# Lagrangian dual framework for conservative neural network solutions of kinetic equations

arXiv: https://arxiv.org/abs/2106.12147 · HTML: https://mlsanity.com/paper/2106.12147/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 23 Jun 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (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)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Imposing Hard Constraints on Deep Networks: Promises and Limitations](https://mlsanity.com/paper/1706.02025/) (2017)
- [Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach](https://mlsanity.com/paper/1911.09843/) (2019)
- [Deep Neural Network Approach to Forward-Inverse Problems](https://mlsanity.com/paper/1907.12925/) (2019)
- [Deep Ritz revisited](https://mlsanity.com/paper/1912.03937/) (2019)
- [The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach](https://mlsanity.com/paper/2009.13280/) (2020)

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

- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [Achieving Constraints in Neural Networks: A Stochastic Augmented Lagrangian Approach](https://mlsanity.com/paper/2310.16647/) (2023)
- [WANCO: Weak Adversarial Networks for Constrained Optimization problems](https://mlsanity.com/paper/2407.03647/) (2024)
- [A Two-Stage Training Method for Modeling Constrained Systems With Neural Networks](https://mlsanity.com/paper/2403.02730/) (2024)
- [Sample-wise Constrained Learning via a Sequential Penalty Approach with Applications in Image Processing](https://mlsanity.com/paper/2601.16812/) (2026)

## Abstract (arXiv)

In this paper, we propose a novel conservative formulation for solving kinetic equations via neural networks. More precisely, we formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. The constraints are relaxed toward the residual loss function by the Lagrangian duality. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions in terms of errors and the conservation laws, for the kinetic Fokker-Planck equation and the homogeneous Boltzmann equation.