mlsanityData to 7 Oct 2026

Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networks

arXiv 2309.02589 ↗First posted 5 Sep 2023math.AP cs.AIcited by 5

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Abstract (arXiv)

In this paper, we compute numerical approximations of the minimal surfaces, an essential type of Partial Differential Equation (PDE), in higher dimensions. Classical methods cannot handle it in this case because of the Curse of Dimensionality, where the computational cost of these methods increases exponentially fast in response to higher problem dimensions, far beyond the computing capacity of any modern supercomputers. Only in the past few years have machine learning researchers been able to mitigate this problem. The solution method chosen here is a model known as a Physics-Informed Neural Network (PINN) which trains a deep neural network (DNN) to solve the minimal surface PDE. It can be scaled up into higher dimensions and trained relatively quickly even on a laptop with no GPU. Due to the inability to view the high-dimension output, our data is presented as snippets of a higher-dimension shape with enough fixed axes so that it is viewable with 3-D graphs. Not only will the functionality of this method be tested, but we will also explore potential limitations in the method's performance.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 16 of its 29 references that are in the map. First posted on arXiv 5 Sep 2023.

Builds on

16 of 29 references.

References · 16

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  3. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  4. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  5. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  6. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  7. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 75
  8. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  9. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
  10. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  11. 2022Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodologycited by 43
  12. 2020nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applicationscited by 36
4 more
  1. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  2. 2020Numerical Solution of Inverse Problems by Weak Adversarial Networkscited by 29
  3. 2018A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equationscited by 23
  4. 2018Neural Networks Trained to Solve Differential Equations Learn General Representationscited by 16

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 5.

5 papers.

  1. 2026A Machine Learning Approach to the Nirenberg Problemcited by 5
  2. 2026Minimising Willmore Energy via Neural Flowcited by 4
  3. 2026Minimal surfaces, Knots, and Neural Networkscited by 2
  4. 2026PINNs in More General Geometrycited by 1
  5. 2026A user's guide to PINNs in geometric analysis: lessons from the asymptotic Plateau problemcited by 0