Due to this property, there should exist a corresponding Hamiltonian function, i.e., energy function, As such, the discrete gradient method has achieved great success. As shown above, a discrete gradient is defined in Definition 1. The target equations for this study are the differential equations with a certain geometric structure. A (null u)null v (16) with a matrix A (null u); hence, Eq. (15) is shown to be equivalent to null w This is our target equation in Eq. (1). This section provides the proofs of the Theorems in the main text.

Arabshahi, Singh, and Anandkumar (2018) propose a method for creating a dataset of symbolic mathematical equations for the tasks of symbolic equation verification and equation completion. Unfortunately, a dataset constructed using the method they propose will suffer from two serious flaws. First, the class of true equations that the procedure can generate will be very limited. Second, because true and false equations are generated in completely different ways, there are likely to be artifactual features that allow easy discrimination. Moreover, over the class of equations they consider, there is an extremely simple probabilistic procedure that solves the problem of equation verification with extremely high reliability. The usefulness of this problem in general as a testbed for AI systems is therefore doubtful.