Data-driven discovery of free energy function through functional derivatives

To determine which governing equations (GEs) can best describe complex nonlinear physical systems is an essential task in science and engineering, as simulations often fail if some parts of the GEs are unclear or misspecified. In this study, we propose a novel method of equation discovery based on the concept of the variational principle. Specifically, an unknown local function g(u) of the solution field variable u is introduced, and a functional G is defined as a volume integral of g(u). The unknown terms in the GE are expressed using functional derivatives δG/δu. These functional derivatives can be calculated by automatic differentiation using neural networks. Among the scientific machine learning methods, the proposed method has two fundamental advantages: generalization capability and interpretability. For the former, the function trained on one dataset can be applied to other datasets that are governed by the same equation. For the latter, this approach can be regarded as the discovery of Gibbs free energy, and the GE corresponds to the Cahn-Hilliard equation, which is formulated so that the field variables evolve as the total free energy of the system decreases. The validity of the proposed method has been confirmed through various types of problems, which involve the Burgers equation, the KdV equation, and the Navier-Stokes equations.