Error bounds for PDE-regularized learning
arXiv:2003.06524
Abstract
In this work we consider the regularization of a supervised learning problem by partial differential equations (PDEs) and derive error bounds for the obtained approximation in terms of a PDE error term and a data error term. Assuming that the target function satisfies an unknown PDE, the PDE error term quantifies how well this PDE is approximated by the auxiliary PDE used for regularization. It is shown that this error term decreases if more data is provided. The data error term quantifies the accuracy of the given data. Furthermore, the PDE-regularized learning problem is discretized by generalized Galerkin discretizations solving the associated minimization problem in subsets of the infinite dimensional functions space, which are not necessarily subspaces. For such discretizations an error bound in terms of the PDE error, the data error, and a best approximation error is derived.
References in corpus (11)
- Adam: A Method for Stochastic Optimization
- Solving high-dimensional partial differential equations using deep learning
- Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
- Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
- A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
- Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions
- Mathematics of Deep Learning
- Nonlinear Approximation via Compositions
- Approximation spaces of deep neural networks
- Deep Ritz revisited
- A note on Poincaré- and Friedrichs-type inequalities