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La diferenciación automática es esencial en el entrenamiento de redes neuronales para resolver ecuaciones

Chuqi Chen1,2, Yahong Yang1, Yang Xiang1,3

  • 1Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.

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Resumen
Este resumen es generado por máquina.

Los métodos de red neuronal para resolver ecuaciones diferenciales parciales (EDP) son prometedores. La diferenciación automática (AD) ofrece ventajas sobre los métodos de diferencia finita (FD) en el entrenamiento de redes neuronales para PDEs.

Palabras clave:
Diferenciación automáticaEcuación diferencialRed neuronalDiferenciación numéricaError de entrenamiento

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Área de la Ciencia:

  • Ciencias e ingeniería computacionales
  • Matemáticas aplicadas
  • Aprendizaje automático para la computación científica

Sus antecedentes:

  • Las redes neuronales se utilizan cada vez más para resolver ecuaciones diferenciales parciales (EDP).
  • Los métodos tradicionales como la diferencia finita (FD) requieren puntos locales para el cálculo de la derivada.
  • La diferenciación automática (DA) ofrece una alternativa utilizando sólo puntos de muestreo.

Objetivo del estudio:

  • Demostrar cuantitativamente las ventajas de entrenamiento de la diferenciación automática (AD) sobre los métodos de diferencia finita (FD) para los solucionadores de PDE basados en redes neuronales.
  • Introducir y validar una nueva métrica, la entropía truncada, para caracterizar las propiedades de entrenamiento de las redes neuronales.
  • Comparar el rendimiento de AD y FD en la resolución de PDE desde una perspectiva de formación.

Principales métodos:

  • Introducción del concepto de entropía truncada para la caracterización de la formación.
  • Análisis experimentales y teóricos de modelos de características aleatorias.
  • Análisis de redes neuronales de dos capas utilizando tanto AD como FD.

Principales resultados:

  • La entropía truncada cuantifica de manera confiable la pérdida residual en modelos de características aleatorias.
  • La entropía truncada sirve como una métrica para la velocidad de entrenamiento de la red neuronal con AD y FD.
  • La evidencia experimental y teórica muestra que AD supera a FD en el entrenamiento de redes neuronales para PDEs.

Conclusiones:

  • La diferenciación automática (AD) presenta un enfoque de capacitación superior para los solucionadores de ecuaciones diferenciales parciales (PDE) basados en redes neuronales en comparación con los métodos de diferencias finitas (FD).
  • La nueva métrica de entropía truncada caracteriza efectivamente la dinámica y el rendimiento del entrenamiento.
  • Los hallazgos apoyan la adopción más amplia de AD en el aprendizaje automático científico para resolver ecuaciones complejas.