Reduced order models from computed states of physical systems using non-local calculus on finite weighted graphs
arXiv:2105.01740
Abstract
Partial differential equation-based numerical solution frameworks for initial and boundary value problems have attained a high degree of complexity. Applied to a wide range of physics with the ultimate goal of enabling engineering solutions, these approaches encompass a spectrum of spatiotemporal discretization techniques that leverage solver technology and high performance computing. While high-fidelity solutions can be achieved using these approaches, they come at a high computational expense and complexity. Systems with billions of solution unknowns are now routine. The expense and complexity do not lend themselves to typical engineering design and decision-making, which must instead rely on reduced-order models. Here we present an approach to reduced-order modelling that builds off of recent graph theoretic work for representation, exploration, and analysis on computed states of physical systems (Banerjee et al., Comp. Meth. App. Mech. Eng., 351, 501-530, 2019). We extend a non-local calculus on finite weighted graphs to build such models by exploiting first order dynamics, polynomial expansions, and Taylor series. Some aspects of the non-local calculus related to consistency of the models are explored. Details on the numerical implementations and the software library that has been developed for non-local calculus on graphs are described. Finally, we present examples of applications to various quantities of interest in mechano-chemical systems.
arXiv comment72 pages, 19 figures
References in corpus (10)
- Scikit-learn: Machine Learning in Python
- Three-dimensional iso-geometric solutions to general boundary value problems of Toupin's gradient elasticity theory at finite strains
- Machine learning materials physics: Multi-resolution neural networks learn the free energy and nonlinear elastic response of evolving microstructures
- Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data
- Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition
- A graph theoretic framework for representation, exploration and analysis on computed states of physical systems
- A variational treatment of material configurations with application to interface motion and microstructural evolution
- On Some Variants of the Gauss Circle Problem
- Unconditionally stable, second-order schemes for gradient-regularized, non-convex, finite-strain elasticity modeling martensitic phase transformations
- The Graph Theoretic Approach for Nodal Cross Section Parameterization