# Artificial neural networks for boundary integral equations

Partial Differential Equations (PDEs) are used to model a variety of phenomena in engineering and natural sciences. Common to most of the PDEs encountered in practical applications is that they cannot be solved analytically but require various approximation techniques. Conventionally, mesh-based methods, such as Finite Element Method (FEM), are the dominant techniques for obtaining approximate solutions. These methods commonly require the domain of interest to be discretized into a set of elements and the solution is approximated at nodal points in the mesh. Such methods typically are very efficient for low-dimensional problems on regular geometries, while for complicated geometries or for evaluating solution at points other than the nodes, these methods may not be reliable since the meshing could be as difficult as the numerical solution of the PDE itself.

An alternative to FEM is Boundary Element Method (BEM), where the PDE is transformed into a Boundary Integral Equation (BIE), which is subsequently discretized and converted to a system of linear algebraic equations. The dimensionality of the problem is reduced by one and hence, it is less complex to solve. In 2012, Simpson et al. introduced the concept of IsoGeometric Analysis (IGA) in the framework of BEM which has led to isogeometric Boundary Element Method (IGABEM).  IGABEM takes advantage of IGA that allows BEM directly read geometrical data from Computer Aided Design (CAD) models hence minimized the error caused by meshing and preserving tight link between design and analysis.

The past several decades brought a revolution in machine learning where deep Artificial Neural Networks (ANNs) are the key component. The success of deep learning is due to a combination of improved theory starting with unsupervised pre-training and deep belief nets, and improved hardware resources such as general-purpose Graphics Processing Units (GPUs). Deep ANNs are now routinely used with impressive results in areas such as image analysis, pattern recognition, object detection, natural language processing, and self-driving cars, to name a few.  Application of DNNs for solving PDEs was first proposed in Rassi et al. in 2018. Later, in 2019, Anitescu et al. proposed an ANN approximation combined with adaptive collocation strategy, and in 2020, a deep energy method was proposed by Samaniego et al. that combines ANN with energy minimization to solve PDEs.

In our previous work (Zhang et al., 2022), we proposed an Artificial Neural Network method for solving boundary integral equations in domains parameterized by NURBS. In this approach, the neural network is trained to approximate the unknown field or its derivative by minimizing the loss function which is obtained from errors in the BIE at a set of collocation points. A good agreement with the analytical solution was demonstrated, but the approach was limited to 2D problems for Laplace and elasticity equations.

In this project, we aim to investigate further potential of ANN-based methods for BIEs, namely:

1. Extension to 3D problems.
2. Extension to other applications, such as acoustics
3. Extension to systems of dual boundary integral equations in elasticity to model static and propagating cracks.

###### References:

Simpson, R., Bordas, S., Trevelyan, J., and Rabczuk, T. A two-dimensional isogeometric boundary element method for elastostatic analysis. 2012.

Raissi, M. Deep hidden physics models: Deep learning of nonlinear partial differential equations. 2018.

Anitescu, C., Atroshchenko, E., Alajlan, N. and Rabczuk, T. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials And Continua. 59, 345-359, 2019.

Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V., Guo, H., Hamdia, K., Zhuang, X. and Rabczuk, T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications. Computer Methods In Applied Mechanics And Engineering. 362, 2020.

Zhang, H., Anitescu, C., Bordas, S., Rabczuk, T., and Atroshchenko, E. Artificial neural network methods for boundary integral equations. 2022. TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.20164769.v1