mlsanityData to 7 Oct 2026

A machine learning framework for data driven acceleration of computations of differential equations

arXiv 1807.09519 ↗First posted 25 Jul 2018math.NA cs.LG cs.NAcited by 18

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Abstract (arXiv)

We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an offline training process by (approximately) minimizing suitable (possibly non-convex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying differential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods.

Key facts. Cited by 18 papers in the map (data through 7 Oct 2026). First posted on arXiv 25 Jul 2018.

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Citing papers per year: 2021: 3; 2022: 3; 2023: 7; 2024: 3; 2025: 2.

18 papers.

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  1. 2022Learning to correct spectral methods for simulating turbulent flowscited by 49
  2. 2024DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systemscited by 29
  3. 2023A Neural PDE Solver with Temporal Stencil Modelingcited by 22
  4. 2021Learning optimal multigrid smoothers via neural networkscited by 17
  5. 2023Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systemscited by 14
  6. 2023Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equationscited by 13
  7. 2024Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEscited by 12
  8. 2022A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithmscited by 8
  9. 2023Invariant preservation in machine learned PDE solvers via error correctioncited by 7
  10. 2023A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEscited by 6
  11. 2023AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarioscited by 4
  12. 2021Multigoal-oriented dual-weighted-residual error estimation using deep neural networkscited by 2
6 more
  1. 2022Learning Relaxation for Multigridcited by 2
  2. 2021STENCIL-NET: Data-driven solution-adaptive discretization of partial differential equationscited by 1
  3. 2023Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equationscited by 1
  4. 2025Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometrycited by 1
  5. 2024Rational-WENO: A lightweight, physically-consistent three-point weighted essentially non-oscillatory schemecited by 0
  6. 2025Learning second-order TVD flux limiters using differentiable solverscited by 0