Error analysis for deep neural network approximations of parametric hyperbolic conservation laws
This work addresses theoretical guarantees for neural network approximations in computational physics, offering rigorous error bounds that are incremental but important for reliability in scientific applications.
The paper tackles the problem of approximating solutions to parametric hyperbolic scalar conservation laws using ReLU neural networks, showing that the approximation error can be made arbitrarily small while avoiding the curse of dimensionality, and provides explicit bounds on generalization error in terms of training error, sample size, and network size.
We derive rigorous bounds on the error resulting from the approximation of the solution of parametric hyperbolic scalar conservation laws with ReLU neural networks. We show that the approximation error can be made as small as desired with ReLU neural networks that overcome the curse of dimensionality. In addition, we provide an explicit upper bound on the generalization error in terms of the training error, number of training samples and the neural network size. The theoretical results are illustrated by numerical experiments.