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

arXiv: https://arxiv.org/abs/1807.09519 · HTML: https://mlsanity.com/paper/1807.09519/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2018.

## Built on by (most cited first, 18 of 18)

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- [STENCIL-NET: Data-driven solution-adaptive discretization of partial differential equations](https://mlsanity.com/paper/2101.06182/) (2021)
- [Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations](https://mlsanity.com/paper/2302.03286/) (2023)
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- [Rational-WENO: A lightweight, physically-consistent three-point weighted essentially non-oscillatory scheme](https://mlsanity.com/paper/2409.09217/) (2024)
- [Learning second-order TVD flux limiters using differentiable solvers](https://mlsanity.com/paper/2503.09625/) (2025)

## 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.