Numerical analysis of non-local calculus on finite weighted graphs, with application to reduced-order modelling of dynamical systems

TL;DR

This paper develops a non-local calculus on finite weighted graphs for reduced-order modeling of dynamical systems, ensuring derivative consistency and high accuracy, demonstrated through applications to PDE-based models.

Contribution

It introduces a novel graph-based non-local calculus with algorithmically computed edge weights for consistent derivatives, enabling accurate reduced-order models without symmetry assumptions.

Findings

Achieves any desired derivative order accuracy in multiple dimensions.

Ensures consistency of non-local derivatives with local derivatives.

Demonstrates effective reduced-order modeling of complex PDEs.

Abstract

We present an approach to reduced-order modelling that builds off recent graph-theoretic work for representation, exploration, and analysis of 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 polynomial expansions and Taylor series. In the general framework for non-local calculus on graphs, the graph edge weights are intricately linked to the embedding of the graph, and consequently to the definition of the derivatives. In a previous communication (Duschenes and Garikipati, arXiv:2105.01740), we have shown that radially symmetric, continuous edge weights derived from, for example Gaussian functions, yield inconsistent results in the resulting non-local derivatives when compared against the corresponding local, differential derivative definitions. Taking inspiration from finite difference methods, we algorithmically compute edge weights, considering the embedding of the local neighborhood of each graph vertex. Given this procedure, we ensure the consistency of the non-local derivatives in this setting, a crucial requirement for numerical applications. We show that we can achieve any desired orders of accuracy of derivatives, in a chosen number of dimensions without symmetry assumptions in the underlying data. Finally, we present two example applications of extracting reduced-order models using this non-local calculus, in the form of ordinary differential equations from parabolic partial differential equations of progressively greater complexity.