Neural networks for boundary value problems

We report here a promising approach to use neural networks for the efficient numerical solution of boundary value problems.

According to the theory of boundary integral equations and the so-called method of fundamental solutions, the principal concept is to approximate the solution by a linear combination of fundamental solutions.
Neural networks come here into the picture to find the optimal coefficients.

An auxiliary or pseudo-boundary is built to avoid the singularities appearing in the integration. For certain source points, the fundamental solution between these points and outer points can be comprehended as the output training data. The input is the fundamental solution between boundary points and outer points proportional to specific boundary-type conditions. When training data is generated, we apply linear neural networks and use up-to-date machine learning tools to estimate the weights. Based on this result, evaluating the numerical solution is done by injecting the actual boundary function into the constructed network.

At this stage, the 2D Laplace equation was investigated to verify the correctness of the method. We presented two approaches to generate input-output data that work accurately on irregular domains. Nodal distribution of auxiliary boundary was studied in details together with the optimal choice for parameters including the number of nodal points, learning rates and epochs. Also, the distance between boundary and auxiliary boundary has a crucial role.

Figure 1 illustrates the numerical solution compared to the analytical solution in the case domain is an Amoeba-like shape. For such an accurate approximation, it was sufficient to usea dense neural network with a single input and output layer of size 90 and 60, respectively. We could completely get rid of assembling or solving any other linear systems and no computational mesh should be generated.

Our main objective is to apply a similar approach to a real-life 3D acoustic problem, where the significant reduction of computational costs is very essential.

This research is conducted in the AI Research Group at the Institute of Mathematics, Eötvös Loránd University.

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