Can physics-informed neural networks beat the finite element method?

Partial differential equations (PDEs) play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of PDEs. So far, physics-informed neural networks and the finite element method have mainly been…

• N
  Nvidia

  Awards: P6000, Quadro P6000
• WT
  Wellcome Trust

  Awards: 215733/Z/19/Z, 221633/Z/20/Z
• AT
  Alan Turing Institute
• LT
  Leverhulme Trust
• EC
  European Commission

  Award: 777826
• EA
  Engineering and Physical Sciences Research Council

  Awards: EP/T017961/1, EP/N014588/1, EP/T003553/1, EP/S026045/1, EP/T017961/1, EP/N014588/1, EP/V029428/1, EP/V029428/1, EP/S026045/1, EP/S515334/1