Physics-Constrained, Data-Driven Discovery of Coarse-Grained Dynamics

Felsberger, Lukas; Koutsourelakis, Phaedon‐Stelios

doi:10.4208/cicp.oa-2018-0174

Cited by 19 publications

(24 citation statements)

References 72 publications

(81 reference statements)

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“…We assume that, given the CG state X X X t , the coordinates of the particles x x x t are conditionally independent. This does not imply that they move independently nor that they cannot exhibit coherent behavior [22]. The consequence of Equation (17) is that for this example no parameters need to be learned for p c f .…”

Section: Cg Model Specificationsmentioning

confidence: 96%

Physics-Aware, Deep Probabilistic Modeling of Multiscale Dynamics in the Small Data Regime

Kaltenbach¹,

Koutsourelakis²

2021

14th WCCM-ECCOMAS Congress

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The data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a probabilistic perspective that simultaneously identifies predictive, lower-dimensional coarse-grained (CG) variables as well as their dynamics. We make use of the expressive ability of deep neural networks in order to represent the right-hand side of the CG evolution law. Furthermore, we demonstrate how domain knowledge that is very often available in the form of physical constraints (e.g. conservation laws) can be incorporated with the novel concept of virtual observables. Such constraints, apart from leading to physically realistic predictions, can significantly reduce the requisite amount of training data which enables reducing the amount of required, computationally expensive multiscale simulations (Small Data regime). The proposed state-space model is trained using probabilistic inference tools and, in contrast to several other techniques, does not require the prescription of a fine-to-coarse (restriction) projection nor time-derivatives of the state variables. The formulation adopted is capable of quantifying the predictive uncertainty as well as of reconstructing the evolution of the full, fine-scale system which allows to select the quantities of interest a posteriori. We demonstrate the efficacy of the proposed framework in a high-dimensional system of moving particles.

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Section: Cg Model Specificationsmentioning

confidence: 96%

Physics-Aware, Deep Probabilistic Modeling of Multiscale Dynamics in the Small Data Regime

Kaltenbach¹,

Koutsourelakis²

2021

14th WCCM-ECCOMAS Congress

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“…Clearly, this interest has not been limited to just quantum systems with recent developments included a relationship between stochastic processes and nonlocal versions of classical models [237], as well as new probabilistic approaches to known nonlocal models [238]. Based on the Stochastic Variational Inference technique, Bayesian learning coarse-grained methodologies with probabilistic state-space have also been receiving attention in the analysis of complex systems dynamics in data-driven environments [239] which could present interest for nonlocal models. Many parameterization methods related to the Mori-Zwanzig formalism have been a natural development of control-theoretical ideas such as Kalman-filtering which allowed the construction of reduced models for both classical and quantum systems, including those based on Schrödinger equations [240].…”

Section: Modelling With Nonlocality In Data-driven Environmentsmentioning

confidence: 99%

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Sytnyk

Melnik

2021

MCA

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Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

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“…In particular, we have used data from a trajectory for z ∈ [0, 3]. After we trained the GAN generator we used it to predict the solution for z ∈ [0, 9]. This is a severe test of the GAN generator's predictive abilities for three reasons.…”

Section: Nonlinear Systemmentioning

confidence: 99%

“…In recent years, there has been considerable interest in the development of methods that utilize data and physical constraints in order to train predictors for dynamical systems and differential equations e.g. see [4,22,6,13,23,9,25,20] and references therein. Our approach is different, it introduces the novel concept of training on purpose with modified (noisy) data in order to incorporate a restoring force in the dynamics learned by the generator.…”

Section: Introductionmentioning

confidence: 99%

Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networks

Stinis

Hagge

Tartakovsky

et al. 2019

Journal of Computational Physics

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Generative Adversarial Networks (GANs) are becoming popular machine learning choices for training generators. At the same time there is a concerted effort in the machine learning community to expand the range of tasks in which learning can be applied as well as to utilize methods from other disciplines to accelerate learning. With this in mind, in the current work we suggest ways to enforce given constraints in the output of a GAN generator both for interpolation and extrapolation (prediction). For the case of dynamical systems, given a time series, we wish to train GAN generators that can be used to predict trajectories starting from a given initial condition. In this setting, the constraints can be in algebraic and/or differential form. Even though we are predominantly interested in the case of extrapolation, we will see that the tasks of interpolation and extrapolation are related. However, they need to be treated differently.For the case of interpolation, the incorporation of constraints is built into the training of the GAN. The incorporation of the constraints respects the primary gametheoretic setup of a GAN so it can be combined with existing algorithms. However, it can exacerbate the problem of instability during training that is well-known for GANs. We suggest adding small noise to the constraints as a simple remedy that has performed well in our numerical experiments.The case of extrapolation (prediction) is more involved. During training, the GAN generator learns to interpolate a noisy version of the data and we enforce the constraints. This approach has connections with model reduction that we can utilize to improve the efficiency and accuracy of the training. Depending on the form of the constraints, we may enforce them also during prediction through a projection step. We provide examples of linear and nonlinear systems of differential equations to illustrate the various constructions.

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Physics-Constrained, Data-Driven Discovery of Coarse-Grained Dynamics

Cited by 19 publications

References 72 publications

Physics-Aware, Deep Probabilistic Modeling of Multiscale Dynamics in the Small Data Regime

Physics-Aware, Deep Probabilistic Modeling of Multiscale Dynamics in the Small Data Regime

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networks

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