Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Maxwell-Boltzmann Distribution: Problem Solving01:20

Maxwell-Boltzmann Distribution: Problem Solving

2.3K
Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
2.3K
Ampere-Maxwell's Law: Problem-Solving01:17

Ampere-Maxwell's Law: Problem-Solving

1.4K
A parallel-plate capacitor with capacitance C, whose plates have area A and separation distance d, is connected to a resistor R and a battery of voltage V. The current starts to flow at t = 0. What is the displacement current between the capacitor plates at time t? From the properties of the capacitor, what is the corresponding real current?
To solve the problem, we can use the equations from the analysis of an RC circuit and Maxwell's version of Ampère's law.
For the first part of the...
1.4K
Ampere's Law: Problem-Solving01:31

Ampere's Law: Problem-Solving

3.4K
Ampere's law states that for any closed looped path, the line integral of the magnetic field along the path equals the vacuum permeability times the current enclosed in the loop. If the fingers of the right hand curl along the direction of the integration path, the current in the direction of the thumb is considered positive. The current opposite to the thumb direction is considered negative.
Specific steps need to be considered while calculating the symmetric magnetic field distribution...
3.4K
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

438
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
438
Maxwell's Equation Of Electromagnetism01:29

Maxwell's Equation Of Electromagnetism

3.3K
James Clerk Maxwell (1831–1879) was one of the major contributors to physics in the nineteenth century. Although he died young, he made major contributions to the development of the kinetic theory of gases, to the understanding of color vision, and to understanding the nature of Saturn's rings. He is probably best known for having combined existing knowledge on the laws of electricity and magnetism with his insights into a complete overarching electromagnetic theory, which is...
3.3K
Differential Form of Maxwell's Equations01:17

Differential Form of Maxwell's Equations

1.5K
James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and...
1.5K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Freeform Mode-Engineered Metasurfaces.

Nano letters·2026
Same author

3D nanolithography with metalens arrays and spatially adaptive illumination.

Nature·2025
Same author

A multi-agentic framework for real-time, autonomous freeform metasurface design.

Science advances·2025
Same author

Magnetically Tunable Polariton Cavities in van der Waals Heterostructures.

Nano letters·2025
Same author

Hydrogel-to-Aerogel Transitions in Polymer-Particle Hydrogels Expand the Wildfire Defense Window.

ACS applied materials & interfaces·2025
Same author

Roadmap for Optical Metasurfaces.

ACS photonics·2024
Same journal

In This Issue.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Correction for Otsuki et al., Extracellular sulfatases support cartilage homeostasis by regulating BMP and FGF signaling pathways.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Hive mind: Microbial communities and the making of memory.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Targets for disease modification in schizophrenia: New findings add to evidence for the involvement of the immune complement system.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Correction for Wang et al., The role of reduced aerosol masking from air pollutant emission reductions in recent global warming acceleration (2013-2023).

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Correction for Mishra, Ecology is not yet ready for AI-and why that matters.

Proceedings of the National Academy of Sciences of the United States of America·2026
See all related articles

Related Experiment Video

Updated: Apr 29, 2026

Deep Neural Networks for Image-Based Dietary Assessment
13:19

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

11.0K

Accurate and scalable deep Maxwell solvers.

Chenkai Mao1, Jonathan A Fan1

  • 1Department of Electrical Engineering, Stanford University, Stanford, CA 94305.

Proceedings of the National Academy of Sciences of the United States of America
|April 27, 2026
PubMed
Summary
This summary is machine-generated.

Neural networks combined with iterative algorithms can accurately solve complex partial differential equation (PDE) problems. This approach enables scalable and accurate multiphysics surrogate solvers for large-scale scientific challenges.

Keywords:
GMRESdomain decompositioniterative methodsneural operatorsurrogate models

Related Experiment Videos

Last Updated: Apr 29, 2026

Deep Neural Networks for Image-Based Dietary Assessment
13:19

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

11.0K

Area of Science:

  • Computational mathematics
  • Applied physics
  • Machine learning

Background:

  • Neural networks show potential for solving partial differential equations (PDEs).
  • Challenges remain in achieving high accuracy and scalability with neural network-based PDE solvers.
  • Integrating neural networks with iterative algorithms is an active area of research.

Purpose of the Study:

  • To develop accurate and scalable neural network surrogate solvers for partial differential equations (PDEs).
  • To demonstrate the effectiveness of combining neural network surrogates with iterative algorithms.
  • To enable the solution of complex, large-scale multiphysics problems.

Main Methods:

  • Developed a subdomain neural operator model supporting arbitrary Robin-type boundary conditions.
  • Utilized the model as a flexible preconditioner for iterative subdomain problem solving.
  • Constructed global coarse spaces for accelerated, large-scale PDE solving via multilevel domain decomposition.
  • Trained a single neural network on 2D Maxwell's equations for diverse problem instances.

Main Results:

  • Neural network surrogates combined with iterative algorithms accurately solve PDE problems across different scales, resolutions, and boundary conditions.
  • The subdomain neural operator model provides bounded accuracy for iterative subdomain solutions.
  • The approach enables accelerated, large-scale PDE problem solving.
  • A single trained network simulated large-scale problems with varying parameters (size, resolution, wavelength, dielectric distribution).
  • Demonstrated accurate inverse design of multiwavelength nanophotonic devices.

Conclusions:

  • Neural network surrogates integrated with iterative methods offer a promising path to accurate and scalable PDE solutions.
  • The developed subdomain neural operator model is flexible and effective for complex boundary conditions.
  • This framework facilitates efficient large-scale multiphysics simulations and inverse design applications.