@article{29240,
  abstract     = {{The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,
we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.}},
  author       = {{Ober-Blöbaum, Sina and Offen, Christian}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  keywords     = {{Lagrangian learning, variational backward error analysis, modified Lagrangian, variational integrators, physics informed learning}},
  pages        = {{114780}},
  publisher    = {{Elsevier}},
  title        = {{{Variational Learning of Euler–Lagrange Dynamics from Data}}},
  doi          = {{10.1016/j.cam.2022.114780}},
  volume       = {{421}},
  year         = {{2023}},
}

@article{34633,
  author       = {{Hesse, Kerstin and Le Gia, Quoc Thong}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  keywords     = {{Applied Mathematics, Computational Mathematics}},
  publisher    = {{Elsevier BV}},
  title        = {{{L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data}}},
  doi          = {{10.1016/j.cam.2022.114118}},
  volume       = {{408}},
  year         = {{2022}},
}

@article{34629,
  author       = {{Hesse, Kerstin and Sloan, Ian H. and Womersley, Robert S.}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  keywords     = {{Applied Mathematics, Computational Mathematics}},
  publisher    = {{Elsevier BV}},
  title        = {{{Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data}}},
  doi          = {{10.1016/j.cam.2020.113061}},
  volume       = {{382}},
  year         = {{2021}},
}

@article{16624,
  author       = {{Klus, Stefan and Sahai, Tuhin and Liu, Cong and Dellnitz, Michael}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  pages        = {{3053--3062}},
  title        = {{{An efficient algorithm for the parallel solution of high-dimensional differential equations}}},
  doi          = {{10.1016/j.cam.2010.12.026}},
  year         = {{2011}},
}

@article{17023,
  author       = {{Dellnitz, Michael and Schütze, Oliver and Zheng, Qinghua}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  pages        = {{325--333}},
  title        = {{{Locating all the zeros of an analytic function in one complex variable}}},
  doi          = {{10.1016/s0377-0427(01)00371-5}},
  year         = {{2002}},
}

@article{40197,
  author       = {{Rösler, Margit and Voit, Michael}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  keywords     = {{Applied Mathematics, Computational Mathematics}},
  number       = {{1-2}},
  pages        = {{337--351}},
  publisher    = {{Elsevier BV}},
  title        = {{{Biorthogonal polynomials associated with reflection groups and a formula of Macdonald}}},
  doi          = {{10.1016/s0377-0427(98)00168-x}},
  volume       = {{99}},
  year         = {{1998}},
}

@article{40207,
  author       = {{Rösler, Margit}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  keywords     = {{Applied Mathematics, Computational Mathematics}},
  number       = {{1-3}},
  pages        = {{357--368}},
  publisher    = {{Elsevier BV}},
  title        = {{{Trigonometric convolution structures on Z derived from Jacobi polynomials}}},
  doi          = {{10.1016/0377-0427(95)00122-0}},
  volume       = {{65}},
  year         = {{1995}},
}

@article{16541,
  author       = {{Dellnitz, Michael and Melbourne, Ian}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  pages        = {{249--259}},
  title        = {{{Generic movement of eigenvalues for equivariant self-adjoint matrices}}},
  doi          = {{10.1016/0377-0427(94)90032-9}},
  year         = {{1994}},
}

@article{16682,
  author       = {{Dellnitz, Michael and Werner, Bodo}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  pages        = {{97--123}},
  title        = {{{Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points}}},
  doi          = {{10.1016/0377-0427(89)90150-7}},
  year         = {{1989}},
}