insight - Computational Complexity - # Solving Poisson Equations with Geometric Singularities using Singularity Enriched Physics Informed Neural Networks

Core Concepts

Singularity enriched physics informed neural networks (SEPINN) can effectively solve Poisson problems with geometric singularities, such as corners, mixed boundary conditions, and edges, by explicitly incorporating the singular behavior of the analytical solution into the neural network ansatz.

Abstract

The content discusses the development of a novel numerical method called singularity enriched physics informed neural networks (SEPINN) for solving Poisson problems with geometric singularities in two and three dimensions.
Key highlights:
Poisson problems with geometric singularities, such as corners, mixed boundary conditions, and edges, exhibit limited solution regularity, which poses challenges for standard numerical methods including neural network-based solvers.
The authors leverage the known singular function representation of the solution to develop SEPINN, which explicitly incorporates the singular behavior into the neural network ansatz.
For the 2D case, SEPINN splits the solution into a regular part approximated by a neural network and a singular part with known analytical form. The parameters associated with the singular part are learned along with the neural network.
For the 3D case with edge singularities, SEPINN employs a truncated Fourier series representation of the singular part.
Theoretical error bounds are provided for the SEPINN approximation.
Extensive numerical experiments in 2D and 3D demonstrate the flexibility and accuracy of SEPINN compared to standard PINN and other existing approaches.

Stats

The solution u of the Poisson problem (1.1) belongs to H1+π/ω*-ε(Ω) for every ε > 0, where ω* = max1≤j≤M ωj is the maximum interior angle of the polygonal domain Ω.
The regular part w of the solution satisfies the estimate: ∥w∥H2(Ω) ≤ C∥f∥L2(Ω), where C is a constant independent of f.

Quotes

"Physics-Informed Neural Networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their effectiveness is somewhat diminished when addressing issues involving singularities, such as point sources or geometric irregularities, where the approximations they provide often suffer from reduced accuracy due to the limited regularity of the exact solution."
"To the best of our knowledge, it is the first work systematically exploring the use of singularity enrichment in a neural PDE solver."

Key Insights Distilled From

by Tianhao Hu,B... at **arxiv.org** 04-18-2024

Deeper Inquiries

The SEPINN approach can be extended to handle other types of singularities, such as crack singularities or singularities due to discontinuous coefficients, by incorporating the specific behavior of these singularities into the ansatz space of the neural network. For crack singularities, the neural network can be enriched with functions that capture the discontinuities along the crack surfaces. This can involve using special basis functions or activation functions that can represent the discontinuities accurately. Additionally, for singularities due to discontinuous coefficients, the neural network can be trained to adapt to these variations by adjusting the network architecture or the loss function to account for the discontinuities in the coefficients. By incorporating the knowledge of these singularities into the neural network design, SEPINN can effectively handle a wider range of singularities in Poisson-type problems in polygonal domains.

Applying SEPINN to time-dependent or nonlinear Poisson-type problems may pose several challenges. In time-dependent problems, the evolution of the solution over time introduces an additional dimension to the problem, requiring the neural network to capture the dynamics accurately. This may involve training the network on time sequences of data and incorporating time derivatives into the loss function. Nonlinear Poisson-type problems introduce complexities in the solution behavior, requiring the neural network to handle nonlinearities effectively. This can be challenging as the network needs to learn the nonlinear relationships between the variables and adapt to the changing behavior of the solution. Additionally, the convergence and stability of the neural network in handling nonlinearities and time-dependent terms need to be carefully addressed to ensure accurate and reliable solutions.

The ideas of SEPINN can be combined with other neural network-based PDE solvers, such as the deep Ritz method or deep Galerkin method, to further improve the performance for problems with geometric singularities. By integrating the singularity enrichment techniques of SEPINN with the solution representations of these methods, a more robust and accurate solver can be developed. The deep Ritz method, which formulates the PDE as a variational problem, can benefit from the singularity enrichment approach of SEPINN to handle geometric singularities more effectively. Similarly, the deep Galerkin method, which uses a trial space to approximate the solution, can be enhanced by incorporating the singularity behavior directly into the trial space. By combining the strengths of SEPINN with these methods, a more versatile and powerful solver can be created for challenging problems with geometric singularities.

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