Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Probabilistic learning on manifolds constrained by nonlinear partial
  differential equations for small datasets   A novel extension of the Probabilistic Learning on Manifolds (PLoM) is
presented. It makes it possible to synthesize solutions to a wide range of
nonlinear stochastic boundary value problems described by partial differential
equations (PDEs) for which a stochastic computational model (SCM) is available
and depends on a vector-valued random control parameter. The cost of a single
numerical evaluation of this SCM is assumed to be such that only a limited
number of points can be computed for constructing the training dataset (small
data). Each point of the training dataset is made up realizations from a
vector-valued stochastic process (the stochastic solution) and the associated
random control parameter on which it depends. The presented PLoM constrained by
PDE allows for generating a large number of learned realizations of the
stochastic process and its corresponding random control parameter. These
learned realizations are generated so as to minimize the vector-valued random
residual of the PDE in the mean-square sense. Appropriate novel methods are
developed to solve this challenging problem. Three applications are presented.
The first one is a simple uncertain nonlinear dynamical system with a
nonstationary stochastic excitation. The second one concerns the 2D nonlinear
unsteady Navier-Stokes equations for incompressible flows in which the Reynolds
number is the random control parameter. The last one deals with the nonlinear
dynamics of a 3D elastic structure with uncertainties. The results obtained
make it possible to validate the PLoM constrained by stochastic PDE but also
provide further validation of the PLoM without constraint.

12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol