Tensor methods to learn the Green's function to solve high-dimensional PDE

Kishan Wimalawarne, Taiji Suzuki

Primary Area: general machine learning (i.e., none of the above)

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Keywords: Green's function, high-dimensional PDE, neural networks, tensor-train, low-rank, tensor product grid

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Abstract: The method of Green's function plays an important role in solving PDEs. Recently deep learning models have been used to explicitly learn the Green's function to parameterize solutions of PDEs. DecGreenNet uses low-rank decomposition of the Green's function to obtain computational efficiency by separated learning from training data and Monte-Carlo samples. However, learning from a large number of Monte-Carlo samples for a high-dimensional PDE can lead to slow training and large memory requirements. As a solution we investigate on learning the Green's function by using tensor product grids generated by random partitions of dimensions. We propose DecGreenNet-TT by applying tensor-train structured low-rank decomposition to the Green's function and replace its components with neural networks that learn from partitions of each dimensions instead of all grid elements. We further propose DecGreenNet-TT-C to learn with a reduced number of neural networks by combining dimensions to generate combined tensor product grids. We further show that for the special case of separable source functions the Green's function can be constructed without multiplication of all tensor-train component neural networks leading to memory and computational efficiency. Using several Poisson equations we show that the proposed methods can learn with a collection of smaller neural networks compared to DecGreenNet to efficiently parameterize solutions with faster training times and low errors.

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Submission Number: 3116