Publication: Propagación de ondas mediante aprendizaje automático
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2023-06
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Abstract
El propósito de este trabajo es investigar en qué medida se pueden usar redes neuronales para aproximar resultados que, en principio, requerirían resolver numéricamente ecuaciones de ondas. Nos centraremos en geometrías unidimensionales prefijadas gobernadas por pocos parámetros e intentaremos aproximar problemas directos e inversos. En el primer caso, tratamos de estimar mediante redes convolucionales aproximaciones de soluciones de ecuaciones de onda a partir de parámetros materiales. El problema inverso consiste en estimar mediante redes neuronales los parámetros de la ecuación de ondas que representan el material a partir de valores de la solución medidos en un extremo en una secuencia de tiempos. Tras el estudio realizado, pudimos obtener unas estimaciones de las ondas mediante redes convolucionales con un error respecto al rango menor de un 5 % en todos los casos para el problema directo. Por otro lado, en el problema inverso, el modelo presentaba dificultades para estimar uno de los parámetros (la posición del objeto) de los tres estudiados. Finalmente, se comprobó que el problema residía en los conjuntos de datos generados para alimentar el modelo, pues no disponían de suficiente información para representar todo el espacio de parámetros.
The purpose of this work is to investigate the extent to which neural networks can be used to approximate results that would typically require numerically solving wave equations. We will focus on predetermined one-dimensional geometries governed by a few parameters and attempt to approximate both direct and inverse problems. In the first case, we attempt to estimate wave equation solutions using convolutional networks based on material parameters. The inverse problem involves estimating the wave equation parameters that represent the material using neural networks, given measured solution values at one end over a sequence of time steps. After the conducted study, we were able to obtain wave estimations using convolutional networks with a deviation of less than 5 % from the range of values in all cases for the direct problem. On the other hand, in the inverse problem, the model faced difficulties in estimating one of the parameters (object position) out of the three studied. Finally, it was determined that the issue resided in the generated data sets used to train the model, as they did not contain sufficient information to represent the entire parameter space.
The purpose of this work is to investigate the extent to which neural networks can be used to approximate results that would typically require numerically solving wave equations. We will focus on predetermined one-dimensional geometries governed by a few parameters and attempt to approximate both direct and inverse problems. In the first case, we attempt to estimate wave equation solutions using convolutional networks based on material parameters. The inverse problem involves estimating the wave equation parameters that represent the material using neural networks, given measured solution values at one end over a sequence of time steps. After the conducted study, we were able to obtain wave estimations using convolutional networks with a deviation of less than 5 % from the range of values in all cases for the direct problem. On the other hand, in the inverse problem, the model faced difficulties in estimating one of the parameters (object position) out of the three studied. Finally, it was determined that the issue resided in the generated data sets used to train the model, as they did not contain sufficient information to represent the entire parameter space.