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2018
In order to give the functionals $$ J(y) = \int ^b_a F(x,y, y') dx $$ a domain of definition, we need to introduce suitable function spaces. First of all we require that the Lagrange function or Lagrangian, $$ F : [a, b] \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}, {\qquad }\text {is continuous.} $$ Here \([a, b] = \{
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In order to give the functionals $$ J(y) = \int ^b_a F(x,y, y') dx $$ a domain of definition, we need to introduce suitable function spaces. First of all we require that the Lagrange function or Lagrangian, $$ F : [a, b] \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}, {\qquad }\text {is continuous.} $$ Here \([a, b] = \{
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DeepXDE: A Deep Learning Library for Solving Differential Equations
SIAM Review, 2021Lu Lu, George E Karniadakis
exaly