023

Mach. Learn.: Sci. Technol.

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The governing equations of nonlinear dynamic systems is of great significance for understanding the internal physical characteristics. In order to learn the governing equations of nonlinear systems from noisy observed data, we propose a novel method named governing equation discovery based on causal graph that combines spatio-temporal graph convolution network with governing equation modeling. The essence of our method is to first devise the causal graph encoding based on transfer entropy to obtain the adjacency matrix with causal significance between variables. Then, the spatio-temporal graph convolutional network is used to obtain approximate solutions for the system variables. On this basis, automatic differentiation is applied to obtain basic derivatives and form a dictionary of candidate algebraic terms. Finally, sparse regression is used to obtain the coefficient matrix and determine the explicit formulation of the governing equations. We also design a novel cross-combinatorial optimization strategy to learn the heterogeneous parameters that include neural network parameters and control equation coefficients. We conduct extensive experiments on seven datasets from different physical fields. The experimental results demonstrate the proposed method can automatically discover the underlying governing equation of the systems, and has great robustness.

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Section: Discussionmentioning

confidence: 99%

Governing equation discovery based on causal graph for nonlinear dynamic systems

Jia,

Zhou,

et al. 2023

Mach. Learn.: Sci. Technol.

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“…2022; Xie et al. 2022). We assume that the scaling can be obtained through superposing appropriate polynomials of the Pi variables.…”

Section: Methodsmentioning

confidence: 99%

Data-driven nonlinear turbulent flow scaling with Buckingham Pi variables

Fukami,

Goto,

Taira

2024

J. Fluid Mech.

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Nonlinear machine learning for turbulent flows can exhibit robust performance even outside the range of training data. This is achieved when machine-learning models can accommodate scale-invariant characteristics of turbulent flow structures. This study presents a data-driven approach to reveal scale-invariant vortical structures across Reynolds numbers that provide insights for supporting nonlinear machine-learning-based studies of turbulent flows. To uncover conditions for which nonlinear models are likely to perform well, we use a Buckingham-Pi-based sparse nonlinear scaling to find the influence of the Pi groups on the turbulent flow data. We consider nonlinear scalings of the invariants of the velocity gradient tensor for an example of three-dimensional decaying isotropic turbulence. The present scaling not only enables the identification of vortical structures that are interpolatory and extrapolatory for the given flow field data but also captures non-equilibrium effects of the energy cascade. As a demonstration, the present findings are applied to machine-learning-based super-resolution analysis of three-dimensional isotropic turbulence. We show that machine-learning models reconstruct vortical structures well in the interpolatory space with reduced performance in the extrapolatory space revealed by the nonlinearly scaled invariants. The present approach enables us to depart from labelling turbulent flow data with a single parameter of Reynolds number and comprehensively examine the flow field to support training and testing of nonlinear machine-learning techniques.

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“…In this case, the newly introduced Ω terms have a number of characteristics that make them valuable for cosmologists. First of all, they are non-dimensional, something that physicists value in many cases, when working with cross-scale problems, since they are scale invariant [33]. Furthermore, the energy density parameters can be calculated using a specific procedure from other observables.…”

Section: Foreground and Backgroundmentioning

confidence: 99%

Rearranging equations to develop physics reasoning

Kapodistrias,

Airey

2024

Eur. J. Phys.

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Researchers generally agree that physics experts use mathematics in a way that blends mathematical knowledge with physics intuition. However, the use of mathematics in physics education has traditionally tended to focus more on the computational aspect (manipulating mathematical operations to get numerical solutions) to the detriment of building conceptual understanding and physics intuition. Several solutions to this problem have been suggested; some authors have suggested building conceptual understanding before mathematics is introduced, while others have argued for the inseparability of the two, claiming instead that mathematics and conceptual physics need to be taught simultaneously. Although there is a body of work looking into how students employ mathematical reasoning when working with equations, the specifics of how physics experts use mathematics blended with physics intuition remain relatively underexplored. In this paper, we describe some components of this blending, by analyzing how physicists perform the rearrangement of a specific equation in cosmology. Our data consist of five consecutive forms of rearrangement of the equation, as observed in three separate higher education cosmology courses. This rearrangement was analyzed from a conceptual reasoning perspective using Sherin’s framework of symbolic forms. Our analysis clearly demonstrates how the number of potential symbolic forms associated with each subsequent rearrangement of the equation decreases as we move from line to line. Drawing on this result, we suggest an underlying mechanism for how physicists reason with equations. This mechanism seems to consist of three components: narrowing down meaning potential, moving aspects between the background and the foreground and purposefully transforming the equation according to the discipline’s questions of interest. In the discussion section we highlight the potential that our work has for generalizability and how being aware of the components of this underlying mechanism can potentially affect physics teachers’ practice when using mathematics in the physics classroom.

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Data-driven discovery of dimensionless numbers and governing laws from scarce measurements

Cited by 36 publications

References 49 publications

Governing equation discovery based on causal graph for nonlinear dynamic systems

Governing equation discovery based on causal graph for nonlinear dynamic systems

Data-driven nonlinear turbulent flow scaling with Buckingham Pi variables

Rearranging equations to develop physics reasoning

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