# Gaussian Process Assisted Active Learning of Physical Laws

arXiv: https://arxiv.org/abs/1910.03120 · HTML: https://mlsanity.com/paper/1910.03120/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2019.

## Built on by (most cited first, 14 of 14)

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- [Learning Governing Equations of Unobserved States in Dynamical Systems](https://mlsanity.com/paper/2404.18572/) (2024)
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- [Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations](https://mlsanity.com/paper/2107.07040/) (2021)
- [PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equations](https://mlsanity.com/paper/2204.02583/) (2022)
- [Inverse Models for Estimating the Initial Condition of Spatio-Temporal Advection-Diffusion Processes](https://mlsanity.com/paper/2302.04134/) (2023)
- [Learning Collective Behaviors from Observation](https://mlsanity.com/paper/2311.00875/) (2023)
- [Data-Driven Model Selections of Second-Order Particle Dynamics via Integrating Gaussian Processes with Low-Dimensional Interacting Structures](https://mlsanity.com/paper/2311.00902/) (2023)
- [BLADE: Bayesian Langevin Active Discovery with Replica Exchange for Identification of Complex Systems](https://mlsanity.com/paper/2503.02983/) (2025)
- [Active Learning for Manifold Gaussian Process Regression](https://mlsanity.com/paper/2506.20928/) (2025)
- [Human-AI Synergy in Adaptive Active Learning for Continuous Lithium Carbonate Crystallization Optimization](https://mlsanity.com/paper/2507.19316/) (2025)
- [Data-driven Learning of Interaction Laws in Multispecies Particle Systems with Gaussian Processes: Convergence Theory and Applications](https://mlsanity.com/paper/2511.02053/) (2025)

## Abstract (arXiv)

In many areas of science and engineering, discovering the governing differential equations from the noisy experimental data is an essential challenge. It is also a critical step in understanding the physical phenomena and prediction of the future behaviors of the systems. However, in many cases, it is expensive or time-consuming to collect experimental data. This article provides an active learning approach to estimate the unknown differential equations accurately with reduced experimental data size. We propose an adaptive design criterion combining the D-optimality and the maximin space-filling criterion. In contrast to active learning for other regression models, the D-optimality here requires the unknown solution of the differential equations and derivatives of the solution. We estimate the Gaussian process (GP) regression models from the available experimental data and use them as the surrogates of these unknown solution functions. The derivatives of the estimated GP models are derived and used to substitute the derivatives of the solution. Variable selection-based regression methods are used to learn the differential equations from the experimental data. Through multiple case studies, we demonstrate the proposed approach outperforms the D-optimality and the maximin space-filling design alone in terms of model accuracy and data economy.