# Gaussian process learning with flow map refinement for parameter estimation in dynamical systems

arXiv: https://arxiv.org/abs/2608.22324 · HTML: https://mlsanity.com/paper/2608.22324/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 32 references that are in the map.
- First posted on arXiv 23 Aug 2026.

## Builds on (most cited first, 10 of 10)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learning](https://mlsanity.com/paper/2403.09110/) (2024)
- [Bayesian operator inference for data-driven reduced-order modeling](https://mlsanity.com/paper/2204.10829/) (2022)
- [Fast Gaussian Process Based Gradient Matching for Parameter Identification in Systems of Nonlinear ODEs](https://mlsanity.com/paper/1804.04378/) (2018)
- [Symplectic Gaussian Process Regression of Hamiltonian Flow Maps](https://mlsanity.com/paper/2009.05569/) (2020)
- [Gaussian process learning of nonlinear dynamics](https://mlsanity.com/paper/2312.12193/) (2023)

## Abstract (arXiv)

Parameter estimation is a central task in data-driven learning of dynamical systems. It aims to recover the underlying physical parameters from observed time-series data, thereby providing interpretable insights into the physical mechanisms governing the system. Gradient/derivative matching methods based on Gaussian process provide an efficient way to perform parameter estimation. Those methods avoid repeated numerical integration and enforce local derivative consistency. However, such local matching may result in global inconsistency with the governing flow map, particularly under scarce and noisy observations. To address this limitation, we propose a framework based on Gaussian process learning with flow map refinement (GPL-FMR), a two-stage parameter estimation framework. The first stage is based on Gaussian process learning algorithm and the posterior obtained from which is transferred as an informative prior to the second stage based on flow-map refinement. The second stage further improves the parameter estimation via optimisation based on global dynamical constraints. We demonstrate and analyse its performance on multiple numerical examples, including the Van der Pol oscillator, the Lotka-Volterra model, and the Lorenz-63 system. The results show that the proposed framework consistently improves parameter estimation accuracy, particularly under scarce and noisy observations.