D-CIPHER: Discovery of Closed-form Partial Differential Equations

TL;DR

D-CIPHER is a robust method for discovering a broad class of closed-form differential equations from data, overcoming limitations of existing approaches especially in noisy and sparse settings.

Contribution

It introduces D-CIPHER, a novel approach capable of uncovering general differential equations without strong assumptions, and a new optimization method, CoLLie, for efficient search.

Findings

Successfully discovers well-known differential equations beyond current methods.

Robust to measurement noise and infrequent sampling.

Demonstrates effectiveness on various real-world datasets.

Abstract

Closed-form differential equations, including partial differential equations and higher-order ordinary differential equations, are one of the most important tools used by scientists to model and better understand natural phenomena. Discovering these equations directly from data is challenging because it requires modeling relationships between various derivatives that are not observed in the data (equation-data mismatch) and it involves searching across a huge space of possible equations. Current approaches make strong assumptions about the form of the equation and thus fail to discover many well-known systems. Moreover, many of them resolve the equation-data mismatch by estimating the derivatives, which makes them inadequate for noisy and infrequently sampled systems. To this end, we propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations. We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently. Finally, we demonstrate empirically that it can discover many well-known equations that are beyond the capabilities of current methods.