Multilayer Neural Network Models and their Application to Solving Nonlinear Partial Differential Equations
Keywords:
BNNM, PDE, Nonlinear Equation, Solution Models, Neural NetworksAbstract
One of the most powerful techniques for modelling and solving complex problems in different fields that are used in day-to-day life, is artificial neural networks (ANNs). Among them, the Multilayer Perceptron (MLP), i.e., feedforward neural network, has been widely used and found to be very successful in learning non-linear relationships in data. Mimicking neurons in the brain of a human, the MLPs process the data input by the user in a series of multiple layers, including weighted connections, bias terms, and an activation function, bit by bit, which then remaps the input to produce the output. The simplest MLP is made up of an input layer, one or more than one(i.e., multiple) hidden layers, and an output layer. This neural network model learns by modifying its internal parameters with the help of the backpropagation technique, which forces the model to iteratively reduce the error that arises between the output obtained and the output that was predicted. This learning involves optimization techniques such as gradient descent.\\ In my work, we consider the multilayer perceptron models applied to the context of nonlinear PDEs. In particular, we investigate the construction of the Bilinear Neural Network Models (BNNMs) and show how they can be utilized to obtain exact analytical solutions for strong nonlinear systems, e.g., the p-gBKP equation. With a tensorized neural network architecture embedded in the Hirota bilinear framework, the work demonstrates the capability of neural architectures for symbolic computing and mathematical modeling.References
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