Abstract : Dynamical systems abound in engineering and science, and their accurate long-time simulation and outer-loop applications such as control, design and uncertainty quantification remains a computational challenge. From first principles modeling of physical systems, it is clear that many of these dynamical systems have a natural geometric structure (e.g., Hamiltonian, Lagrangian, metriplectic) or symmetry (translational, rotational). Exploiting and enforcing this structure in physics-based learning methods remains imperative for capturing the underlying physics accurately. This minisymposium highlights recent developments in physics-preserving learning for dynamical systems, such as: Lagrangian/Hamiltonian neural networks, sparse identification of nonlinear dynamics (SINDy), operator inference, preservation of conservation laws, the incorporation of interconnection and modular structure, structure-preserving system identification and other machine learning approaches.
[01395] Symplectic Model Reduction on Quadratic Manifolds
Format : Talk at Waseda University
Author(s) :
Boris Kramer (University of California San Diego)
Abstract : When Hamiltonian models are used for long-term simulation, constraints on CPU hours need to be met. Structure-preserving model reduction for Hamiltonian systems addresses this computational issue by projecting Hamilton’s equations of the full-order model onto linear symplectic subspaces, which can yield inaccurate results for problems with a slowly decaying Kolmogorov n-width. We present symplectic structure-preserving reduced-order modeling of Hamiltonian systems using quadratic manifolds. We demonstrate the proposed method on wave equations in 1-D and 2-D.
[01373] Identification of variational principles, symmetries, and conservation laws from data
Format : Talk at Waseda University
Author(s) :
Yana Lishkova (University of Oxford)
Paul Scherer (University of Cambridge)
Steffen Ridderbusch (University of Oxford)
Mateja Jamnik (University of Cambridge)
Pietro Lio (University of Cambridge)
Sina Ober-Blöbaum (Paderborn University)
Christian Offen (Paderborn University)
Abstract : The identification of equations of motions of dynamical systems from data as well as dynamical properties such as symmetries and conservation laws is an important task in the context of system identification. I will show a framework based on Lie group theory to learn a variational principle governing a dynamical system which can identify variational symmetries and conservation laws along the way. Identified symmetries prove helpful when the learned equations of motions are integrated numerically.
[01597] Data-driven structure-preserving model reduction for stochastic Hamiltonian systems
Format : Talk at Waseda University
Author(s) :
Tomasz Tyranowski (Max Planck Institute for Plasma Physics)
Abstract : In this work we demonstrate that SVD-based model reduction techniques known for ordinary differential equations, such as the proper orthogonal decomposition, can be extended to stochastic differential equations in order to reduce the computational cost arising from both the high dimension of the considered stochastic system and the large number of independent Monte Carlo runs. We also extend the proper symplectic decomposition method to stochastic Hamiltonian systems, both with and without external forcing, and argue that preserving the underlying symplectic or variational structures results in more accurate and stable solutions that conserve energy better than when the non-geometric approach is used. We validate our proposed techniques with numerical experiments for a semi-discretization of the stochastic nonlinear Schrödinger equation and the Kubo oscillator.
[01793] Learning video models with Lagrangian/Hamiltonian neural networks
Format : Talk at Waseda University
Author(s) :
Christine Allen-Blanchette (Princeton University)
Abstract : The dynamics underlying object and camera motion in a video typically evolve on a low-dimensional manifold with unknown structure and dimension. While prior work has used the Hamiltonian formalism to give a physically meaningful interpretation to this manifold, the problem of discovering the manifold structure and dimension remains unaddressed. We introduce a Hamiltonian neural network for video generation where the structure and dimension of the phase-space are implicitly learned from data. To achieve this we introduce a GAN-based video generation pipeline which embeds a learned transformation from a Gaussian distribution to the phase-space manifold, and captures the underlying dynamics of the video in a Hamiltonian neural network motion model.
[02758] Structure-preserving exterior calculus for GNNs: surrogates, physics discovery, and causality
Format : Online Talk on Zoom
Author(s) :
Nathaniel Trask (Sandia National Laboratories)
Abstract : We present a graph exterior calculus which may be used to design graph neural network which naturally preserve mathematical and physical structure without resorting to physics-informed regularizers. The calculus provides a framework for proving numerical stability, conservation, preservation of geometric symmetries, thermodynamic consistency, gauge conditions, and other properties more typical of traditional PDE discretization. In this setting we discover Whitney forms encoding physically relevant subdomains, their boundaries, and flux conservation laws for multiphysics/multiscale systems. We introduce ongoing applications work using this to discover structure-preserving surrogates which exhibit 100000x speedup for typical problems while guaranteeing mathematical robustness, and introduce recent extensions discovering causal relationships in scientific datasets.
