Enforcing exact boundary and initial conditions in the deep mixed residual method

Understand

In theory, boundary and initial conditions are important for the wellposedness of partial differential equations (PDEs).

• Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method and finite element method.
• Recent years have witnessed growing interests in solving PDEs by deep neural networks (DNNs), especially in the high-dimensional case.
• However, in the generic situation, a careful literature review shows that boundary conditions cannot be enforced exactly for DNNs, which inevitably leads to a modeling error.