Data-Driven Discovery of PDEs via the Adjoint Method
Mohsen Sadr Tony Tohme Kamal Youcef-Toumi
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.