84 Result(s)
within
Chapter
Lu**g Chen
Mathematics
Artificial Intelligence
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Chapter
Combining Stochastic Models with Machine Learning
Machine learning has become a prevalent and powerful tool in many scientific and engineering disciplines. This last chapter presents a few methods that combine stochastic models with machine learning to advanc...
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Basic Stochastic Computational Methods
In this chapter, several fundamental stochastic computational tools are introduced. The chapter starts by presenting the ideas of the Monte Carlo method, a widely utilized technique that exploits the repeated ...
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Instruction Manual for the MATLAB Codes
This chapter includes an instruction manual of the MATLAB codes.
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Data-Driven Low-Order Stochastic Models
Data-driven low-order stochastic models have broad applications in reality. They can be utilized to effectively model the time evolution of each spectral mode of a complex spatially extended system, where the ...
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Parameter Estimation with Uncertainty Quantification
In this chapter, various parameter estimation methods are presented. First, the Markov chain Monte Carlo (MCMC) technique is introduced, which is then applied to estimate model parameters utilizing the Metropo...
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Introduction to Information Theory
One major challenge in modeling, understanding, and predicting complex systems is quantifying the associated uncertainty due to the intrinsic turbulent nature or the external stochastic forcing. Information th...
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Simple Gaussian and Non-Gaussian SDEs
In this chapter, simple Gaussian and non-Gaussian SDEs are presented. They are appropriate paradigms to understand key dynamical and statistical features of general complex systems. They also serve as simple i...
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Prediction
Forecasting complex dynamical systems is one of the most important practical issues. This chapter starts by presenting the method of ensemble forecast, which adopts a probabilistic characterization of the mode...
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Conditional Gaussian Nonlinear Systems
In this chapter, a nonlinear modeling framework, called the conditional Gaussian nonlinear system (CGNS), is introduced. The CGNS contains a rich class of nonlinear models, where the joint and marginal distrib...
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Stochastic Toolkits
This chapter introduces the basic probability concepts and stochastic processes. It includes random variables, probability density function (PDF), moments, Gaussian random variables, Wiener process, Markov jum...
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Data Assimilation
Data assimilation seeks to optimally integrate different information sources to improve the state estimation of a complex system. It is also the prerequisite for effective ensemble forecasts. This chapter aims...
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Concept Mathematics
Part III deals with the very definition of conceptual mathematics and then a discussion of first attempts to generate a systematic approach to conceptual mathematics.
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Semiotics
This chapter gives an overview of semiotics as developed by Charles Sanders Peirce, Ferdinand de Saussure, Louis Hjelmslev, and Roland Barthes.
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Semantic Representations
This chapter opens the question about the semantic “loading” of a mathematical concept. This representation relates to the categories of H-jets, where concept are conceived as “sources” of a semantic extension...
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Applications and Consequences
Music (and more generally the arts) creates semiotic structures independently of ‘external’ reference contents. We discuss this qualification as a conceptual challenge.
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Motivation and Background
We present the motivation for the development of this functorial semiotics as a bridge between Human Intelligence and Artificial Intelligence.
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Functorial Semantics Category
This chapter presents the category of functorial semantics, which formalizes the semiotic objects together with the connecting morphisms. These objects are called H-jets, “H” standing for Hjelmslev, who introd...
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Semantic and Expressive Topology
This chapter presents a (classical) topology on H-jet collections that relates to semantic aspects. Dually, we shall introduce an expressive topology.
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Yoneda
We discuss some global consequences of Yoneda’s Lemma and its philosophy.
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Cech Cohomology
We discuss two approaches to Cech cohomology: function spaces for global filters, and functorial cohomology associated with the semantic topology.
