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1. Variational Pictures

   https://doi.org/10.1007/978-3-319-91376-6_9  

   Erwig, M.; Smeltzer, K. (May 2018, Int. Conf. on the Theory and Application of Diagrams)

   Diagrams and pictures are a powerful medium to communicate ideas, designs, and art. However, authors of pictures are forced to use rudimentary and ad hoc techniques in managing multiple variants of their creations, such as copying and renaming files or abusing layers in an advanced graphical editing tool. We propose a model of variational pictures as a basis for the design of editors and other tools for managing variation in pictures. This model enjoys a number of theoretical properties that support exploratory graphical design and can help systematize picture creators' workflows. 

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2. Variational Convexity of Functions and Variational Sufficiency in Optimization

   https://doi.org/10.1137/22M1519250  

   Khanh, Pham Duy; Mordukhovich, Boris S.; Phat, Vo Thanh (June 2023, SIAM Journal on Optimization)
3. Denoising Diffusion Variational Inference: Diffusion Models as Expressive Variational Posteriors

   Piriyakulkij, Wasu; Wang, Yingheng; Kuleshov, Volodymyr (January 2025, AAAI)
4. Variational microcanonical estimator

   https://doi.org/10.1103/PhysRevResearch.5.033224  

   Pollock, Klée; Orth, Peter P.; Iadecola, Thomas (September 2023, Physical Review Research)
5. Variational Surface Cutting

   Sharp, Nicholas; Crane, Keenan (July 2018, ACM transactions on graphics)

   This paper develops a global variational approach to cutting curved surfaces so that they can be flattened into the plane with low metric distortion. Such cuts are a critical component in a variety of algorithms that seek to parameterize surfaces over flat domains, or fabricate structures from flat materials. Rather than evaluate the quality of a cut solely based on properties of the curve itself (e.g., its length or curvature), we formulate a flow that directly optimizes the distortion induced by cutting and flattening. Notably, we do not have to explicitly parameterize the surface in order to evaluate the cost of a cut, but can instead integrate a simple evolution equation defined on the cut curve itself. We arrive at this flow via a novel application of shape derivatives to the Yamabe equation from conformal geometry. We then develop an Eulerian numerical integrator on triangulated surfaces, which does not restrict cuts to mesh edges and can incorporate user-defined data such as importance or occlusion. The resulting cut curves can be used to drive distortion to arbitrarily low levels, and have a very different character from cuts obtained via purely discrete formulations. We briefly explore potential applications to computational design, as well as connections to space filling curves and the problem of uniform heat distribution. 

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