 Award ID(s):
 1837131
 Publication Date:
 NSFPAR ID:
 10199916
 Journal Name:
 Frontiers in applied mathematics and statistics
 Volume:
 6
 ISSN:
 22974687
 Sponsoring Org:
 National Science Foundation
More Like this

Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and blocktridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistorcapacitorinductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to socalled nitedierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM statespace representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from themore »

Abstract In this article, the recently discovered phenomenon of delayed Hopf bifurcations (DHB) in reaction–diffusion partial differential equations (PDEs) is analysed in the cubic Complex Ginzburg–Landau equation, as an equation in its own right, with a slowly varying parameter. We begin by using the classical asymptotic methods of stationary phase and steepest descents on the linearized PDE to show that solutions, which have approached the attracting quasisteady state (QSS) before the Hopf bifurcation remain near that state for long times after the instantaneous Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linearized PDE has a saddle point, and the Stokes and antiStokes lines are central to the asymptotics. The nonlinear terms are treated by applying an iterative method to the mild form of the PDE given by perturbations about the linear particular solution. This tracks the closeness of solutions near the attracting and repelling QSS in the full, nonlinear PDE. Next, we show that beyond a key Stokes line through the saddle there is a curve in the spacetime plane along which the particular solution of the linear PDE ceases to be exponentially small, causing the solution of the nonlinearmore »

Numerically solving partial differential equations (PDEs) remains a compelling application of supercomputing resources. The next generation of computing resources  exhibiting increased parallelism and deep memory hierarchies  provide an opportunity to rethink how to solve PDEs, especially time dependent PDEs. Here, we consider time as an additional dimension and simultaneously solve for the unknown in large blocks of time (i.e. in 4D spacetime), instead of the standard approach of sequential timestepping. We discretize the 4D spacetime domain using a meshfree kD tree construction that enables good parallel performance as well as onthefly construction of adaptive 4D meshes. To best use the 4D spacetime mesh adaptivity, we invoke concepts from PDE analysis to establish rigorous a posteriori error estimates for a general class of PDEs. We solve canonical linear as well as nonlinear PDEs (heat diffusion, advectiondiffusion, and AllenCahn) in spacetime, and illustrate the following advantages: (a) sustained scaling behavior across a larger processor count compared to sequential timestepping approaches, (b) the ability to capture "localized" behavior in space and time using the adaptive spacetime mesh, and (c) removal of any timestepping constraints like the CourantFriedrichsLewy (CFL) condition, as well as the ability to utilize spatially varying timesteps. We believemore »

We investigate methods for learning partial differential equation (PDE) models from spatiotemporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyse the performance in using previous methods to denoise data for the task of discovering the governing system of PDEs. We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e. the advection–diffusion, classical Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) and nonlinear Fisher–KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and both local and global polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model.

Abstract Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a highly multivariable system with incomplete governing equations. This paper introduces a mechanistic datadriven approach that embeds the principle of dimensional invariance into a twolevel machine learning scheme to automatically discover dominant dimensionless numbers and governing laws (including scaling laws and differential equations) from scarce measurement data. The proposed methodology, called dimensionless learning, is a physicsbased dimension reduction technique. It can reduce highdimensional parameter spaces to descriptions involving only a few physically interpretable dimensionless parameters, greatly simplifying complex process design and system optimization. We demonstrate the algorithm by solving several challenging engineering problems with noisy experimental measurements (not synthetic data) collected from the literature. Examples include turbulent RayleighBénard convection, vapor depression dynamics in laser melting of metals, and porosity formation in 3D printing. Lastly, we show that the proposed approach can identify dimensionally homogeneous differential equations with dimensionless number(s) by leveraging sparsitypromoting techniques.