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Boundary Conditions: Lossless Lines01:21

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Consider a single-phase, two-wire, lossless transmission line terminated by an impedance at the receiving end and a source with Thevenin voltage and impedance at the sending end. The line, with length, has a surge impedance and wave velocity determined by the line's inductance and capacitance.
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For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
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Learning Only on Boundaries: A Physics-Informed Neural Operator for Solving Parametric Partial Differential

Zhiwei Fang1, Sifan Wang2, Paris Perdikaris3

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This study introduces a new physics-informed neural operator method for solving parameterized boundary value problems. It efficiently trains on domain boundaries, reducing data needs and enabling solutions for unbounded problems.

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Area of Science:

  • Computational mathematics
  • Applied physics
  • Machine learning

Background:

  • Deep learning surrogates and neural operators show potential for solving partial differential equations (PDEs).
  • Existing methods often require extensive training data and are restricted to bounded domains.
  • Physics-informed neural networks (PINNs) and neural operators face limitations with unbounded problems.

Purpose of the Study:

  • To develop a novel physics-informed neural operator method for solving parameterized boundary value problems.
  • To overcome limitations of existing methods regarding data requirements and domain boundedness.
  • To enable efficient training and application to complex and unbounded problems.

Main Methods:

  • Reformulating PDEs into boundary integral equations (BIEs).
  • Training neural operator networks solely on the boundary of the domain.
  • Reducing sample point requirements from O(Nd) to O(Nd-1) for dimension d.

Main Results:

  • Significant acceleration of the training process due to reduced sample points.
  • Successful handling of unbounded problems, a limitation for existing methods.
  • Demonstrated effectiveness for parameterized complex geometries and unbounded scenarios.

Conclusions:

  • The proposed physics-informed neural operator method offers an efficient and versatile approach to solving parameterized boundary value problems.
  • This method significantly reduces data requirements and expands applicability to unbounded domains.
  • It represents a promising advancement in the field of scientific machine learning for PDEs.