DataDriven References

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@Article{Brunton2016,
  author       = {Brunton, Steven L. and Proctor, Joshua L. and Kutz, J. Nathan},
  date         = {2016-04},
  journaltitle = {Proceedings of the National Academy of Sciences},
  title        = {Discovering governing equations from data by sparse identification of nonlinear dynamical systems},
  doi          = {10.1073/pnas.1517384113},
  issn         = {0027-8424, 1091-6490},
  language     = {en},
  number       = {15},
  pages        = {3932--3937},
  url          = {https://www.pnas.org/content/113/15/3932},
  urldate      = {2021-08-26},
  volume       = {113},
  abstract     = {Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.},
  chapter      = {Physical Sciences},
  copyright    = {© . Freely available online through the PNAS open access option.},
  file         = {Full Text PDF:https\://www.pnas.org/content/pnas/113/15/3932.full.pdf:application/pdf;PubMed entry:http\://www.ncbi.nlm.nih.gov/pubmed/27035946:text/html},
  keywords     = {dynamical systems, machine learning, sparse regression, system identification, optimization},
  pmid         = {27035946},
  publisher    = {National Academy of Sciences},
}

@Article{Mangan2016,
  author       = {Mangan, Niall M. and Brunton, Steven L. and Proctor, Joshua L. and Kutz, J. Nathan},
  date         = {2016-06},
  journaltitle = {IEEE Transactions on Molecular, Biological and Multi-Scale Communications},
  title        = {Inferring {Biological} {Networks} by {Sparse} {Identification} of {Nonlinear} {Dynamics}},
  doi          = {10.1109/TMBMC.2016.2633265},
  issn         = {2332-7804},
  number       = {1},
  pages        = {52--63},
  volume       = {2},
  abstract     = {Inferring the structure and dynamics of network models is critical to understanding the functionality and control of complex systems, such as metabolic and regulatory biological networks. The increasing quality and quantity of experimental data enable statistical approaches based on information theory for model selection and goodness-of-fit metrics. We propose an alternative data-driven method to infer networked nonlinear dynamical systems by using sparsity-promoting optimization to select a subset of nonlinear interactions representing dynamics on a network. In contrast to standard model selection methods-based upon information content for a finite number of heuristic models (order 10 or less), our model selection procedure discovers a parsimonious model from a combinatorially large set of models, without an exhaustive search. Our particular innovation is appropriate for many biological networks, where the governing dynamical systems have rational function nonlinearities with cross terms, thus requiring an implicit formulation and the equations to be identified in the null-space of a library of mixed nonlinearities, including the state and derivative terms. This method, implicit-SINDy, succeeds in inferring three canonical biological models: 1) Michaelis-Menten enzyme kinetics; 2) the regulatory network for competence in bacteria; and 3) the metabolic network for yeast glycolysis.},
  file         = {IEEE Xplore Full Text PDF:https\://ieeexplore.ieee.org/ielx7/6687308/7809127/07809160.pdf?tp=&arnumber=7809160&isnumber=7809127&ref=aHR0cHM6Ly9pZWVleHBsb3JlLmllZWUub3JnL2RvY3VtZW50Lzc4MDkxNjA=:application/pdf},
  keywords     = {Biological system modeling, Nonlinear dynamical systems, Biochemistry, Machine learning, Computational modeling, Dynamical systems, network inference, nonlinear dynamics, biological networks, machine learning, sparse selection, non-convex optimization},
}

@Article{Kaheman2020,
  author       = {Kaheman, Kadierdan and Kutz, J. Nathan and Brunton, Steven L.},
  date         = {2020},
  journaltitle = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
  title        = {SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics},
  doi          = {10.1098/rspa.2020.0279},
  eprint       = {https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2020.0279},
  number       = {2242},
  pages        = {20200279},
  url          = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2020.0279},
  volume       = {476},
  abstract     = {Accurately modelling the nonlinear dynamics of a system from measurement data is a challenging yet vital topic. The sparse identification of nonlinear dynamics (SINDy) algorithm is one approach to discover dynamical systems models from data. Although extensions have been developed to identify implicit dynamics, or dynamics described by rational functions, these extensions are extremely sensitive to noise. In this work, we develop SINDy-PI (parallel, implicit), a robust variant of the SINDy algorithm to identify implicit dynamics and rational nonlinearities. The SINDy-PI framework includes multiple optimization algorithms and a principled approach to model selection. We demonstrate the ability of this algorithm to learn implicit ordinary and partial differential equations and conservation laws from limited and noisy data. In particular, we show that the proposed approach is several orders of magnitude more noise robust than previous approaches, and may be used to identify a class of ODE and PDE dynamics that were previously unattainable with SINDy, including for the double pendulum dynamics and simplified model for the Belousov–Zhabotinsky (BZ) reaction.},
}

@Article{Brunton2016a,
  author       = {Brunton, Steven L. and Proctor, Joshua L. and Kutz, J. Nathan},
  date         = {2016-01},
  journaltitle = {IFAC-PapersOnLine},
  title        = {Sparse {Identification} of {Nonlinear} {Dynamics} with {Control} ({SINDYc})**{SLB} acknowledges support from the {U}.{S}. {Air} {Force} {Center} of {Excellence} on {Nature} {Inspired} {Flight} {Technologies} and {Ideas} ({FA9550}-14-1-0398). {JLP} thanks {Bill} and {Melinda} {Gates} for their active support of the {Institute} of {Disease} {Modeling} and their sponsorship through the {Global} {Good} {Fund}. {JNK} acknowledges support from the {U}.{S}. {Air} {Force} {Office} of {Scientific} {Research} ({FA9550}-09-0174).},
  doi          = {10.1016/j.ifacol.2016.10.249},
  issn         = {2405-8963},
  language     = {en},
  number       = {18},
  pages        = {710--715},
  series       = {10th {IFAC} {Symposium} on {Nonlinear} {Control} {Systems} {NOLCOS} 2016},
  url          = {https://www.sciencedirect.com/science/article/pii/S2405896316318298},
  urldate      = {2021-08-26},
  volume       = {49},
  abstract     = {Identifying governing equations from data is a critical step in the modeling and control of complex dynamical systems. Here, we investigate the data-driven identification of nonlinear dynamical systems with inputs and forcing using regression methods, including sparse regression. Specifically, we generalize the sparse identification of nonlinear dynamics (SINDY) algorithm to include external inputs and feedback control. This method is demonstrated on examples including the Lotka-Volterra predator-prey model and the Lorenz system with forcing and control. We also connect the present algorithm with the dynamic mode decomposition (DMD) and Koopman operator theory to provide a broader context.},
  file         = {ScienceDirect Full Text PDF:https\://www.sciencedirect.com/science/article/pii/S2405896316318298/pdf?md5=3d87aefd0ce433faeede21e87ac4589c&pid=1-s2.0-S2405896316318298-main.pdf&isDTMRedir=Y:application/pdf},
  keywords     = {Dynamical systems, control, system identification, sparse regression},
}

@Article{Williams2015,
  author       = {Williams, Matthew O. and Kevrekidis, Ioannis G. and Rowley, Clarence W.},
  date         = {2015-12},
  journaltitle = {Journal of Nonlinear Science},
  title        = {A {Data}–{Driven} {Approximation} of the {Koopman} {Operator}: {Extending} {Dynamic} {Mode} {Decomposition}},
  doi          = {10.1007/s00332-015-9258-5},
  issn         = {1432-1467},
  language     = {en},
  number       = {6},
  pages        = {1307--1346},
  url          = {https://doi.org/10.1007/s00332-015-9258-5},
  urldate      = {2021-08-26},
  volume       = {25},
  abstract     = {The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator. We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005). Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions.},
  file         = {Springer Full Text PDF:https\://link.springer.com/content/pdf/10.1007%2Fs00332-015-9258-5.pdf:application/pdf},
  shorttitle   = {A {Data}–{Driven} {Approximation} of the {Koopman} {Operator}},
}

@Article{Proctor2016,
  author       = {Proctor, Joshua L. and Brunton, Steven L. and Kutz, J. Nathan},
  date         = {2016},
  journaltitle = {SIAM Journal on Applied Dynamical Systems},
  title        = {Dynamic Mode Decomposition with Control},
  doi          = {10.1137/15M1013857},
  eprint       = {https://doi.org/10.1137/15M1013857},
  number       = {1},
  pages        = {142-161},
  url          = {https://doi.org/10.1137/15M1013857},
  volume       = {15},
}

@Article{Schmid2010,
  author       = {Schmid, Peter J.},
  date         = {2010-08},
  journaltitle = {Journal of Fluid Mechanics},
  title        = {Dynamic mode decomposition of numerical and experimental data},
  doi          = {10.1017/S0022112010001217},
  issn         = {1469-7645, 0022-1120},
  language     = {en},
  pages        = {5--28},
  url          = {https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/dynamic-mode-decomposition-of-numerical-and-experimental-data/AA4C763B525515AD4521A6CC5E10DBD4},
  urldate      = {2021-08-26},
  volume       = {656},
  abstract     = {The description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information from flow fields that are either generated by a (direct) numerical simulation or visualized/measured in a physical experiment. The extracted dynamic modes, which can be interpreted as a generalization of global stability modes, can be used to describe the underlying physical mechanisms captured in the data sequence or to project large-scale problems onto a dynamical system of significantly fewer degrees of freedom. The concentration on subdomains of the flow field where relevant dynamics is expected allows the dissection of a complex flow into regions of localized instability phenomena and further illustrates the flexibility of the method, as does the description of the dynamics within a spatial framework. Demonstrations of the method are presented consisting of a plane channel flow, flow over a two-dimensional cavity, wake flow behind a flexible membrane and a jet passing between two cylinders.},
  file         = {Full Text PDF:https\://www.cambridge.org/core/services/aop-cambridge-core/content/view/AA4C763B525515AD4521A6CC5E10DBD4/S0022112010001217a.pdf/div-class-title-dynamic-mode-decomposition-of-numerical-and-experimental-data-div.pdf:application/pdf},
  publisher    = {Cambridge University Press},
}

@Article{Brunton2016b,
  author       = {Brunton, Steven L. and Brunton, Bingni W. and Proctor, Joshua L. and Kutz, J. Nathan},
  date         = {2016-02},
  journaltitle = {PLOS ONE},
  title        = {Koopman {Invariant} {Subspaces} and {Finite} {Linear} {Representations} of {Nonlinear} {Dynamical} {Systems} for {Control}},
  doi          = {10.1371/journal.pone.0150171},
  issn         = {1932-6203},
  language     = {en},
  number       = {2},
  pages        = {e0150171},
  url          = {https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0150171},
  urldate      = {2021-08-26},
  volume       = {11},
  abstract     = {In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ1-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.},
  file         = {Full Text PDF:https\://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0150171&type=printable:application/pdf},
  keywords     = {Dynamical systems, Nonlinear dynamics, Nonlinear systems, Polynomials, Algorithms, Eigenvalues, Fluid dynamics, Eigenvectors},
  publisher    = {Public Library of Science},
}

@Article{Tu2014,
  author       = {Tu, Jonathan H. and Rowley, Clarence W. and Luchtenburg, Dirk M. and Brunton, Steven L. and Kutz, J. Nathan},
  date         = {2014},
  journaltitle = {Journal of Computational Dynamics},
  title        = {On dynamic mode decomposition: {Theory} and applications},
  doi          = {10.3934/jcd.2014.1.391},
  language     = {en},
  number       = {2},
  pages        = {391},
  url          = {https://www.aimsciences.org/article/doi/10.3934/jcd.2014.1.391},
  urldate      = {2021-08-26},
  volume       = {1},
  abstract     = {Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems.
    However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken.
    We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator.
    This generalizes DMD to a larger class of datasets, including nonsequential time series.
    We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively.
    We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples.
    Such computations are not considered in the existing literature but can be understood using our more general framework.
    In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory.
    It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science.
    We show that under certain conditions, DMD is equivalent to LIM.},
  copyright    = {http://creativecommons.org/licenses/by/3.0/},
  file         = {Full Text PDF:https\://www.aimsciences.org/article/exportPdf?id=1dfebc20-876d-4da7-8034-7cd3c7ae1161:application/pdf},
  publisher    = {American Institute of Mathematical Sciences},
  shorttitle   = {On dynamic mode decomposition},
}

@Article{Klus2020,
  author       = {Klus, Stefan and Nüske, Feliks and Peitz, Sebastian and Niemann, Jan-Hendrik and Clementi, Cecilia and Schütte, Christof},
  date         = {2020-05},
  journaltitle = {Physica D: Nonlinear Phenomena},
  title        = {Data-driven approximation of the {Koopman} generator: {Model} reduction, system identification, and control},
  doi          = {10.1016/j.physd.2020.132416},
  issn         = {0167-2789},
  language     = {en},
  pages        = {132416},
  url          = {https://www.sciencedirect.com/science/article/pii/S0167278919306086},
  urldate      = {2021-08-26},
  volume       = {406},
  abstract     = {We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.},
  keywords     = {Data-driven methods, Koopman operator, Infinitesimal generator, System identification, Coarse graining, Control},
  shorttitle   = {Data-driven approximation of the {Koopman} generator},
}

@Book{Mauroy2020,
  date       = {2020},
  title      = {The {Koopman} {Operator} in {Systems} and {Control}: {Concepts}, {Methodologies}, and {Applications}},
  doi        = {10.1007/978-3-030-35713-9},
  editor     = {Mauroy, Alexandre and Mezic, Igor and Susuki, Yoshihiko},
  isbn       = {9783030357122},
  language   = {en},
  publisher  = {Springer International Publishing},
  series     = {Lecture {Notes} in {Control} and {Information} {Sciences}},
  url        = {https://www.springer.com/de/book/9783030357122},
  urldate    = {2021-08-26},
  abstract   = {This book provides a broad overview of state-of-the-art research at the intersection of the Koopman operator theory and control theory. It also reviews novel theoretical results obtained and efficient numerical methods developed within the framework of Koopman operator theory.The contributions discuss the latest findings and techniques in several areas of control theory, including model predictive control, optimal control, observer design, systems identification and structural analysis of controlled systems, addressing both theoretical and numerical aspects and presenting open research directions, as well as detailed numerical schemes and data-driven methods. Each contribution addresses a specific problem. After a brief introduction of the Koopman operator framework, including basic notions and definitions, the book explores numerical methods, such as the dynamic mode decomposition (DMD) algorithm and Arnoldi-based methods, which are used to represent the operator in a finite-dimensional basis and to compute its spectral properties from data. The main body of the book is divided into three parts:theoretical results and numerical techniques for observer design, synthesis analysis, stability analysis, parameter estimation, and identification;data-driven techniques based on DMD, which extract the spectral properties of the Koopman operator from data for the structural analysis of controlled systems; andKoopman operator techniques with specific applications in systems and control, which range from heat transfer analysis to robot control.A useful reference resource on the Koopman operator theory for control theorists and practitioners, the book is also of interest to graduate students, researchers, and engineers looking for an introduction to a novel and comprehensive approach to systems and control, from pure theory to data-driven methods.},
  shorttitle = {The {Koopman} {Operator} in {Systems} and {Control}},
}

@Book{Brunton2019,
  author     = {Brunton, Steven L. and Kutz, J. Nathan},
  date       = {2019},
  title      = {Data-{Driven} {Science} and {Engineering}: {Machine} {Learning}, {Dynamical} {Systems}, and {Control}},
  doi        = {10.1017/9781108380690},
  isbn       = {9781108422093},
  location   = {Cambridge},
  publisher  = {Cambridge University Press},
  url        = {https://www.cambridge.org/core/books/datadriven-science-and-engineering/77D52B171B60A496EAFE4DB662ADC36E},
  urldate    = {2021-08-26},
  abstract   = {Data-driven discovery is revolutionizing the modeling, prediction, and control of complex systems. This textbook brings together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science. It highlights many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, such as turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy. Aimed at advanced undergraduate and beginning graduate students in the engineering and physical sciences, the text presents a range of topics and methods from introductory to state of the art.},
  shorttitle = {Data-{Driven} {Science} and {Engineering}},
}

@Book{Kutz2016,
  author    = {Kutz, J. Nathan and Brunton, Steven L. and Brunton, Bingni W. and Proctor, Joshua L.},
  date      = {2016},
  title     = {Dynamic Mode Decomposition},
  doi       = {10.1137/1.9781611974508},
  eprint    = {https://epubs.siam.org/doi/pdf/10.1137/1.9781611974508},
  location  = {Philadelphia, PA},
  publisher = {Society for Industrial and Applied Mathematics},
  url       = {https://epubs.siam.org/doi/abs/10.1137/1.9781611974508},
}

@Article{Cranmer2020,
  author       = {Cranmer, Miles and Sanchez-Gonzalez, Alvaro and Battaglia, Peter and Xu, Rui and Cranmer, Kyle and Spergel, David and Ho, Shirley},
  date         = {2020-11},
  journaltitle = {arXiv:2006.11287 [astro-ph, physics:physics, stat]},
  title        = {Discovering {Symbolic} {Models} from {Deep} {Learning} with {Inductive} {Biases}},
  note         = {arXiv: 2006.11287},
  url          = {http://arxiv.org/abs/2006.11287},
  urldate      = {2021-08-26},
  abstract     = {We develop a general approach to distill symbolic representations of a learned deep model by introducing strong inductive biases. We focus on Graph Neural Networks (GNNs). The technique works as follows: we first encourage sparse latent representations when we train a GNN in a supervised setting, then we apply symbolic regression to components of the learned model to extract explicit physical relations. We find the correct known equations, including force laws and Hamiltonians, can be extracted from the neural network. We then apply our method to a non-trivial cosmology example-a detailed dark matter simulation-and discover a new analytic formula which can predict the concentration of dark matter from the mass distribution of nearby cosmic structures. The symbolic expressions extracted from the GNN using our technique also generalized to out-of-distribution data better than the GNN itself. Our approach offers alternative directions for interpreting neural networks and discovering novel physical principles from the representations they learn.},
  annotation   = {Comment: Accepted to NeurIPS 2020. 9 pages content + 16 pages appendix/references. Supporting code found at https://github.com/MilesCranmer/symbolic\_deep\_learning},
  file         = {arXiv Fulltext PDF:https\://arxiv.org/pdf/2006.11287.pdf:application/pdf},
  keywords     = {Computer Science - Machine Learning, Astrophysics - Cosmology and Nongalactic Astrophysics, Astrophysics - Instrumentation and Methods for Astrophysics, Physics - Computational Physics, Statistics - Machine Learning},
}

@Article{Cranmer2019,
  author       = {Cranmer, Miles D. and Xu, Rui and Battaglia, Peter and Ho, Shirley},
  date         = {2019-11},
  journaltitle = {arXiv:1909.05862 [astro-ph, physics:physics, stat]},
  title        = {Learning {Symbolic} {Physics} with {Graph} {Networks}},
  note         = {arXiv: 1909.05862},
  url          = {http://arxiv.org/abs/1909.05862},
  urldate      = {2021-08-26},
  abstract     = {We introduce an approach for imposing physically motivated inductive biases on graph networks to learn interpretable representations and improved zero-shot generalization. Our experiments show that our graph network models, which implement this inductive bias, can learn message representations equivalent to the true force vector when trained on n-body gravitational and spring-like simulations. We use symbolic regression to fit explicit algebraic equations to our trained model's message function and recover the symbolic form of Newton's law of gravitation without prior knowledge. We also show that our model generalizes better at inference time to systems with more bodies than had been experienced during training. Our approach is extensible, in principle, to any unknown interaction law learned by a graph network, and offers a valuable technique for interpreting and inferring explicit causal theories about the world from implicit knowledge captured by deep learning.},
  annotation   = {Comment: 6 pages; references added + improvements to writing and clarity; accepted for an oral presentation at Machine Learning and the Physical Sciences Workshop @ NeurIPS 2019},
  file         = {arXiv Fulltext PDF:https\://arxiv.org/pdf/1909.05862.pdf:application/pdf},
  keywords     = {Computer Science - Machine Learning, Astrophysics - Instrumentation and Methods for Astrophysics, Physics - Computational Physics, Statistics - Machine Learning},
}

@Article{Costa2021,
  author       = {Costa, Allan and Dangovski, Rumen and Dugan, Owen and Kim, Samuel and Goyal, Pawan and Soljačić, Marin and Jacobson, Joseph},
  date         = {2021-07},
  journaltitle = {arXiv:2007.10784 [cs, stat]},
  title        = {Fast {Neural} {Models} for {Symbolic} {Regression} at {Scale}},
  note         = {arXiv: 2007.10784},
  url          = {http://arxiv.org/abs/2007.10784},
  urldate      = {2021-08-26},
  abstract     = {Deep learning owes much of its success to the astonishing expressiveness of neural networks. However, this comes at the cost of complex, black-boxed models that extrapolate poorly beyond the domain of the training dataset, conflicting with goals of finding analytic expressions to describe science, engineering and real world data. Under the hypothesis that the hierarchical modularity of such laws can be captured by training a neural network, we introduce OccamNet, a neural network model that finds interpretable, compact, and sparse solutions for fitting data, {\textbackslash}`\{a\} la Occam's razor. Our model defines a probability distribution over a non-differentiable function space. We introduce a two-step optimization method that samples functions and updates the weights with backpropagation based on cross-entropy matching in an evolutionary strategy: we train by biasing the probability mass toward better fitting solutions. OccamNet is able to fit a variety of symbolic laws including simple analytic functions, recursive programs, implicit functions, simple image classification, and can outperform noticeably state-of-the-art symbolic regression methods on real world regression datasets. Our method requires minimal memory footprint, does not require AI accelerators for efficient training, fits complicated functions in minutes of training on a single CPU, and demonstrates significant performance gains when scaled on a GPU. Our implementation, demonstrations and instructions for reproducing the experiments are available at https://github.com/druidowm/OccamNet\_Public.},
  file         = {arXiv Fulltext PDF:https\://arxiv.org/pdf/2007.10784.pdf:application/pdf},
  keywords     = {Computer Science - Machine Learning, Computer Science - Neural and Evolutionary Computing, Statistics - Machine Learning},
}

@Comment{jabref-meta: databaseType:biblatex;}