# Monotonic Gaussian process for physics-constrained machine learning with materials science applications

arXiv: https://arxiv.org/abs/2209.00628 · HTML: https://mlsanity.com/paper/2209.00628/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 50 references that are in the map.
- First posted on arXiv 31 Aug 2022.

## Builds on (most cited first, 12 of 12)

- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Gaussian processes with linear operator inequality constraints](https://mlsanity.com/paper/1901.03134/) (2019)
- [Linearly Constrained Gaussian Processes with Boundary Conditions](https://mlsanity.com/paper/2002.00818/) (2020)
- [Multi-fidelity machine-learning with uncertainty quantification and Bayesian optimization for materials design: Application to ternary random alloys](https://mlsanity.com/paper/2006.00139/) (2020)
- [Nonnegativity-Enforced Gaussian Process Regression](https://mlsanity.com/paper/2004.04632/) (2020)
- [Monotone Function Estimation for Computer Experiments](https://mlsanity.com/paper/1309.3802/) (2013)
- [Solving stochastic inverse problems for property-structure linkages using data-consistent inversion and machine learning](https://mlsanity.com/paper/2010.03603/) (2020)
- [Monotonic Gaussian Process Flow](https://mlsanity.com/paper/1905.12930/) (2019)
- [When Bifidelity Meets CoKriging: An Efficient Physics-Informed Multifidelity Method](https://mlsanity.com/paper/1812.02919/) (2018)

## Built on by (most cited first, 2 of 2)

- [Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial](https://mlsanity.com/paper/2305.04933/) (2023)
- [Featuremetric benchmarking: Quantum computer benchmarks based on circuit features](https://mlsanity.com/paper/2504.12575/) (2025)

## Abstract (arXiv)

Physics-constrained machine learning is emerging as an important topic in the field of machine learning for physics. One of the most significant advantages of incorporating physics constraints into machine learning methods is that the resulting model requires significantly less data to train. By incorporating physical rules into the machine learning formulation itself, the predictions are expected to be physically plausible. Gaussian process (GP) is perhaps one of the most common methods in machine learning for small datasets. In this paper, we investigate the possibility of constraining a GP formulation with monotonicity on three different material datasets, where one experimental and two computational datasets are used. The monotonic GP is compared against the regular GP, where a significant reduction in the posterior variance is observed. The monotonic GP is strictly monotonic in the interpolation regime, but in the extrapolation regime, the monotonic effect starts fading away as one goes beyond the training dataset. Imposing monotonicity on the GP comes at a small accuracy cost, compared to the regular GP. The monotonic GP is perhaps most useful in applications where data is scarce and noisy, and monotonicity is supported by strong physical evidence.