# An Inverse Scattering Inspired Fourier Neural Operator for Time-Dependent PDE Learning

arXiv: https://arxiv.org/abs/2512.19439 · HTML: https://mlsanity.com/paper/2512.19439/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 54 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 15 of 15)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [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)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [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)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods](https://mlsanity.com/paper/2311.11798/) (2023)
- [Parametric Learning of Time-Advancement Operators for Unstable Flame Evolution](https://mlsanity.com/paper/2402.10238/) (2024)
- [Koopman Theory-Inspired Method for Learning Time Advancement Operators in Unstable Flame Front Evolution](https://mlsanity.com/paper/2412.08426/) (2024)
- [Learning Flame Evolution Operator under Hybrid Darrieus Landau and Diffusive Thermal Instability](https://mlsanity.com/paper/2405.07067/) (2024)

## Built on by (most cited first, 1 of 1)

- [Local gradient neural operator](https://mlsanity.com/paper/2609.07752/) (2026)

## Abstract (arXiv)

Learning accurate and stable time-advancement operators for nonlinear partial differential equations (PDEs) remains challenging, particularly for chaotic, stiff, and long-horizon dynamical systems. While neural operator methods such as the Fourier Neural Operator (FNO) and Koopman-inspired extensions achieve good short-term accuracy, their long-term stability is often limited by unconstrained latent representations and cumulative rollout errors. In this work, we introduce an inverse scattering inspired Fourier Neural Operator(IS-FNO), motivated by the reversibility and spectral evolution structure underlying the classical inverse scattering transform. The proposed architecture enforces a near-reversible pairing between lifting and projection maps through an explicitly invertible neural transformation, and models latent temporal evolution using exponential Fourier layers that naturally encode linear and nonlinear spectral dynamics. We systematically evaluate IS-FNO against baseline FNO and Koopman-based models on a range of benchmark PDEs, including the Michelson-Sivashinsky and Kuramoto-Sivashinsky equations (in one and two dimensions), as well as the integrable Korteweg-de Vries and Kadomtsev-Petviashvili equations. The results demonstrate that IS-FNO achieves lower short-term errors and substantially improved long-horizon stability in non-stiff regimes. For integrable systems, reduced IS-FNO variants that embed analytical scattering structure retain competitive long-term accuracy despite limited model capacity. Overall, this work shows that incorporating physical structure -- particularly reversibility and spectral evolution -- into neural operator design significantly enhances robustness and long-term predictive fidelity for nonlinear PDE dynamics.