# Parametric Learning of Time-Advancement Operators for Unstable Flame Evolution

arXiv: https://arxiv.org/abs/2402.10238 · HTML: https://mlsanity.com/paper/2402.10238/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 50 references that are in the map.
- First posted on arXiv 14 Feb 2024.

## Builds on (most cited first, 12 of 12)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)

## Built on by (most cited first, 5 of 5)

- [CompNO: A Novel Foundation Model approach for solving Partial Differential Equations](https://mlsanity.com/paper/2601.07384/) (2026)
- [Learning Flame Evolution Operator under Hybrid Darrieus Landau and Diffusive Thermal Instability](https://mlsanity.com/paper/2405.07067/) (2024)
- [Data-driven quantum Koopman method for simulating nonlinear dynamics](https://mlsanity.com/paper/2507.21890/) (2025)
- [An Inverse Scattering Inspired Fourier Neural Operator for Time-Dependent PDE Learning](https://mlsanity.com/paper/2512.19439/) (2025)
- [Compositional Neural Operators for Multi-Dimensional Fluid Dynamics](https://mlsanity.com/paper/2605.11691/) (2026)

## Abstract (arXiv)

This study investigates the application of machine learning, specifically Fourier Neural Operator (FNO) and Convolutional Neural Network (CNN), to learn time-advancement operators for parametric partial differential equations (PDEs). Our focus is on extending existing operator learning methods to handle additional inputs representing PDE parameters. The goal is to create a unified learning approach that accurately predicts short-term solutions and provides robust long-term statistics under diverse parameter conditions, facilitating computational cost savings and accelerating development in engineering simulations. We develop and compare parametric learning methods based on FNO and CNN, evaluating their effectiveness in learning parametric-dependent solution time-advancement operators for one-dimensional PDEs and realistic flame front evolution data obtained from direct numerical simulations of the Navier-Stokes equations.