# Machine Learning of Linear Differential Equations using Gaussian Processes

## TL;DR

This paper introduces a probabilistic machine learning approach using Gaussian processes to discover and infer parameters of linear differential equations, including complex operators, from limited and noisy data.

## Contribution

It develops a novel method that adapts Gaussian process priors to various linear operators for effective parameter inference in differential equations.

## Key findings

- Successfully infers parameters from scarce data
- Handles noisy observations effectively
- Applies to diverse linear operators including fractional derivatives

## Abstract

This work leverages recent advances in probabilistic machine learning to discover conservation laws expressed by parametric linear equations. Such equations involve, but are not limited to, ordinary and partial differential, integro-differential, and fractional order operators. Here, Gaussian process priors are modified according to the particular form of such operators and are employed to infer parameters of the linear equations from scarce and possibly noisy observations. Such observations may come from experiments or "black-box" computer simulations.