An official website of the United States government
Abstract 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.
more »
« less
Data-driven discovery of Green’s functions with human-understandable deep learning
Abstract 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
- 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
More Like this
-
-
Abstract The control of complex systems is often challenging due to high-dimensional nonlinear models, unmodeled phenomena, and parameter uncertainty. The increasing ubiquity of sensors measuring such systems and increased computational resources has led to an interest in purely data-driven control methods, particularly using the Koopman operator. In this paper, we elucidate the construction of a linear predictor based on a sequence of time realizations of observables drawn from a data archive of different trajectories combined with subspace identification methods for linear systems. This approach is free of any predefined set of basis functions but instead depends on the time realization of these basis functions. The prediction and control are demonstrated with examples. The basis functions can be constructed using time-delayed coordinates of the outputs, enabling the application to purely data-driven systems. The paper thus shows the link between Koopman operator-based control methods and classical subspace identification methods. The approach in this paper can be extended to adaptive online learning and control.more » « less
-
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.more » « less
-
Abstract Near-time conservation palaeobiology uses palaeontological, archaeological and other geohistorical records to study the late Quaternary transition of the biosphere from its pristine past to its present-day, human-altered state. Given the scarcity of data on recent extinctions in the oceans, geohistorical records are critical for documenting human-driven extinctions and extinction threats in the marine realm. The historical perspective can provide two key insights. First, geohistorical records archive the state of pre-industrial oceans at local, regional and global scales, thus enabling the detection of recent extinctions and extirpations as well as shifts in species distribution, abundance, body size and ecosystem function. Second, we can untangle the contributions of natural and anthropogenic processes by documenting centennial-to-millennial changes in the composition and diversity of marine ecosystems before and after the onset of major human impacts. This long-term perspective identifies recently emerging patterns and processes that are unprecedented, thus allowing us to better assess human threats to marine biodiversity. Although global-scale extinctions are not well documented for brackish and marine invertebrates, geohistorical studies point to numerous extirpations, declines in ecosystem functions, increases in range fragmentation and dwindling abundance of previously widespread species, indicating that marine ecosystems are accumulating a human-driven extinction debt.more » « less
-
Summary Many real‐world scientific processes are governed by complex non‐linear dynamic systems that can be represented by differential equations. Recently, there has been an increased interest in learning, or discovering, the forms of the equations driving these complex non‐linear dynamic systems using data‐driven approaches. In this paper, we review the current literature on data‐driven discovery for dynamic systems. We provide a categorisation to the different approaches for data‐driven discovery and a unified mathematical framework to show the relationship between the approaches. Importantly, we discuss the role of statistics in the data‐driven discovery field, describe a possible approach by which the problem can be cast in a statistical framework and provide avenues for future work.more » « less
