Uniform convergence guarantees for the deep Ritz method for nonlinear problems.

Area of Science:

• Numerical Analysis
• Partial Differential Equations
• Computational Mathematics

Background:

• Variational methods are fundamental in solving differential equations.
• The Deep Ritz Method (DRM) is a machine learning-based approach for solving variational problems.
• Convergence guarantees are crucial for validating numerical methods.

Purpose of the Study:

• To establish convergence guarantees for the Deep Ritz Method (DRM) applied to abstract variational energies.
• To demonstrate the applicability of DRM to a range of nonlinear problems.
• To analyze the behavior of DRM under different boundary conditions and parameter variations.

Main Methods:

• The study employs theoretical analysis to derive convergence rates.
• Abstract variational formulations are used to generalize the problem setting.
• The analysis covers both essential and natural boundary conditions.

Main Results:

• Convergence guarantees are provided for the Deep Ritz Method for abstract variational energies.
• The method is shown to be effective for nonlinear problems, including the p-Laplace equation and Modica-Mortola energy.
• Uniform convergence is demonstrated across bounded families of right-hand sides under specific assumptions.

Conclusions:

• The Deep Ritz Method is a theoretically sound approach for solving complex variational problems.
• The established convergence properties enhance the reliability of DRM in computational mathematics.
• This work contributes to the theoretical foundation of using deep learning for solving differential equations.