Deep Ritz Method [Weinan E & Bing Yu (2018)]

Equivalent formulation: variational PINN [Kharazmi, Zhang and Karniadakis (2021)]

\[ \min_{u\in H} \int_{\Omega} \left(\frac{1}{2} |\nabla u(x)|^2 - f(x)u(x) \right) \mathrm{d}x \iff \Delta u(x) = f(x) \]

main idea: replace the trial function by a neural network \(u(x;\theta)\), search in \(\theta\in\mathbb{R}^d\).

key advantage:

• unified method against complexity in boundary geometry \(\partial\Omega\)