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Related Concept Videos

Potential Energy00:52

Potential Energy

The energy stored by a structure and location of matter in space is called potential energy. For instance, raising a kettlebell changes its spatial location and increases its potential energy. Similarly, a stretched rubber band contains potential energy which, under certain conditions, can be converted into other forms of energy, such as kinetic energy.
Chemical bonds that form attractive forces between atoms also contain potential energy, called chemical energy. When a chemical reaction...
Elastic Potential Energy01:01

Elastic Potential Energy

Elastic potential energy is the energy stored as a result of the deformation of an elastic object, such as the stretching of a spring. An object is elastic if it returns to its original shape and size after being deformed. 
Potential energy is also associated with the elastic force exerted by an ideal spring. The work done by this force can be represented as a change in the elastic potential energy of the spring. Thus, the work done by a perfectly elastic spring, in one dimension, depends only...
Force and Potential Energy in One Dimension01:13

Force and Potential Energy in One Dimension

Force can be calculated from the expression for potential energy, which is a function of position. The component of a conservative force, in a particular direction, equals the negative of the derivative of the corresponding potential energy with respect to the displacement in that direction. For regions where potential energy changes rapidly with displacement, the work done and force is maximum. Also, when force is applied along the positive coordinate axis, the potential energy decreases with...
Force and Potential Energy in Three Dimensions01:04

Force and Potential Energy in Three Dimensions

Consider a particle moving under the action of a conservative force that has components along each coordinate axis. Each component of force is a function of the coordinates. The potential energy function U is also a function of all three spatial coordinates. Force in one dimension can be written as the negative ratio of potential energy change to the displacement along that coordinate. For minimal displacement, the ratios become derivatives. If a function has many variables, the derivative only...
Potential Energy01:09

Potential Energy

A conservative force, such as a gravitational or elastic force, gives the body the capacity to do work. This capacity, measured as the potential energy, depends on the body's location or “position” relative to a fixed reference position or datum. The gravitational potential energy is considered zero at the reference point. Suppose a body is located at some vertical distance above a fixed horizontal reference or datum. In that case, the weight of the body has positive gravitational potential...
Parametric Surfaces01:30

Parametric Surfaces

A parametric surface in three-dimensional space is defined through a vector-valued function\begin{equation*}\mathbf{r}(u, v) = x(u, v)\mathbf{i} + y(u, v)\mathbf{j} + z(u, v)\mathbf{k}\end{equation*}where u and v are parameters within a specified domain D in the uv-plane. The functions x(u, v), y(u, v), and z(u, v) define the coordinates of points on the surface. As u and v vary over D, the position vector r(u, v) traces a continuous surface in space. This parametric representation is essential...

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Statistical Modelling of Cortical Connectivity Using Non-invasive Electroencephalograms
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A random-sampling high dimensional model representation neural network for building potential energy surfaces.

Sergei Manzhos1, Tucker Carrington

  • 1Département de chimie, Université de Montréal, Case postale 6128, succursale Centre-ville, Montréal (Québec) H3C 3J7, Canada. sergei.manzhos@umontreal.de

The Journal of Chemical Physics
|September 13, 2006
PubMed
Summary

This study integrates high dimensional model representation (HDMR) with neural networks (NNs) to construct accurate multidimensional potentials. The method enables efficient quantum dynamics calculations by building potentials from low-dimensional fits.

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Area of Science:

  • Computational Chemistry
  • Quantum Mechanics
  • Machine Learning

Background:

  • Accurate multidimensional potentials are crucial for quantum dynamics simulations.
  • Traditional methods for potential energy surface construction can be computationally expensive and complex.
  • High dimensional model representation (HDMR) offers a way to decompose complex potentials into simpler, lower-dimensional terms.

Purpose of the Study:

  • To develop an effective method for building accurate multidimensional potentials.
  • To facilitate quantum dynamics calculations through an efficient potential representation.
  • To leverage neural networks (NNs) within the HDMR framework.

Main Methods:

  • Combining the high dimensional model representation (HDMR) concept with neural network (NN) fitting.
  • Using NNs to represent HDMR component functions, minimizing errors iteratively.
  • Constructing the final potential as a sum of terms, each dependent on a subset of coordinates.

Main Results:

  • Demonstrated the feasibility of determining accurate many-dimensional potentials from low-dimensional fits.
  • Successfully employed NNs to create high-order HDMR component functions.
  • Showcased that the number of available potential data points dictates the required HDMR order.

Conclusions:

  • The integrated HDMR-NN approach provides an effective strategy for constructing multidimensional potentials.
  • The resulting potential form is amenable to quantum dynamics calculations.
  • The methodology offers a scalable and efficient alternative for potential energy surface generation.