# Lagrangian Flow Networks for Conservation Laws

arXiv: https://arxiv.org/abs/2305.16846 · HTML: https://mlsanity.com/paper/2305.16846/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 59 references that are in the map.
- First posted on arXiv 26 May 2023.

## Builds on (most cited first, 30 of 31)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [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)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Measuring the Effects of Data Parallelism on Neural Network Training](https://mlsanity.com/paper/1811.03600/) (2018)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Neural Flows: Efficient Alternative to Neural ODEs](https://mlsanity.com/paper/2110.13040/) (2021)
- [Neural Spatio-Temporal Point Processes](https://mlsanity.com/paper/2011.04583/) (2020)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Autoregressive Energy Machines](https://mlsanity.com/paper/1904.05626/) (2019)
- [Invertible DenseNets with Concatenated LipSwish](https://mlsanity.com/paper/2102.02694/) (2021)
- [Self-Consistent Velocity Matching of Probability Flows](https://mlsanity.com/paper/2301.13737/) (2023)
- [Semi-Conditional Normalizing Flows for Semi-Supervised Learning](https://mlsanity.com/paper/1905.00505/) (2019)
- [Jacobian Determinant of Normalizing Flows](https://mlsanity.com/paper/2102.06539/) (2021)
- [Mesh-free Eulerian Physics-Informed Neural Networks](https://mlsanity.com/paper/2206.01545/) (2022)

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

- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Injective flows for star-like manifolds](https://mlsanity.com/paper/2406.09116/) (2024)
- [Two-Parameter Flows for Learning Population Dynamics of Physical Systems](https://mlsanity.com/paper/2605.26285/) (2026)
- [Learning and Transferring Physical Models through Derivatives](https://mlsanity.com/paper/2505.01391/) (2025)

## Abstract (arXiv)

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.