# DeepSPoC: A Deep Learning Based Sequential Propagation of Chaos

arXiv: https://arxiv.org/abs/2408.16403 · HTML: https://mlsanity.com/paper/2408.16403/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 36 references that are in the map.
- First posted on arXiv 29 Aug 2024.

## Builds on (most cited first, 11 of 11)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Adaptive deep density approximation for Fokker-Planck equations](https://mlsanity.com/paper/2103.11181/) (2021)
- [Certified machine learning: A posteriori error estimation for physics-informed neural networks](https://mlsanity.com/paper/2203.17055/) (2022)
- [Adaptive deep density approximation for fractional Fokker-Planck equations](https://mlsanity.com/paper/2210.14402/) (2022)

## Abstract (arXiv)

Classical particle methods based on propagation of chaos (PoC) have been developed for solving mean-field stochastic differential equations and their associated nonlinear Fokker--Planck equations. However, direct PoC implementations are difficult to apply to high-dimensional problems because they require simulating and storing large numbers of interacting particles, often with high particle-particle interaction costs. Motivated by these limitations, we build on the recently proposed sequential propagation of chaos (SPoC) framework, which replaces the fully interacting particle system in PoC with a sequential interaction mechanism. Based on this structure, we present DeepSPoC, a neural particle method that embeds a neural density representation into the sequential particle dynamics. DeepSPoC simulates particles batch by batch, while the neural network represents the evolving empirical law and is substituted into the coefficients of the mean-field SDE, thereby replacing direct particle-particle interactions with particle-network interactions. In DeepSPoC, a recently developed normalizing flow model called KRnet is used to approximate the empirical measure of particles. Compared with direct particle implementations, DeepSPoC substantially reduces memory consumption and evaluates interaction terms more efficiently, thereby improving scalability for high-dimensional problems. We apply DeepSPoC to a wide range of mean-field equations and verify its effectiveness and computational advantages.