Publications
10 results found
Gabel A, Klein V, Valperga R, et al., 2023, Learning lie group symmetry transformations with neural networks, 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML) at the 40th International Conference on Machine Learning, Publisher: ML Research Press, Pages: 50-59
The problem of detecting and quantifying the presence of symmetries in datasets is useful for model selection, generative modeling, and data analysis, amongst others. While existing methods for hard-coding transformations in neural networks require prior knowledge of the symmetries of the task at hand, this work focuses on discovering and characterizing unknown symmetries present in the dataset, namely, Lie group symmetry transformations beyond the traditional ones usually considered in the field (rotation, scaling, and translation). Specifically, we consider a scenario in which a dataset has been transformed by a one-parameter subgroup of transformations with different parameter values for each data point. Our goal is to characterize the transformation group and the distribution of the parameter values. The results showcase the effectiveness of the approach in both these settings.
Webster K, 2022, Low-Rank kernel approximation of Lyapunov functions using neural networks, Journal of Computational Dynamics, ISSN: 2158-2505
Valperga R, Webster K, Klein V, et al., 2022, Learning reversible symplectic dynamics, 4th Annual Learning for Dynamics and Control Conference, Publisher: PLMR
Time-reversal symmetry arises naturally as a structural property in manydynamical systems of interest. While the importance of hard-wiring symmetry isincreasingly recognized in machine learning, to date this has eludedtime-reversibility. In this paper we propose a new neural network architecturefor learning time-reversible dynamical systems from data. We focus inparticular on an adaptation to symplectic systems, because of their importancein physics-informed learning.
Giesl P, Hamzi B, Rasmussen M, et al., 2020, Approximation of Lyapunov functions from noisy data, Journal of Computational Dynamics, Vol: 7, Pages: 57-81, ISSN: 2158-2491
Methods have previously been developed for the approximation of Lyapunovfunctions using radial basis functions. However these methods assume that theevolution equations are known. We consider the problem of approximating a givenLyapunov function using radial basis functions where the evolution equationsare not known, but we instead have sampled data which is contaminated withnoise. We propose an algorithm in which we first approximate the underlyingvector field, and use this approximation to then approximate the Lyapunovfunction. Our approach combines elements of machine learning/statisticallearning theory with the existing theory of Lyapunov function approximation.Error estimates are provided for our algorithm.
Rasmussen M, Rieger J, Webster KN, 2018, A reinterpretation of set differential equations as differential equations in a Banach space, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, Vol: 148, Pages: 429-446, ISSN: 1473-7124
Set differential equations are usually formulated in terms of theHukuhara differential. As a consequence, the theory of set differentialequations is perceived as an independent subject, in which all resultsare proved within the framework of the Hukuhara calculus.We propose to reformulate set differential equations as ordinarydifferential equations in a Banach space by identifying the convex andcompact subsets ofRdwith their support functions. Using this rep-resentation, standard existence and uniqueness theorems for ordinarydifferential equations can be applied to set differential equations. Weprovide a geometric interpretation of the main result, and we demon-strate that our approach overcomes the heavy restrictions the use ofthe Hukuhara differential implies for the nature of a solution.
knobloch J, Lamb JSW, webster KN, 2017, Shift dynamics near non-elementary T-points with real eigenvalues, Journal of Difference Equations and Applications, Vol: 24, Pages: 609-654, ISSN: 1563-5120
We consider non-elementary T-points in reversible systems in R2n+1. We assume that the leading eigenvalues are real. We prove the existence of shift dynamics in the unfolding of this T-point. Furthermore, we study local bifurcations of symmetric periodic orbits occurring in the process of dissolution of the chaotic dynamics.
Rasmussen M, Rieger J, Webster KN, 2016, Approximation of reachable sets using optimal control and support vector machines, Journal of Computational and Applied Mathematics, Vol: 311, Pages: 68-83, ISSN: 0377-0427
We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples.
Knobloch J, Lamb JSW, Webster KN, 2014, Using Lin's method to solve Bykov's problems, JOURNAL OF DIFFERENTIAL EQUATIONS, Vol: 257, Pages: 2984-3047, ISSN: 0022-0396
- Author Web Link
- Cite
- Citations: 16
Lamb JSW, Teixeira MA, Webster KN, 2005, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R<SUP>3</SUP>, JOURNAL OF DIFFERENTIAL EQUATIONS, Vol: 219, Pages: 78-115, ISSN: 0022-0396
- Author Web Link
- Cite
- Citations: 56
Webster KN, Elgin JN, 2003, Asymptotic analysis of the Michelson system, NONLINEARITY, Vol: 16, Pages: 2149-2162, ISSN: 0951-7715
- Author Web Link
- Cite
- Citations: 9
This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters.