Machine Learning of Linear Differential Equations using Gaussian Processes

This generality was demostrated using various bechmark problems with utterly different attributes along with an example application in functional genomics. Furthermore, the current methodology can be applied to inverse problems involving characterization of materials, tomography and electrophysiology, design of effective metamaterials, etc. The methodology can be straightforwardly generalized to address data with multiple levels of fidelity [24, 39] and equations with variable coefficients and complex geometries. Non-Gaussian and input-dependent noise models (e.g., student-t, heteroscedastic, etc.) [3] can also be accommodated. Moreover, systems of linear integro-differential equations can be addressed using multi-output Gaussian process regressions [40, 23, 22]. These scenarios are all feasible because they do not affect the key observation that any linear transformation of a Gaussian process is still a Gaussian process. In its current form, despite its generality regarding linear equations, the proposed framework cannot deal with nonlinear equations. However, some specific nonlinear operators can be addressed with extensions of the current framework by transforming such equations into systems of linear equations [41, 42] - albeit in high dimensions. In the end, the proposed methodology in this work, being essentially a regression technology, is suitable for resolving such high-dimensional problems.