2,419 research outputs found
Functional expansion representations of artificial neural networks
In the past few years, significant interest has developed in using artificial neural networks to model and control nonlinear dynamical systems. While there exists many proposed schemes for accomplishing this and a wealth of supporting empirical results, most approaches to date tend to be ad hoc in nature and rely mainly on heuristic justifications. The purpose of this project was to further develop some analytical tools for representing nonlinear discrete-time input-output systems, which when applied to neural networks would give insight on architecture selection, pruning strategies, and learning algorithms. A long term goal is to determine in what sense, if any, a neural network can be used as a universal approximator for nonliner input-output maps with memory (i.e., realized by a dynamical system). This property is well known for the case of static or memoryless input-output maps. The general architecture under consideration in this project was a single-input, single-output recurrent feedforward network
SISO Output Affine Feedback Transformation Group and Its Faa di Bruno Hopf Algebra
The general goal of this paper is to identify a transformation group that can
be used to describe a class of feedback interconnections involving subsystems
which are modeled solely in terms of Chen-Fliess functional expansions or
Fliess operators and are independent of the existence of any state space
models. This interconnection, called an output affine feedback connection, is
distinguished from conventional output feedback by the presence of a multiplier
in an outer loop. Once this transformation group is established, three basic
questions are addressed. How can this transformation group be used to provide
an explicit Fliess operator representation of such a closed-loop system? Is it
possible to use this feedback scheme to do system inversion purely in an
input-output setting? In particular, can feedback input-output linearization be
posed and solved entirely in this framework, i.e., without the need for any
state space realization? Lastly, what can be said about feedback invariants
under this transformation group? A final objective of the paper is to describe
the Lie algebra of infinitesimal characters associated with the group in terms
of a pre-Lie product.Comment: revised manuscript; title and abstract changed; new material adde
Entropy of Generating Series for Nonlinear Input-Output Systems and their Interconnections
This paper has two main objectives. The first is to introduce a notion of entropy that is well suited for the analysis of nonlinear input-output systems that have a Chen-Fliess series representation. The latter is defined in terms of its generating series over a noncommutative alphabet. The idea is to assign an entropy to a generating series as an element of a graded vector space. The second objective is to describe the entropy of generating series originating from interconnected systems of Chen-Fliess series that arise in the context of control theory. It is shown that one set of interconnections can never increase entropy as defined here, while a second set has the potential to do so. The paper concludes with a brief introduction to an entropy ultrametric space and some open questions
AN APPLIED PROCEDURE FOR ESTIMATING AND SIMULATING MULTIVARIATE EMPIRICAL (MVE) PROBABILITY DISTRIBUTIONS IN FARM-LEVEL RISK ASSESSMENT AND POLICY ANALYSIS
Research Methods/ Statistical Methods,
Dendriform-Tree Setting for Fully Non-commutative Fliess Operators
This paper provides a dendriform-tree setting for Fliess operators with
matrix-valued inputs. This class of analytic nonlinear input-output systems is
convenient, for example, in quantum control. In particular, a description of
such Fliess operators is provided using planar binary trees. Sufficient
conditions for convergence of the defining series are also given
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