We extend the Adaptive Antoulas-Anderson () algorithm to develop a data-driven modeling framework for linear systems with quadratic output (). Such systems are characterized by two transfer functions: one corresponding to the linear part of the output and another one to the quadratic part. We first establish the joint barycentric representations and the interpolation theory for the two transfer functions of systems. This analysis leads to the proposed algorithm. We show that by interpolating the transfer function values on a subset of samples together with imposing a least-squares minimization on the rest, we construct reliable data-driven models. Two numerical test cases illustrate the efficiency of the proposed method.
Data-driven discovery of Green’s functions with human-understandable deep learning
There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. To do this, we develop a novel data-driven approach for creating a human–machine partnership to accelerate scientific discovery. By collecting physical system responses under excitations drawn from a Gaussian process, we train rational neural networks to learn Green’s functions of hidden linear partial differential equations. These functions reveal human-understandable properties and features, such as linear conservation laws and symmetries, along with shock and singularity locations, boundary effects, and dominant modes. We illustrate the technique on several examples and capture a range of physics, including advection–diffusion, viscous shocks, and Stokes flow in a lid-driven cavity.
more » « less- Award ID(s):
- 2045646
- NSF-PAR ID:
- 10364146
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Scientific Reports
- Volume:
- 12
- Issue:
- 1
- ISSN:
- 2045-2322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Machine learning (ML) has become commonplace in educational research and science education research, especially to support assessment efforts. Such applications of machine learning have shown their promise in replicating and scaling human‐driven codes of students' work. Despite this promise, we and other scholars argue that machine learning has not yet achieved its transformational potential. We argue that this is because our field is currently lacking frameworks for supporting creative, principled, and critical endeavors to use machine learning in science education research. To offer considerations for science education researchers' use of ML, we present a framework, Distributing Epistemic Functions and Tasks (DEFT), that highlights the functions and tasks that pertain to generating knowledge that can be carried out by either trained researchers or machine learning algorithms. Such considerations are critical decisions that should occur alongside those about, for instance, the type of data or algorithm used. We apply this framework to two cases, one that exemplifies the cutting‐edge use of machine learning in science education research and another that offers a wholly different means of using machine learning and human‐driven inquiry together. We conclude with strategies for researchers to adopt machine learning and call for the field to rethink how we prepare science education researchers in an era of great advances in computational power and access to machine learning methods.
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Summary Citrus come in diverse sizes and shapes, and play a key role in world culture and economy. Citrus oil glands in particular contain essential oils which include plant secondary metabolites associated with flavor and aroma. Capturing and analyzing nuanced information behind the citrus fruit shape and its oil gland distribution provide a morphology‐driven path to further our insight into phenotype–genotype interactions.
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