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Physics-constrained machine learning is emerging as an important topic in thefield of machine learning for physics. One of the most significant advantagesof incorporating physics constraints into machine learning methods is that theresulting model requires significantly less data to train. By incorporatingphysical rules into the machine learning formulation itself, the predictionsare expected to be physically plausible. Gaussian process (GP) is perhaps oneof the most common methods in machine learning for small datasets. In thispaper, we investigate the possibility of constraining a GP formulation withmonotonicity on three different material datasets, where one experimental andtwo computational datasets are used. The monotonic GP is compared against theregular GP, where a significant reduction in the posterior variance isobserved. The monotonic GP is strictly monotonic in the interpolation regime,but in the extrapolation regime, the monotonic effect starts fading away as onegoes beyond the training dataset. Imposing monotonicity on the GP comes at asmall accuracy cost, compared to the regular GP. The monotonic GP is perhapsmost useful in applications where data is scarce and noisy, and monotonicity issupported by strong physical evidence.

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