In many mechanistic medical, biological, physical, and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs), especially for diffusion, fluid flow and mechanical relaxation, can make simulations impractically slow. Biological models of tissues and organs often require the simultaneous calculation of the spatial variation of concentration of dozens of diffusing chemical species. One clinical example where rapid calculation of a diffusing field is of use is the estimation of oxygen gradients in the retina, based on imaging of the retinal vasculature, to guide surgical interventions in diabetic retinopathy. Furthermore, the ability to predict blood perfusion and oxygenation may one day guide clinical interventions in diverse settings, i.e., from stent placement in treating heart disease to BOLD fMRI interpretation in evaluating cognitive function (Xie et al., 2019 ; Lee et al., 2020 ). Since the quasisteadystate solutions required for fastdiffusing chemical species like oxygen are particularly computationally costly, we consider the use of a neural network to provide an approximate solution to the steadystate diffusion equation. Machine learning surrogates, neural networks trained to provide approximate solutions to such complicated numerical problems, can often provide speedups of several orders of magnitude compared to direct calculation. Surrogates of PDEs couldmore »
This content will become publicly available on December 3, 2022
Spectral stability of patternforming fronts in the complex Ginzburg–Landau equation with a quenching mechanism
Abstract We consider patternforming fronts in the complex Ginzburg–Landau equation with a traveling spatial heterogeneity which destabilises, or quenches, the trivial ground state while progressing through the domain. We consider the regime where the heterogeneity propagates with speed c just below the linear invasion speed of the patternforming front in the associated homogeneous system. In this situation, the front locks to the interface of the heterogeneity leaving a long intermediate state lying near the unstable ground state, possibly allowing for growth of perturbations. This manifests itself in the spectrum of the linearisation about the front through the accumulation of eigenvalues onto the absolute spectrum associated with the unstable ground state. As the quench speed c increases towards the linear invasion speed, the absolute spectrum stabilises with the same rate at which eigenvalues accumulate onto it allowing us to rigorously establish spectral stability of the front in L 2 ( R ) . The presence of unstable absolute spectrum poses a technical challenge as spatial eigenvalues along the intermediate state no longer admit a hyperbolic splitting and standard tools such as exponential dichotomies are unavailable. Instead, we projectivise the linear flow, and use Riemann surface unfolding in combination with a superposition more »
 Award ID(s):
 2006887
 Publication Date:
 NSFPAR ID:
 10335777
 Journal Name:
 Nonlinearity
 Volume:
 35
 Issue:
 1
 Page Range or eLocationID:
 170 to 244
 ISSN:
 09517715
 Sponsoring Org:
 National Science Foundation
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