Data-Driven Discovery of PDEs via the Adjoint Method
This addresses the challenge of data-driven PDE discovery for researchers in computational science and engineering, representing a novel method for a known bottleneck rather than an incremental improvement.
The authors tackled the problem of discovering governing partial differential equations (PDEs) from data by proposing an adjoint-based method that formulates a PDE-constrained optimization problem, achieving identification up to machine accuracy and outperforming an existing method (PDE-FIND) on large datasets.
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.