Multiple general sigmoids based Banach space valued neural network multivariate approximation

• Autores: George A. Anastassiou
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 25, Nº. 3, 2023, págs. 411-439
• Idioma: inglés
• DOI: 10.56754/0719-0646.2503.411
• Enlaces
  • Texto completo
• Resumen
  • español

    Resumen Presentamos aproximaciones multivariadas cuantitativas de funciones multivariadas continuas con valores en un espacio de Banach definidas en una caja o en RN, N ∈ N, a través de operadores de redes neuronales multivariados normalizados, de cuasi-interpolación, de tipo Kantorovich y de tipo cuadratura. También tratamos el caso de aproximación usando operadores iterados de los últimos cuatro tipos. Estas aproximaciones se derivan estableciendo desigualdades multidimensionales de tipo Jackson que involucran el módulo de continuidad multivariado de la función comprometida o sus derivadas de Fréchet de alto orden. Nuestros operadores multivariados son definidos usando una función de densidad multidimensional inducida por varias funciones sigmoidales generales diferentes entre sí. Esto se hace con el propósito de activar la mayor cantidad de neuronas posible. Las aproximaciones son puntutales y uniformes. La red neuronal prealimentada relacionada tiene un nivel oculto. Concluimos con aproximaciones Lp relacionadas.

  • English

    Abstract Here we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or RN, N ∈ N, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by several different among themselves general sigmoid functions. This is done on the purpose to activate as many as possible neurons. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer. We finish with related Lp approximations.

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