mlsanityData to 14 Oct 2026

The Deep Ritz Method for Parametric p-Dirichlet Problems

arXiv 2207.01894 ↗First posted 5 Jul 2022math.NA cs.LG cs.NAcited by 3

Not on the map. Why?

Abstract (arXiv)

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 for parametric problems. Finally, we present numerical examples to illustrate potential applications.

Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 13 of its 49 references that are in the map. First posted on arXiv 5 Jul 2022.

Builds on

13 of 49 references.

References · 13

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2016Gaussian Error Linear Units (GELUs)cited by 3,502
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  4. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  5. 2016TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systemscited by 664
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  7. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  8. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 262
  9. 2017Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanicscited by 214
  10. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
  11. 2020NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation frameworkcited by 58
  12. 2019The Barron Space and the Flow-induced Function Spaces for Neural Network Modelscited by 39
1 more
  1. 2018A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientscited by 31

Built on by

Cited by year

Citing papers per year: 2022: 1; 2023: 0; 2024: 0; 2025: 2.

3 papers.

  1. 2022Deep learning for gradient flows using the Brezis-Ekeland principlecited by 0
  2. 2025Engineering application of physics-informed neural networks for Saint-Venant torsioncited by 0
  3. 2025A decomposition-based robust training of physics-informed neural networks for nearly incompressible linear elasticitycited by 0