Potsentials

Potsentials are a class of potential-like scalar fields used to model energy landscapes in mathematics, physics, and engineering. They are defined as modular combinations of simpler base components, allowing a flexible and tunable shaping of the landscape on a given domain.

Formally, a potsental V is typically constructed as V(x) = sum_{k=1}^m w_k ψ_k(x), where ψ_k are base potential shapes drawn from a prescribed family (for example quadratic wells, barrier terms, sinusoidal waves, or Gaussian wells) and w_k are adjustable coefficients. More generally, potsentials may be formed by composing base components through nonnegative multipliers or by piecewise definitions to reflect domain constraints. The design aims to produce desirable features such as local minima, ridges, and plateaus that guide dynamics or optimization.

Domains include continuous spaces (R^n) and discrete structures such as graphs. In discrete settings, ψ_k are defined on nodes or edges and V assigns costs to states. The resulting landscapes support both analytical study and numerical optimization, with differentiability and smoothness depending on the chosen base components.

Applications of potsentials span robotics, optimization, machine learning, and physical sciences. In robotics, potsentials inform potential-field methods to steer agents toward goals while repelling obstacles. In optimization and ML, they serve as flexible loss or energy landscapes for training or regularization. In physics and chemistry, they model complex energy surfaces by combining simple interaction terms.

Advantages include modularity and interpretability of components; limitations include sensitivity to parameter choices, potential for unwanted local minima, and computational challenges in high dimensions. See also potential, energy landscape, potential field, and Morse theory.