variationally

Variationally is an adverb describing actions or results that arise from a variational approach—one that optimizes an objective by varying candidate solutions rather than solving a fixed set of equations directly.

In the calculus of variations and in many areas of physics, problems are formulated in terms of functionals, objects that assign numbers to functions. One seeks an extremal function by considering small variations and requiring the first variation to vanish, yielding equations such as the Euler–Lagrange equations. The outcome is often described as being obtained variationally, or via a variational principle, and is common in deriving equations of motion or energy minima.

In quantum mechanics and chemistry, the variational method uses a trial wavefunction and minimizes the expectation value of the Hamiltonian to bound ground-state energies. These procedures are described as variational when the result arises from this optimization over a chosen trial set.

In statistics and machine learning, variational methods replace intractable posteriors with tractable approximations by optimizing a lower bound on the marginal likelihood, a process called variational inference or variational Bayes. Training variational models, such as variational autoencoders, is likewise described as variationally learning the model.

Etymology: variational derives from variation, with the adverb variationally formed to describe methods or results linked to variation-based optimization.