mlsanityData to 7 Oct 2026

A Low Rank Neural Representation of Entropy Solutions

arXiv 2406.05694 ↗First posted 9 Jun 2024math.NA cs.LG cs.NAcited by 3

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Abstract (arXiv)

We construct a new representation of entropy solutions to nonlinear scalar conservation laws with a smooth convex flux function in a single spatial dimension. The representation is a generalization of the method of characteristics and posseses a compositional form. While it is a nonlinear representation, the embedded dynamics of the solution in the time variable is linear. This representation is then discretized as a manifold of implicit neural representations where the feedforward neural network architecture has a low rank structure. Finally, we show that the low rank neural representation with a fixed number of layers and a small number of coefficients can approximate any entropy solution regardless of the complexity of the shock topology, while retaining the linearity of the embedded dynamics.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 20 of its 80 references that are in the map. First posted on arXiv 9 Jun 2024.

Builds on

20 of 80 references.

References · 20

  1. 2019DeepSDF: Learning Continuous Signed Distance Functions for Shape Representationcited by 2,682
  2. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
  3. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  4. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  5. 2016Error bounds for approximations with deep ReLU networkscited by 567
  6. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  7. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
  8. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  9. 2019A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Casecited by 113
  10. 2020A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEscited by 105
  11. 2017Optimal Approximation with Sparsely Connected Deep Neural Networkscited by 90
  12. 2020A fast and accurate physics-informed neural network reduced order model with shallow masked autoencodercited by 80
8 more
  1. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
  2. 2018Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systemscited by 54
  3. 2023Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networkscited by 42
  4. 2020Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networkscited by 27
  5. 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
  6. 2020Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularitiescited by 20
  7. 2022Ridges, Neural Networks, and the Radon Transformcited by 18
  8. 2020Depth separation for reduced deep networks in nonlinear model reduction: Distilling shock waves in nonlinear hyperbolic problemscited by 3

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 1; 2026: 1.

3 papers.

  1. 2024FastLRNR and Sparse Physics Informed Backpropagationcited by 3
  2. 2026Nonlinear model reduction for transport-dominated problemscited by 1
  3. 2025Learning Low Rank Neural Representations of Hyperbolic Wave Dynamics from Datacited by 0