Fachhochschule Nordwestschweiz FHNW

Neural-Network based Engineering physical Simulations as an Alternative to standard Techniques

Summary

We solve linear Partial Differential Equations (PDEs) with the help of an Extreme Learning Machine (ELM). The PDE itself is providing the loss function. We compare two strategies, first fitting the model directly against the base function (BF) of the PDE and second, fitting the model (loss model) solely against the loss and boundary conditions of the PDE. The best accuracy achieved by the loss model is around 0.1. Because we use the loss matrix to train, we can avoid iterative training and gained a significant speed up, around 5.3 compared to other implementations using back propagation.

Keywords

python, Extreme Learning Machine, tensorflow, DeepXDE

Goal

The goal here is to reevaluate the state of the art of solving PDEs by means of machine learning. Here an ELM will be fitted against the exact solution of a linear PDE to pove that it works. After that an ELM will be trained only on the loss of the PDE, since in real applications the exact solution is usualy not known. Then we try to find optimal parameters for such a model and compare different methods of solving PDEs in terms of run time and resources used.

Initial position

We were given a research paper in which a lot of research into solving PDEs with machine learning has been done. Here methods such as Finite Element Method (FEM), Neural Networks (NN) and more specifically extreme learning machines (ELM) have been analyzed and studied on how good they are at solving the given problem. We also received a Jupyter Notebook containing all the code related to the aforementioned paper.

Results

In this study we compared ELM models and two different training concepts in order to solve a linear PDE problem. Fitting a model against the loss of the PDE and fitting it against known points on the BF. We measured the run time of these models with timeit and showed that an ELM with matrix inversion is much faster to train back than propagation algorithms. But this comes with the expense of the quality of the predictions. Why there is a limitation in accuracy around 0.1 is unclear, but there have been similar experiences the preceding paper.