# Lagrangian Density Space-Time Deep Neural Network Topology

arXiv: https://arxiv.org/abs/2207.12209 · HTML: https://mlsanity.com/paper/2207.12209/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 95 references that are in the map.
- First posted on arXiv 30 Jun 2022.

## Builds on (most cited first, 25 of 25)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model](https://mlsanity.com/paper/1911.08265/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Learning Dexterous In-Hand Manipulation](https://mlsanity.com/paper/1808.00177/) (2018)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Learning Hand-Eye Coordination for Robotic Grasping with Deep Learning and Large-Scale Data Collection](https://mlsanity.com/paper/1603.02199/) (2016)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [PHYRE: A New Benchmark for Physical Reasoning](https://mlsanity.com/paper/1908.05656/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [DScribe: Library of Descriptors for Machine Learning in Materials Science](https://mlsanity.com/paper/1904.08875/) (2019)
- [A General Framework for Structured Learning of Mechanical Systems](https://mlsanity.com/paper/1902.08705/) (2019)
- [WACSF - Weighted Atom-Centered Symmetry Functions as Descriptors in Machine Learning Potentials](https://mlsanity.com/paper/1712.05861/) (2017)
- [Space-time deep neural network approximations for high-dimensional partial differential equations](https://mlsanity.com/paper/2006.02199/) (2020)
- [Machine learning and serving of discrete field theories -- when artificial intelligence meets the discrete universe](https://mlsanity.com/paper/1910.10147/) (2019)

## Abstract (arXiv)

As a network-based functional approximator, we have proposed a "Lagrangian Density Space-Time Deep Neural Networks" (LDDNN) topology. It is qualified for unsupervised training and learning to predict the dynamics of underlying physical science governed phenomena. The prototypical network respects the fundamental conservation laws of nature through the succinctly described Lagrangian and Hamiltonian density of the system by a given data-set of generalized nonlinear partial differential equations. The objective is to parameterize the Lagrangian density over a neural network and directly learn from it through data instead of hand-crafting an exact time-dependent "Action solution" of Lagrangian density for the physical system. With this novel approach, can understand and open up the information inference aspect of the "Black-box deep machine learning representation" for the physical dynamics of nature by constructing custom-tailored network interconnect topologies, activation, and loss/cost functions based on the underlying physical differential operators. This article will discuss statistical physics interpretation of neural networks in the Lagrangian and Hamiltonian domains.