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Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

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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.
Consider a scalar function. The curl of its...
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Differential Form of Maxwell's Equations01:17

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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...
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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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When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the appropriate kinematic equations to solve for the unknowns. Either one or two kinematic equations are needed to solve for the unknowns, depending on the known and unknown quantities. Generally, the number of equations required is the same as the number of unknown quantities in the given example. Two-body pursuit problems always require two...
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Bernoulli's Equation: Problem Solving01:16

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A Venturi meter is essential for measuring fluid flow rates in pipelines. It utilizes the relationship between fluid velocity and pressure described by Bernoulli's equation. When installed in a sewage system, the Venturi meter accurately determines the wastewater flow rate by measuring pressure differences.
The first step is to compute the cross-sectional areas of the pipe and the Venturi throat to analyze the pressure difference indicated by the pressure gauge. Next, the continuity...
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Newtonian fluids exhibit a constant viscosity, meaning their shear stress and shear strain rate are directly proportional. This property ensures a predictable and stable response to applied forces, maintaining a linear relationship between force and flow. Examples include water, air, and light oils, consistently demonstrating this proportional behavior regardless of external conditions.
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Related Experiment Video

Updated: Oct 18, 2025

Author Spotlight: Computing the Effects of a Local Radiofrequency Hyperthermia Intervention on Tumor Biomechanics
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Author Spotlight: Computing the Effects of a Local Radiofrequency Hyperthermia Intervention on Tumor Biomechanics

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Learning the solution operator of parametric partial differential equations with physics-informed DeepONets.

Sifan Wang1, Hanwen Wang1, Paris Perdikaris2

  • 1Graduate Group in Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, PA 19104, USA.

Science Advances
|September 29, 2021
PubMed
Summary
This summary is machine-generated.

Physics-informed DeepONets offer a novel deep learning approach for solving partial differential equations (PDEs). This method rapidly predicts PDE solutions without paired data, accelerating scientific discovery.

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Area of Science:

  • Mathematics
  • Computer Science
  • Physics

Background:

  • Partial differential equations (PDEs) are fundamental in science and engineering.
  • Solving PDEs often involves computationally intensive methods, especially for multiple scenarios.
  • Existing methods face challenges in modeling complex nonlinear and nonequilibrium processes.

Purpose of the Study:

  • Introduce a novel deep learning framework, physics-informed DeepONets.
  • Enable learning of solution operators for arbitrary PDEs.
  • Overcome limitations of traditional PDE solvers.

Main Methods:

  • Developed a deep learning framework integrating physical laws into neural networks.
  • Trained DeepONets to learn the mapping from PDE parameters to solutions.
  • Demonstrated capability without requiring paired input-output training data.

Main Results:

  • Physics-informed DeepONets effectively learn PDE solution operators.
  • Achieved prediction speeds up to three orders of magnitude faster than conventional solvers.
  • Showcased applicability to various types of parametric PDEs.

Conclusions:

  • Physics-informed DeepONets present a paradigm shift in PDE modeling and simulation.
  • This framework accelerates the investigation of complex dynamic processes.
  • Enables efficient exploration of nonlinear and nonequilibrium phenomena.