Data-Driven Discovery of PDEs via the Adjoint Method

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 the optimization problem that minimizes the error of 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, for a family of parameterized and nonlinear PDEs, we show how the corresponding adjoint equations can be derived. Here, we show that given smooth data set, the proposed adjoint method can recover the true PDE up to machine accuracy. However, in the presence of noise, the accuracy of the adjoint method becomes comparable to the famous PDE Functional Identification of Nonlinear Dynamics method known as PDE-FIND (Rudy et al., 2017). Even though the presented 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.