A physics-based Reduced Order Model capturing the topology of dynamical manifolds
presented at Lawrence Livermore National Laboratory, DDPS Seminar
7 November 2023
and presented at Courant Institute
7 December 2023
PDF with slides: http://www.qitech.biz/tech_papers/LAN...
Video of presentation at LLNL: • DDPS | A physics-based Reduced Order Model...
Inspired by the Scattering Transformation constructed by Mallat [1], we develop a transformational approach to dynamics [2-3]. It is based on a canonical (or Heisenberg approach), in contrast to the more common Lagrangian (Schrodinger or Feynman path integral) approach. It is a logarithmic generating functional of the Hamiltonian which operates on the state or statistical distribution function of the system. This logarithmic generating functional is a deep convolutional network with specified structure and weights. This transformation projects the dynamics onto the Renormalization Basis (or scattering matrix, S-matrix basis) with basis vectors of a Hilbert space that are all potential solutions to the Renormalization Group Equations (that is, how the physics changes as a function of scale). The dynamics is constrained to a low dimensional complex linear subspace of this Hilbert space that can be identified with a Principal Component Analysis (that is the solutions of the RGEs). Furthermore, the topology can be quantified with a Multi Layer Perceptron (with a decoder/encoder structure), which approximates the analytic function H(z) on the low dimensional complex linear subspace. This approach, which we call the Heisenberg Scattering Transformation, is applied to form a fast surrogate model for 2D pulsed power liner implosions (MagLIF).
[1] Stéphane Mallat, "Group invariant scattering," Communications on Pure and Applied Mathematics 65, 1331 (2012). https://arxiv.org/abs/1101.2286.
[2] Michael E. Glinsky and Kathryn Maupin, "Mallat Scattering Transformation based surrogate for MagnetoHydroDynamics”, Computational Mechanics 72, 291 (2023). https://arxiv.org/abs/2302.10243.
[3] Michael E. Glinsky and Sharon Sievert, "A new economic and financial theory of money", to be submitted to Journal of Economic Affairs (2023). https://arxiv.org/abs/2310.04986.