Wrapping ScimBa and Feel++ Internship

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Introduction

In this project, we explore the numerical solution of the Poisson equation, a fundamental partial differential equation (PDE) that appears in various scientific and engineering disciplines. The Laplacian operator, which involves the second derivative with respect to spatial variables, is critical in modeling phenomena such as heat conduction, electrostatics, and fluid flow.

The goal of this internship is build on the project of streamlining communication between ScimBa and Feel++ to implement and compare different numerical solvers for the Poisson equation, evaluating their accuracy and efficiency on the same graph.

Tools and Methods

Tools Used

• ScimBa: A Python-based solver that utilizes Physics-Informed Neural Networks (PINNs) to approximate solutions to PDEs. ScimBa using machine learning
• Feel++: A C++ and Python library designed for finite element analysis. It offers powerful tools for solving PDEs using the Galerkin method, which is particularly useful for problems defined over complex spatial domains. We will be using the CFPDE toolbox.

Methods

1. Problem Definition: The Laplacian equation is defined on a specified domain with appropriate boundary conditions.
2. Solver Implementation: We implemented two solvers:
   • Solver A (Feel++): This solver uses the finite element method (FEM) to discretize the domain and solve the Poisson equation.
   • Solver B (ScimBa): This solver uses PINNs, where the neural network is trained to minimize the residual of the Poisson equation across the domain.
3. Error Analysis: We computed both absolute and relative errors for the solutions obtained from each solver, comparing them to the exact solution of the Poisson equation.

Case Test: Laplacian Equation

Problem Setup

We considered a Laplacian problem on a square domain with Dirichlet boundary conditions. The exact solution is known, allowing us to accurately assess the performance of the solvers.

$$ - \Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f(x, y) $$

$$ u(x, y) = g(x, y) = 0 \quad \text{on} \quad \partial \Omega $$

Solver Comparison

• Absolute Errors: The absolute error is calculated as the difference between the exact solution and the numerical solution provided by each solver.
• Relative Errors: The relative error normalizes the absolute error by the magnitude of the exact solution, providing a clearer comparison in regions where the solution magnitude varies significantly.

Results

• Absolute Error:
  
  

Discussion

ScimBa: The absolute error from ScimBa's solution exhibits a distinct shape, reflecting the solver's structured approach to approximating the solution. This pattern indicates that ScimBa consistently deviates from the exact solution in a predictable manner, likely due to the underlying physics-informed neural network (PINN) methodology that incorporates physical laws into the learning process. While this approach may capture certain global trends, it can also result in systematic errors that form a recognizable shape.

Feel++: In contrast, the absolute error from Feel++ appears more random and resembles visual noise. This randomness suggests that Feel++'s finite element method (FEM) solver, while highly accurate in many regions, may introduce small, localized errors that do not follow a clear pattern. These errors can arise from the discretization process or numerical approximations inherent in FEM, leading to a scatter of deviations that lack a discernible shape.

• Relative Error:
  

Discussion

Feel++: The relative error for Feel++ is generally close to zero across most of the domain, indicating that the solver accurately approximates the exact solution in these regions. However, there is a significant outlier near the center of the domain, where the relative error sharply increases to a value much greater than 1. This spike suggests a localized area where Feel++'s approximation deviates substantially from the exact solution, possibly due to challenges in capturing the behavior of the solution in that specific region. The rest of the domain, however, remains well-approximated with minimal error.

ScimBa: The relative error distribution for ScimBa is more spread out, with many points near the boundaries and in the middle of the domain having errors close to 1. This pattern indicates that ScimBa's solution, while consistent, deviates more uniformly from the exact solution across the domain. The presence of higher errors near the boundaries and the center may reflect limitations in how the neural network model handles boundary conditions or captures complex behaviors within the domain.

Conclusion

The contrast between these error patterns underscores the different strengths of the solvers: Feel++ provides high accuracy over most of the domain with occasional significant errors, while ScimBa exhibits a more consistent but slightly less accurate performance across the entire domain.

This case test demonstrates the effectiveness of both solvers in approximating the Laplacian equation. While both solvers perform well, the choice between them may depend on the specific requirements of the problem, such as the need for higher accuracy in regions with low solution magnitudes.

References

Feel++ Documentation: https://docs.feelpp.org/user/latest/index.html
Feel++ GitHub Repository: https://github.com/feelpp/feelpp
ScimBa Repository: https://gitlab.inria.fr/scimba/scimba
SciML - ScimBa: https://sciml.gitlabpages.inria.fr/scimba/