# SMS: Spiking Marching Scheme for Efficient Long Time Integration of Differential Equations

arXiv: https://arxiv.org/abs/2211.09928 · HTML: https://mlsanity.com/paper/2211.09928/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 44 references that are in the map.
- First posted on arXiv 17 Nov 2022.

## Builds on (most cited first, 16 of 16)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Class-Balanced Loss Based on Effective Number of Samples](https://mlsanity.com/paper/1901.05555/) (2019)
- [Surrogate Gradient Learning in Spiking Neural Networks](https://mlsanity.com/paper/1901.09948/) (2019)
- [Going Deeper With Directly-Trained Larger Spiking Neural Networks](https://mlsanity.com/paper/2011.05280/) (2020)
- [Deep Learning in Spiking Neural Networks](https://mlsanity.com/paper/1804.08150/) (2018)
- [Training Spiking Neural Networks Using Lessons From Deep Learning](https://mlsanity.com/paper/2109.12894/) (2021)
- [Training Deep Spiking Neural Networks using Backpropagation](https://mlsanity.com/paper/1608.08782/) (2016)
- [Enabling Deep Spiking Neural Networks with Hybrid Conversion and Spike Timing Dependent Backpropagation](https://mlsanity.com/paper/2005.01807/) (2020)
- [Optimal Conversion of Conventional Artificial Neural Networks to Spiking Neural Networks](https://mlsanity.com/paper/2103.00476/) (2021)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Efficient Neuromorphic Signal Processing with Loihi 2](https://mlsanity.com/paper/2111.03746/) (2021)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [Beyond Classification: Directly Training Spiking Neural Networks for Semantic Segmentation](https://mlsanity.com/paper/2110.07742/) (2021)
- [Conversion of Artificial Recurrent Neural Networks to Spiking Neural Networks for Low-power Neuromorphic Hardware](https://mlsanity.com/paper/1601.04187/) (2016)
- [Spiking Neural Operators for Scientific Machine Learning](https://mlsanity.com/paper/2205.10130/) (2022)

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

- [Operator Learning for Reconstructing Flow Fields from Sparse Measurements: an Energy Transformer Approach](https://mlsanity.com/paper/2501.08339/) (2025)
- [Understanding and Mitigating Extrapolation Failures in Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.09478/) (2023)
- [Neuroscience inspired scientific machine learning (Part-1): Variable spiking neuron for regression](https://mlsanity.com/paper/2311.09267/) (2023)
- [Neuroscience inspired scientific machine learning (Part-2): Variable spiking wavelet neural operator](https://mlsanity.com/paper/2311.14710/) (2023)
- [Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach](https://mlsanity.com/paper/2605.23712/) (2026)

## Abstract (arXiv)

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for long time integration of time-dependent Ordinary and Partial Differential Equations (ODEs, PDEs). The core element of the method is a SNN, trained to use spike-encoded information about the solution at previous timesteps to predict spike-encoded information at the next timestep. After the network has been trained, it operates as an explicit numerical scheme that can be used to compute the solution at future timesteps, given a spike-encoded initial condition. A decoder is used to transform the evolved spiking-encoded solution back to function values. We present results from numerical experiments of using the proposed method for ODEs and PDEs of varying complexity.