# DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method

arXiv: https://arxiv.org/abs/2111.01356 · HTML: https://mlsanity.com/paper/2111.01356/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 76 references that are in the map.
- First posted on arXiv 2 Nov 2021.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [On the Expressive Power of Deep Learning: A Tensor Analysis](https://mlsanity.com/paper/1509.05009/) (2015)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Alternating the Population and Control Neural Networks to Solve High-Dimensional Stochastic Mean-Field Games](https://mlsanity.com/paper/2002.10113/) (2020)
- [Numerical Solution of Inverse Problems by Weak Adversarial Networks](https://mlsanity.com/paper/2002.11340/) (2020)
- [Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs](https://mlsanity.com/paper/1911.02109/) (2019)
- [Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation](https://mlsanity.com/paper/2105.11632/) (2021)
- [Deep regularization and direct training of the inner layers of Neural Networks with Kernel Flows](https://mlsanity.com/paper/2002.08335/) (2020)
- [Deep Multiscale Model Learning](https://mlsanity.com/paper/1806.04830/) (2018)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: I -- The Ergodic Case](https://mlsanity.com/paper/1907.05980/) (2019)
- [Global Convergence and Geometric Characterization of Slow to Fast Weight Evolution in Neural Network Training for Classifying Linearly Non-Separable Data](https://mlsanity.com/paper/2002.12563/) (2020)

## Built on by (most cited first, 9 of 9)

- [Mathematical analysis of singularities in the diffusion model under the submanifold assumption](https://mlsanity.com/paper/2301.07882/) (2023)
- [A DeepParticle method for learning and generating aggregation patterns in multi-dimensional Keller-Segel chemotaxis systems](https://mlsanity.com/paper/2209.00109/) (2022)
- [A variational neural network approach for glacier modelling with nonlinear rheology](https://mlsanity.com/paper/2209.02088/) (2022)
- [Accelerating hypersonic reentry simulations using deep learning-based hybridization (with guarantees)](https://mlsanity.com/paper/2209.13434/) (2022)
- [PODNO: Proper Orthogonal Decomposition Neural Operators](https://mlsanity.com/paper/2504.18513/) (2025)
- [DPOT: A DeepParticle method for Computation of Optimal Transport with convergence guarantee](https://mlsanity.com/paper/2506.23429/) (2025)
- [COFM: Consistent Optimal Transport Flow Matching via Partially Input Convex Neural Networks](https://mlsanity.com/paper/2511.06042/) (2025)
- [Deep Eigenspace Network for Parametric Non-self-adjoint Eigenvalue Problems](https://mlsanity.com/paper/2512.20058/) (2025)
- [On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures](https://mlsanity.com/paper/2605.21388/) (2026)

## Abstract (arXiv)

We introduce the so called DeepParticle method to learn and generate invariant measures of stochastic dynamical systems with physical parameters based on data computed from an interacting particle method (IPM). We utilize the expressiveness of deep neural networks (DNNs) to represent the transform of samples from a given input (source) distribution to an arbitrary target distribution, neither assuming distribution functions in closed form nor a finite state space for the samples. In training, we update the network weights to minimize a discrete Wasserstein distance between the input and target samples. To reduce computational cost, we propose an iterative divide-and-conquer (a mini-batch interior point) algorithm, to find the optimal transition matrix in the Wasserstein distance. We present numerical results to demonstrate the performance of our method for accelerating IPM computation of invariant measures of stochastic dynamical systems arising in computing reaction-diffusion front speeds through chaotic flows. The physical parameter is a large Pecl'et number reflecting the advection dominated regime of our interest.