Data-Driven Discovery of PDEs via the Adjoint Method

TL;DR

This paper introduces an adjoint-based method for discovering PDEs from data, leveraging variational calculus to efficiently compute gradients, and demonstrates superior performance over existing methods like PDE-FIND on various datasets.

Contribution

The paper presents a novel adjoint method for PDE discovery that improves gradient computation and outperforms existing techniques on large and noisy datasets.

Findings

Accurately identifies PDE form up to machine precision.

Outperforms PDE-FIND on large and noisy data sets.

Uses variational calculus for efficient gradient computation.

Abstract

In this work, we present an adjoint-based method for discovering the underlying governing partial differential equations (PDEs) given data. The idea is to consider a parameterized PDE in a general form and formulate a PDE-constrained optimization problem aimed at minimizing the error of the PDE solution from data. Using variational calculus, we obtain an evolution equation for the Lagrange multipliers (adjoint equations) allowing us to compute the gradient of the objective function with respect to the parameters of PDEs given data in a straightforward manner. In particular, we consider a family of parameterized PDEs encompassing linear, nonlinear, and spatial derivative candidate terms, and elegantly derive the corresponding adjoint equations. We show the efficacy of the proposed approach in identifying the form of the PDE up to machine accuracy, enabling the accurate discovery of PDEs from data. We also compare its performance with the famous PDE Functional Identification of Nonlinear Dynamics method known as PDE-FIND (Rudy et al., 2017), on both smooth and noisy data sets. Even though the proposed adjoint method relies on forward/backward solvers, it outperforms PDE-FIND for large data sets thanks to the analytic expressions for gradients of the cost function with respect to each PDE parameter.