Variational

Variational refers to methods and principles that involve varying functions or functionals to obtain optimal or stationary quantities. The common thread is the optimization of a functional, a mapping from a function space to real numbers, rather than the optimization of a finite-dimensional objective.

In mathematics and physics, variational methods derive equations of motion or field equations by requiring a functional, such as the action or energy, to be stationary under small variations of the function. The Euler–Lagrange equation is a central result. In classical mechanics, the action integral stationary condition yields dynamics; in optics, Fermat’s principle selects paths of stationary optical length.

In numerical analysis, variational formulations recast differential equations as optimization problems for functionals, enabling approximate solutions by finite-dimensional function spaces. Finite element methods, for example, rely on weak (variational) formulations to discretize partial differential equations.

In statistics and machine learning, variational inference uses a family of approximating distributions to approximate a complex posterior by minimizing a divergence to the true posterior. This yields tractable optimization problems for large models. Variational autoencoders extend this idea to neural networks, parameterizing encoder and decoder distributions and optimizing a lower bound on the marginal likelihood via stochastic gradient methods.

In quantum mechanics, the variational method provides upper bounds on ground-state energies by optimizing the expectation value of the Hamiltonian over trial wavefunctions, a standard tool in quantum chemistry. Variational principles thus span theoretical and applied disciplines, offering a unifying approach based on evaluating functionals through variation.