35,371 research outputs found
A machine learning framework for data driven acceleration of computations of differential equations
We propose a machine learning framework to accelerate numerical computations
of time-dependent ODEs and PDEs. Our method is based on recasting
(generalizations of) existing numerical methods as artificial neural networks,
with a set of trainable parameters. These parameters are determined in an
offline training process by (approximately) minimizing suitable (possibly
non-convex) loss functions by (stochastic) gradient descent methods. The
proposed algorithm is designed to be always consistent with the underlying
differential equation. Numerical experiments involving both linear and
non-linear ODE and PDE model problems demonstrate a significant gain in
computational efficiency over standard numerical methods
A probabilistic algorithm approximating solutions of a singular PDE of porous media type
The object of this paper is a one-dimensional generalized porous media
equation (PDE) with possibly discontinuous coefficient , which is
well-posed as an evolution problem in . In some recent papers
of Blanchard et alia and Barbu et alia, the solution was represented by the
solution of a non-linear stochastic differential equation in law if the initial
condition is a bounded integrable function. We first extend this result, at
least when is continuous and the initial condition is only integrable
with some supplementary technical assumption. The main purpose of the article
consists in introducing and implementing a stochastic particle algorithm to
approach the solution to (PDE) which also fits in the case when is
possibly irregular, to predict some long-time behavior of the solution and in
comparing with some recent numerical deterministic techniques
Matrix-equation-based strategies for convection-diffusion equations
We are interested in the numerical solution of nonsymmetric linear systems
arising from the discretization of convection-diffusion partial differential
equations with separable coefficients and dominant convection. Preconditioners
based on the matrix equation formulation of the problem are proposed, which
naturally approximate the original discretized problem. For certain types of
convection coefficients, we show that the explicit solution of the matrix
equation can effectively replace the linear system solution. Numerical
experiments with data stemming from two and three dimensional problems are
reported, illustrating the potential of the proposed methodology
Approximating a similarity matrix by a latent class model: A reappraisal of additive fuzzy clustering
Let Q be a given nĂ—n square symmetric matrix of nonnegative elements between 0 and 1, similarities. Fuzzy clustering results in fuzzy assignment of individuals to K clusters. In additive fuzzy clustering, the nĂ—K fuzzy memberships matrix P is found by least-squares approximation of the off-diagonal elements of Q by inner products of rows of P. By contrast, kernelized fuzzy c-means is not least-squares and requires an additional fuzziness parameter. The aim is to popularize additive fuzzy clustering by interpreting it as a latent class model, whereby the elements of Q are modeled as the probability that two individuals share the same class on the basis of the assignment probability matrix P. Two new algorithms are provided, a brute force genetic algorithm (differential evolution) and an iterative row-wise quadratic programming algorithm of which the latter is the more effective. Simulations showed that (1) the method usually has a unique solution, except in special cases, (2) both algorithms reached this solution from random restarts and (3) the number of clusters can be well estimated by AIC. Additive fuzzy clustering is computationally efficient and combines attractive features of both the vector model and the cluster mode
Developing Clean Technology through Approximate Solutions of Mathematical Models
In this paper, the role of mathematical modeling in the development of clean technology has been considered.
One method each for obtaining approximate solutions of mathematical models by ordinary differential equations
and partial differential equations respectively arising from the modeling of systems and physical phenomena has
been considered. The construction of continuous hybrid methods for the numerical approximation of the solutions
of initial value problems of ordinary differential equations as well as homotopy analysis method, an approximate
analytical method, for the solution of nonlinear partial differential equations are discussed
Analysis of Petri Net Models through Stochastic Differential Equations
It is well known, mainly because of the work of Kurtz, that density dependent
Markov chains can be approximated by sets of ordinary differential equations
(ODEs) when their indexing parameter grows very large. This approximation
cannot capture the stochastic nature of the process and, consequently, it can
provide an erroneous view of the behavior of the Markov chain if the indexing
parameter is not sufficiently high. Important phenomena that cannot be revealed
include non-negligible variance and bi-modal population distributions. A
less-known approximation proposed by Kurtz applies stochastic differential
equations (SDEs) and provides information about the stochastic nature of the
process. In this paper we apply and extend this diffusion approximation to
study stochastic Petri nets. We identify a class of nets whose underlying
stochastic process is a density dependent Markov chain whose indexing parameter
is a multiplicative constant which identifies the population level expressed by
the initial marking and we provide means to automatically construct the
associated set of SDEs. Since the diffusion approximation of Kurtz considers
the process only up to the time when it first exits an open interval, we extend
the approximation by a machinery that mimics the behavior of the Markov chain
at the boundary and allows thus to apply the approach to a wider set of
problems. The resulting process is of the jump-diffusion type. We illustrate by
examples that the jump-diffusion approximation which extends to bounded domains
can be much more informative than that based on ODEs as it can provide accurate
quantity distributions even when they are multi-modal and even for relatively
small population levels. Moreover, we show that the method is faster than
simulating the original Markov chain
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