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

TL;DR

This paper explores the use of monotonic Gaussian processes constrained by physical laws to improve predictions in materials science, especially with limited and noisy data, demonstrating reduced uncertainty and physical plausibility.

Contribution

It introduces a monotonic Gaussian process formulation constrained by physical rules, applied to materials datasets, showing benefits over regular Gaussian processes in physics-constrained machine learning.

Findings

Monotonic GP reduces posterior variance compared to regular GP.

Monotonicity holds in interpolation but fades in extrapolation.

Imposing monotonicity incurs a small accuracy cost.

Abstract

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.