# Stability of implicit neural networks for long-term forecasting in dynamical systems

arXiv: https://arxiv.org/abs/2305.17155 · HTML: https://mlsanity.com/paper/2305.17155/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 20 references that are in the map.
- First posted on arXiv 26 May 2023.

## Builds on (most cited first, 14 of 14)

- [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)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow](https://mlsanity.com/paper/1802.10123/) (2018)
- [Learning Dynamical Systems from Partial Observations](https://mlsanity.com/paper/1902.11136/) (2019)
- [IMEXnet: A Forward Stable Deep Neural Network](https://mlsanity.com/paper/1903.02639/) (2019)
- [Comparing recurrent and convolutional neural networks for predicting wave propagation](https://mlsanity.com/paper/2002.08981/) (2020)
- [Forward Stability of ResNet and Its Variants](https://mlsanity.com/paper/1811.09885/) (2018)
- [Robust learning with implicit residual networks](https://mlsanity.com/paper/1905.10479/) (2019)
- [Implicit Euler ODE Networks for Single-Image Dehazing](https://mlsanity.com/paper/2007.06443/) (2020)
- [Simulating Surface Wave Dynamics with Convolutional Networks](https://mlsanity.com/paper/2012.00718/) (2020)

## Abstract (arXiv)

Forecasting physical signals in long time range is among the most challenging tasks in Partial Differential Equations (PDEs) research. To circumvent limitations of traditional solvers, many different Deep Learning methods have been proposed. They are all based on auto-regressive methods and exhibit stability issues. Drawing inspiration from the stability property of implicit numerical schemes, we introduce a stable auto-regressive implicit neural network. We develop a theory based on the stability definition of schemes to ensure the stability in forecasting of this network. It leads us to introduce hard constraints on its weights and propagate the dynamics in the latent space. Our experimental results validate our stability property, and show improved results at long-term forecasting for two transports PDEs.