# Weak Form Generalized Hamiltonian Learning

arXiv: https://arxiv.org/abs/2104.05096 · HTML: https://mlsanity.com/paper/2104.05096/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 34 references that are in the map.
- First posted on arXiv 11 Apr 2021.

## Builds on (most cited first, 14 of 14)

- [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)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Synthesis of recurrent neural networks for dynamical system simulation](https://mlsanity.com/paper/1512.05702/) (2015)
- [Discovering conservation laws from data for control](https://mlsanity.com/paper/1811.00961/) (2018)

## Built on by (most cited first, 8 of 8)

- [Amortized Reparametrization: Efficient and Scalable Variational Inference for Latent SDEs](https://mlsanity.com/paper/2312.10550/) (2023)
- [FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities](https://mlsanity.com/paper/2210.00272/) (2022)
- [Physics perception in sloshing scenes with guaranteed thermodynamic consistency](https://mlsanity.com/paper/2106.13301/) (2021)
- [Learning Neural Hamiltonian Dynamics: A Methodological Overview](https://mlsanity.com/paper/2203.00128/) (2022)
- [Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs](https://mlsanity.com/paper/2205.12550/) (2022)
- [Modeling Nonlinear Dynamics in Continuous Time with Inductive Biases on Decay Rates and/or Frequencies](https://mlsanity.com/paper/2212.13033/) (2022)
- [Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs](https://mlsanity.com/paper/2402.09018/) (2024)
- [Mesh Field Theory: Port-Hamiltonian Formulation of Mesh-Based Physics](https://mlsanity.com/paper/2605.00394/) (2026)

## Abstract (arXiv)

We present a method for learning generalized Hamiltonian decompositions of ordinary differential equations given a set of noisy time series measurements. Our method simultaneously learns a continuous time model and a scalar energy function for a general dynamical system. Learning predictive models in this form allows one to place strong, high-level, physics inspired priors onto the form of the learnt governing equations for general dynamical systems. Moreover, having shown how our method extends and unifies some previous work in deep learning with physics inspired priors, we present a novel method for learning continuous time models from the weak form of the governing equations which is less computationally taxing than standard adjoint methods.