Nonlinear intrinsic variables and state reconstruction in multiscale simulations

Dsilva, Carmeline J.; Talmon, Ronen; Rabin, Neta; Coifman, Ronald R.; Kevrekidis, Ioannis G.

doi:10.1063/1.4828457

Cited by 32 publications

(18 citation statements)

References 43 publications

(40 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The idea has been, in principle, proposed in the past [31] in the context of navigating effective free energy surfaces, and its incorporation in our Diffusion Map-based approach here is worth pursuing.In this paper the initially collected data came from a detailed set of kinetic ordinary differential equations, and the effective model for the reduced variables was also assumed to be a set of ODEs in these variables. The approach, however, can also, in principle, be used when the data do not need to come from a large set of ODEs, but, for example, from multiscale Stochastic Simulation Algorithm descriptions of chemical kinetic schemes (for such a recent application, see [32]). Moreover, as demonstrated also in the presented combustion example, the method should be able to cope with manifolds, whose dimensions possibly vary across distinct regions of the phase-space (see [23]); how to consistently express and solve reduced systems across manifolds with disparate dimensions remains out of reach of the present method, requiring further investigation.…”

Section: Discussionmentioning

confidence: 99%

Reduced Models in Chemical Kinetics via Nonlinear Data-Mining

et al. 2014

Self Cite

View full text Add to dashboard Cite

The adoption of detailed mechanisms for chemical kinetics often poses two 1 types of severe challenges: First, the number of degrees of freedom is large; and second, 2 the dynamics is characterized by widely disparate time scales. As a result, reactive flow 3 solvers with detailed chemistry often become intractable even for large clusters of CPUs, 4 especially when dealing with direct numerical simulation (DNS) of turbulent combustion 5 problems. This has motivated the development of several techniques for reducing the 6 complexity of such kinetics models, where eventually only a few variables are considered 7 in the development of the simplified model. Unfortunately, no generally applicable a priori 8 recipe for selecting suitable parameterizations of the reduced model is available, and the 9 choice of slow variables often relies upon intuition and experience. We present an automated 10 approach to this task, consisting of three main steps. First, the low dimensional manifold 11 of slow motions is (approximately) sampled by brief simulations of the detailed model, 12 starting from a rich enough ensemble of admissible initial conditions. Second, a global 13 parametrization of the manifold is obtained through the Diffusion Map (DMAP) approach, 14 which has recently emerged as a powerful tool in data analysis/machine learning. Finally, a 15 simplified model is constructed and solved on the fly in terms of the above reduced (slow) 16 variables. Clearly, closing this latter model requires nontrivial interpolation calculations, 17 enabling restriction (mapping from the full ambient space to the reduced one) and lifting 18 (mapping from the reduced space to the ambient one). This is a key step in our approach, 19 and a variety of interpolation schemes are reported and compared. The scope of the proposed 20 procedure is presented and discussed by means of an illustrative combustion example. 21 arXiv:1307.6849v1 [math.DS] 22 29the development of a plethora of approaches aiming at reducing the computational complexity of such 30 detailed combustion models, ideally by recasting them in terms of only a few new reduced variables. 31(see e.g.[1] and references therein). The implementation of many of these techniques typically involves 32 three successive steps. First, a large set of stiff ordinary differential equations (ODEs) is considered 33 for modeling the temporal evolution of a spatially homogenous mixture of chemical species under 34 specified stoichiometric and thermodynamic conditions (usually fixed total enthalpy and pressure for 35 combustion in the low Mach regime). It is well known that, due to the presence of fast and slow 36 dynamics, the above systems are characterized by low dimensional manifolds in the concentration 37 space (or phase-space), where a typical solution trajectory is initially rapidly attracted towards the 38 manifold, while afterwards it proceeds to the thermodynamic equilibrium point always remaining in 39 close proximity to the manifold. Clearly, the presence of a manifold for...

show abstract

Section: Discussionmentioning

confidence: 99%

Reduced Models in Chemical Kinetics via Nonlinear Data-Mining

et al. 2014

Self Cite

View full text Add to dashboard Cite

show abstract

“…We have shown that (invertible) functions of our measurements give rise to homeomorphic (and possibly isometric) embeddings, thereby conveying the same prototypical behavior [in the spirit of Whitney and the even stronger Nash embedding theorems and more specifically, Takens embedding for dynamical systems (36)(37)(38)(39)]. This points toward the feasibility of "gaugeinvariant" data mining (22,24,(40)(41)(42) through algorithms that do not depend on the measurement modality.…”

Section: Summary and Discussionmentioning

confidence: 69%

Reconstruction of normal forms by learning informed observation geometries from data

Yair

Talmon

Coifman

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

View full text Add to dashboard Cite

The discovery of physical laws consistent with empirical observations is at the heart of (applied) science and engineering. These laws typically take the form of nonlinear differential equations depending on parameters; dynamical systems theory provides, through the appropriate normal forms, an "intrinsic" prototypical characterization of the types of dynamical regimes accessible to a given model. Using an implementation of data-informed geometry learning, we directly reconstruct the relevant "normal forms": a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics. Interestingly, the state variables and the parameters of these realizations are inferred from the empirical observations; without prior knowledge or understanding, they parametrize the dynamics intrinsically without explicit reference to fundamental physical quantities.

show abstract

“…Furthermore, images taken from different viewing directions can be analyzed, as the vector diffusion maps parametrization will organize the images according to the viewing angle . Another direction for future work is related to the joint analysis of data sets provided by different imaging approaches, such as merging live imaging data of tissue morphogenesis with snapshots of cell signaling and gene expression from fixed embryos (Rübel et al, 2010; Krzic et al, 2012;Dsilva et al, 2013;Ichikawa et al, 2014). It would also be interesting to explore the connections between our proposed approach and recently developed methods for the ordering and classification of face images (Kemelmacher-Shlizerman et al, 2011.…”

Section: Discussionmentioning

confidence: 99%

Temporal ordering and registration of images in studies of developmental dynamics

Dsilva

Lim

et al. 2015

Development

Self Cite

View full text Add to dashboard Cite

Progress of development is commonly reconstructed from imaging snapshots of chemical or mechanical processes in fixed tissues. As a first step in these reconstructions, snapshots must be spatially registered and ordered in time. Currently, image registration and ordering are often done manually, requiring a significant amount of expertise with a specific system. However, as the sizes of imaging data sets grow, these tasks become increasingly difficult, especially when the images are noisy and the developmental changes being examined are subtle. To address these challenges, we present an automated approach to simultaneously register and temporally order imaging data sets. The approach is based on vector diffusion maps, a manifold learning technique that does not require a priori knowledge of image features or a parametric model of the developmental dynamics. We illustrate this approach by registering and ordering data from imaging studies of pattern formation and morphogenesis in three model systems. We also provide software to aid in the application of our methodology to other experimental data sets.

show abstract

Nonlinear intrinsic variables and state reconstruction in multiscale simulations

Cited by 32 publications

References 43 publications

Reduced Models in Chemical Kinetics via Nonlinear Data-Mining

Reduced Models in Chemical Kinetics via Nonlinear Data-Mining

Reconstruction of normal forms by learning informed observation geometries from data

Temporal ordering and registration of images in studies of developmental dynamics

Contact Info

Product

Resources

About