16,478 research outputs found
Fast Neural Network Predictions from Constrained Aerodynamics Datasets
Incorporating computational fluid dynamics in the design process of jets,
spacecraft, or gas turbine engines is often challenged by the required
computational resources and simulation time, which depend on the chosen
physics-based computational models and grid resolutions. An ongoing problem in
the field is how to simulate these systems faster but with sufficient accuracy.
While many approaches involve simplified models of the underlying physics,
others are model-free and make predictions based only on existing simulation
data. We present a novel model-free approach in which we reformulate the
simulation problem to effectively increase the size of constrained pre-computed
datasets and introduce a novel neural network architecture (called a cluster
network) with an inductive bias well-suited to highly nonlinear computational
fluid dynamics solutions. Compared to the state-of-the-art in model-based
approximations, we show that our approach is nearly as accurate, an order of
magnitude faster, and easier to apply. Furthermore, we show that our method
outperforms other model-free approaches
Uncertainty quantification for kinetic models in socio-economic and life sciences
Kinetic equations play a major rule in modeling large systems of interacting
particles. Recently the legacy of classical kinetic theory found novel
applications in socio-economic and life sciences, where processes characterized
by large groups of agents exhibit spontaneous emergence of social structures.
Well-known examples are the formation of clusters in opinion dynamics, the
appearance of inequalities in wealth distributions, flocking and milling
behaviors in swarming models, synchronization phenomena in biological systems
and lane formation in pedestrian traffic. The construction of kinetic models
describing the above processes, however, has to face the difficulty of the lack
of fundamental principles since physical forces are replaced by empirical
social forces. These empirical forces are typically constructed with the aim to
reproduce qualitatively the observed system behaviors, like the emergence of
social structures, and are at best known in terms of statistical information of
the modeling parameters. For this reason the presence of random inputs
characterizing the parameters uncertainty should be considered as an essential
feature in the modeling process. In this survey we introduce several examples
of such kinetic models, that are mathematically described by nonlinear Vlasov
and Fokker--Planck equations, and present different numerical approaches for
uncertainty quantification which preserve the main features of the kinetic
solution.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic
Equations
Boosting Bayesian Parameter Inference of Nonlinear Stochastic Differential Equation Models by Hamiltonian Scale Separation
Parameter inference is a fundamental problem in data-driven modeling. Given
observed data that is believed to be a realization of some parameterized model,
the aim is to find parameter values that are able to explain the observed data.
In many situations, the dominant sources of uncertainty must be included into
the model, for making reliable predictions. This naturally leads to stochastic
models. Stochastic models render parameter inference much harder, as the aim
then is to find a distribution of likely parameter values. In Bayesian
statistics, which is a consistent framework for data-driven learning, this
so-called posterior distribution can be used to make probabilistic predictions.
We propose a novel, exact and very efficient approach for generating posterior
parameter distributions, for stochastic differential equation models calibrated
to measured time-series. The algorithm is inspired by re-interpreting the
posterior distribution as a statistical mechanics partition function of an
object akin to a polymer, where the measurements are mapped on heavier beads
compared to those of the simulated data. To arrive at distribution samples, we
employ a Hamiltonian Monte Carlo approach combined with a multiple time-scale
integration. A separation of time scales naturally arises if either the number
of measurement points or the number of simulation points becomes large.
Furthermore, at least for 1D problems, we can decouple the harmonic modes
between measurement points and solve the fastest part of their dynamics
analytically. Our approach is applicable to a wide range of inference problems
and is highly parallelizable.Comment: 15 pages, 8 figure
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