mlsanityData to 7 Oct 2026

Sparse Deep Neural Network for Nonlinear Partial Differential Equations

arXiv 2207.13266 ↗First posted 27 Jul 2022math.NA cs.AI cs.NAcited by 5

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Abstract (arXiv)

More competent learning models are demanded for data processing due to increasingly greater amounts of data available in applications. Data that we encounter often have certain embedded sparsity structures. That is, if they are represented in an appropriate basis, their energies can concentrate on a small number of basis functions. This paper is devoted to a numerical study of adaptive approximation of solutions of nonlinear partial differential equations whose solutions may have singularities, by deep neural networks (DNNs) with a sparse regularization with multiple parameters. Noting that DNNs have an intrinsic multi-scale structure which is favorable for adaptive representation of functions, by employing a penalty with multiple parameters, we develop DNNs with a multi-scale sparse regularization (SDNN) for effectively representing functions having certain singularities. We then apply the proposed SDNN to numerical solutions of the Burgers equation and the Schr"odinger equation. Numerical examples confirm that solutions generated by the proposed SDNN are sparse and accurate.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 13 of its 47 references that are in the map. First posted on arXiv 27 Jul 2022.

Builds on

13 of 47 references.

References · 13

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1 more
  1. 2021Convergence of Deep Convolutional Neural Networkscited by 13

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Cited by year

Citing papers per year: 2023: 4; 2024: 1.

5 papers.

  1. 2023Multi-Grade Deep Learningcited by 12
  2. 2023Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equationcited by 5
  3. 2024Deep Neural Network Solutions for Oscillatory Fredholm Integral Equationscited by 5
  4. 2023Successive Affine Learning for Deep Neural Networkscited by 4
  5. 2023PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equationscited by 4