Error Estimates for the Deep Ritz Method with Boundary Penalty
This work addresses the theoretical analysis of neural network-based numerical methods for partial differential equations, offering foundational error estimates that are incremental but crucial for validating such methods in scientific computing.
The paper tackles the error estimation for the Deep Ritz Method applied to linear elliptic equations with Dirichlet boundary conditions using a boundary penalty method, providing general error bounds that depend on optimization accuracy, approximation capabilities, and penalization strength, and showing that neural networks inherit favorable approximation properties for high-dimensional problems.
We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary sets of ansatz functions and estimate the error in dependence of the optimization accuracy, the approximation capabilities of the ansatz class and -- in the case of Dirichlet boundary values -- the penalization strength $λ$. To the best of our knowledge, our results are presently the only ones in the literature that treat the case of Dirichlet boundary conditions in full generality, i.e., without a lower order term that leads to coercivity on all of $H^1(Ω)$. Further, we discuss the implications of our results for ansatz classes which are given through ReLU networks and the relation to existing estimates for finite element functions. For high dimensional problems our results show that the favourable approximation capabilities of neural networks for smooth functions are inherited by the Deep Ritz Method.