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Poisson's And Laplace's Equation01:25

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The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
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The electric field and electric potential are related to each other. If the electric field at various points in the region of interest is known, it can be used to calculate the electric potential difference between any two points. Similarly, if the electric potential is known for various points, then it is possible to calculate the electric field.
In general, regardless of whether the electric field is uniform, it points in the direction of decreasing potential because the force on a positive...
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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...
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Thermodynamic potentials are state functions that are extremely useful in analyzing a thermodynamic system. They have dimensions of energy. The four important thermodynamic potentials are internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. These thermodynamic potentials can be expressed using two of the following variables: pressure, volume, temperature, and entropy. These two variables are expressed as the rate of change of the thermodynamic potential with respect to other...
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Second Derivatives and Laplace Operator01:22

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The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
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Calculations of Electric Potential II01:27

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An electric dipole is a system of two equal but opposite charges, separated by a fixed distance. This system is used to model many real-world systems, including atomic and molecular interactions. One of these systems is the water molecule, but only under certain circumstances. These circumstances are met inside a microwave oven, where electric fields with alternating directions make the water molecules change orientation. This vibration is equivalent to heat at the molecular level.
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Updated: Jan 13, 2026

Computational Modeling of Retinal Neurons for Visual Prosthesis Research - Fundamental Approaches
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Neural operators for forward and inverse potential-density mappings in classical density functional theory.

Runtong Pan1, Xinyi Fang2, Kamyar Azizzadenesheli3

  • 1Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521, USA.

The Journal of Chemical Physics
|October 28, 2025
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Summary

Neural operators effectively model complex relationships in density functional theory. Fourier Neural Operator (FNO) and DeepONet variants show promise, with FNO excelling in predicting excess free energy for hard-rod fluids.

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Area of Science:

  • Computational Physics
  • Statistical Mechanics
  • Machine Learning

Background:

  • Neural operators offer a data-driven approach to model complex functional relationships.
  • Classical density functional theory (DFT) involves intricate mappings between physical quantities.

Purpose of the Study:

  • Evaluate neural operator architectures for learning relationships in one-dimensional hard-rod fluids.
  • Benchmark Deep Operator Network (DeepONet) and Fourier Neural Operator (FNO) against dense neural networks.

Main Methods:

  • Trained DeepONet and FNO variants on data from analytical solutions of hard-rod fluids.
  • Assessed interpolation and extrapolation capabilities using cross-validation.
  • Compared mean squared error and excess free energy prediction accuracy.

Main Results:

  • FNO demonstrated superior accuracy in predicting excess free energy, especially with squared ReLU activation.
  • GK-RMSCNN-DeepONet performed best among DeepONet variants.
  • The neural-operator mapping ρ ↦ c1 proved effective for solving density profiles, outperforming direct Vext ↦ ρ mapping for extrapolation.

Conclusions:

  • Neural operators, particularly FNO, are highly effective for modeling functional relationships in DFT.
  • The ρ ↦ c1 mapping offers a robust approach for density profile prediction, with advantages in extrapolation.
  • Specialized neural operator features enhance prediction accuracy and flexibility.