Abstract
In this chapter, we start with basic elements of Boolean expressions and circuit theory to define a background to study neural networks as computational models. We present the truth tables and algebra of propositions. Then, we consider the circuit theory since neural networks can be cast into the computational model formalized by circuit families. Specifically, the neural computation process can be organized in a graph or circuit whose gates (neurons) are arranged in layers. In this way, owing to the equivalence between circuits and Boolean expression approaches, we can discuss the universality of neural computing. We end this chapter with a discussion about elements from differential geometry and dynamical systems that are part of the mathematical background for our presentation. Specifically, we formalize a data model based on the concept of differentiable manifolds. Hence, the database samples and the associated probability density distribution occupy the natural place in the proposed data model. Dynamical systems foundations are necessary to visualize such data model in the context of Hamiltonian formalism when representing fluids as particle systems.
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Antonio Giraldi, G., Almeida, L.R.d., Lopes Apolinário Jr., A., Silva, L.T.d. (2023). Neural Networks Universality, Geometric, and Dynamical System Elements for Fluid Animation. In: Deep Learning for Fluid Simulation and Animation. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-031-42333-8_4
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