# Machine Learning and Computational Mathematics

arXiv: https://arxiv.org/abs/2009.14596 · HTML: https://mlsanity.com/paper/2009.14596/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2020.

## Built on by (most cited first, 22 of 22)

- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs](https://mlsanity.com/paper/2107.03673/) (2021)
- [Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations](https://mlsanity.com/paper/2203.08501/) (2022)
- [Solving time dependent Fokker-Planck equations via temporal normalizing flow](https://mlsanity.com/paper/2112.14012/) (2021)
- [Self-Validation: Early Stopping for Single-Instance Deep Generative Priors](https://mlsanity.com/paper/2110.12271/) (2021)
- [Failure-informed adaptive sampling for PINNs, Part II: combining with re-sampling and subset simulation](https://mlsanity.com/paper/2302.01529/) (2023)
- [Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems](https://mlsanity.com/paper/2211.01873/) (2022)
- [Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEs](https://mlsanity.com/paper/2112.06660/) (2021)
- [Thermodynamics of learning physical phenomena](https://mlsanity.com/paper/2207.12749/) (2022)
- [A Mathematical Analysis of Neural Operator Behaviors](https://mlsanity.com/paper/2410.21481/) (2024)
- [Differentiable physics-enabled closure modeling for Burgers' turbulence](https://mlsanity.com/paper/2209.11614/) (2022)
- [A Novel Adaptive Causal Sampling Method for Physics-Informed Neural Networks](https://mlsanity.com/paper/2210.12914/) (2022)
- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
- [Solving the Dirichlet problem for the Monge-Amp`ere equation using neural networks](https://mlsanity.com/paper/2110.03310/) (2021)
- [Leveraging Interpolation Models and Error Bounds for Verifiable Scientific Machine Learning](https://mlsanity.com/paper/2404.03586/) (2024)
- [Data-informed Deep Optimization](https://mlsanity.com/paper/2107.08166/) (2021)
- [Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/2210.16196/) (2022)
- [MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling](https://mlsanity.com/paper/2212.12984/) (2022)
- [Capturing Shock Waves by Relaxation Neural Networks](https://mlsanity.com/paper/2404.01163/) (2024)
- [LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process](https://mlsanity.com/paper/2507.22493/) (2025)
- [Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2602.18515/) (2026)
- [A foundation model of numerical intelligence with cross-disciplinary generalization](https://mlsanity.com/paper/2607.28432/) (2026)

## Abstract (arXiv)

Neural network-based machine learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This has opened up many exciting new possibilities, not just in traditional areas of artificial intelligence, but also in scientific computing and computational science. At the same time, machine learning has also acquired the reputation of being a set of "black box" type of tricks, without fundamental principles. This has been a real obstacle for making further progress in machine learning. In this article, we try to address the following two very important questions: (1) How machine learning has already impacted and will further impact computational mathematics, scientific computing and computational science? (2) How computational mathematics, particularly numerical analysis, {can} impact machine learning? We describe some of the most important progress that has been made on these issues. Our hope is to put things into a perspective that will help to integrate machine learning with computational mathematics.