# The Deep Ritz Method for Parametric p-Dirichlet Problems

@article{Kaltenbach2022TheDR, title={The Deep Ritz Method for Parametric p-Dirichlet Problems}, author={Alex Kaltenbach and Marius Zeinhofer}, journal={ArXiv}, year={2022}, volume={abs/2207.01894} }

We establish error estimates for the approximation of parametric p-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, e.g., varying geometries and exponents p ∈ (1,∞). Combining the derived error estimates with quantitative approximation theorems yields error decay rates and establishes that the Deep Ritz Method retains the favorable approximation capabilities of neural networks in the approximation of high dimensional functions which makes the method attractive…

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## References

SHOWING 1-10 OF 48 REFERENCES

### Optimal Convergence for the Implicit Space-Time Discretization of Parabolic Systems with p-Structure

- MathematicsSIAM J. Numer. Anal.
- 2007

A fully discrete scheme is studied using $C^0$-piecewise linear finite elements in space and the backward Euler difference scheme in time, and optimal convergence rates are obtained.

### The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

- Computer ScienceArXiv
- 2017

A deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations, which is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions.

### TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems

- Computer ScienceArXiv
- 2016

The TensorFlow interface and an implementation of that interface that is built at Google are described, which has been used for conducting research and for deploying machine learning systems into production across more than a dozen areas of computer science and other fields.

### Universal approximation bounds for superpositions of a sigmoidal function

- Computer ScienceIEEE Trans. Inf. Theory
- 1993

The approximation rate and the parsimony of the parameterization of the networks are shown to be advantageous in high-dimensional settings and the integrated squared approximation error cannot be made smaller than order 1/n/sup 2/d/ uniformly for functions satisfying the same smoothness assumption.

### Monotone operator theory for unsteady problems on non-cylindrical domains

- Ph.d. thesis, Institute of Applied Mathematics, University of Freiburg.
- 2015

### Convergence analysis of a Local Discontinuous Galerkin approximation for nonlinear systems with Orlicz-structure

- Mathematics, Computer ScienceArXiv
- 2022

A new numerical flux is proposed, which yields optimal convergence rates for linear ansatz functions and a unified treatment for problems with (p, δ)structure for arbitrary p ∈ (1,∞) and δ ≥ 0,.

### Convergence estimates of finite elements for a class of quasilinear elliptic problems

- MathematicsComput. Math. Appl.
- 2021

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

- Mathematics, Computer ScienceAdvances in continuous and discrete models
- 2022

This work provides convergence guarantees for the Deep Ritz Method for abstract variational energies with essential or natural boundary conditions for nonlinear variational problems such as the p -Laplace equation or the Modica–Mortola energy.

### Analysis of Deep Ritz Methods for Laplace Equations with Dirichlet Boundary Conditions

- Materials Science, Computer ScienceArXiv
- 2021

A convergence rate in H norm for deep Ritz methods for Laplace equations with Dirichlet boundary condition is presented, where the error depends on the depth and width in the deep neural networks and the number of samples explicitly.

### Error Analysis of Deep Ritz Methods for Elliptic Equations

- Computer ScienceArXiv
- 2021

This paper establishes the first nonasymptotic convergence rate in H norm for DRM using deep networks with smooth activation functions including logistic and hyperbolic tangent functions.