A Theoretical Analysis of Deep Neural Networks and Parametric PDEs (1904.00377v3)

Abstract: We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.

• The paper demonstrates that ReLU neural networks efficiently approximate parametric PDE solution maps using a reduced basis-dependent framework.
• It introduces a novel method based on the Neumann series for matrix inversion, significantly cutting computational costs.
• The study establishes rigorous bounds, proving that the network size depends mainly on the intrinsic dimension of the solution manifold.

Analysis of ReLU Neural Networks for Parametric PDEs

The paper by Kutyniok et al. investigates the potential of leveraging deep neural networks (DNNs), specifically those utilizing ReLU activation functions, to efficiently approximate the solution maps of parametric partial differential equations (PDEs). This research situates itself at the intersection of deep learning and numerical analysis, providing a theoretical framework that supports the empirical success of neural networks in solving high-dimensional, parameter-dependent problems.

Neural Networks and Parametric PDEs

The paper begins by outlining the context in which parametric PDEs are typically encountered. Parametric problems involve a solution space and a parameter space connected via a parametric map, where the parameters are often associated with physical or geometric constraints of the PDE. Solving such parametric PDEs is computationally intensive, primarily due to the high dimensionality of the parameter space and the complexity of solving the PDEs for each parameter.

To address this issue, the paper leverages reduced basis methods (RBMs), which exploit the inherent low-dimensionality of the solution manifold of the PDEs. A significant contribution of this work is the development of a theoretical framework that enables the efficient approximation of these parametric maps by small-sized DNNs. The networks are constructed such that their size essentially depends only on the dimension of a reduced basis, rather than the higher dimension of a full discretization space.

Key Contributions and Theoretical Insights

The authors make several key theoretical contributions:

1. Matrix Inversion via Neural Networks: The paper details a method to construct ReLU neural networks that efficiently approximate matrix inversion, a critical component in solving linear system representations of PDEs. This construction is based on the Neumann series and uses the compositional nature of neural networks.
2. Efficient Representation: The research shows that DNNs can approximate the parametric maps efficiently with a size that depends polylogarithmically on the approximation accuracy and essentially scales with the size of the reduced basis, rather than the discretization size. This demonstrates that the networks can capture the low-dimensional structure of the problem space.
3. Approximation Theoretical Results: The paper provides rigorous bounds on the approximation capabilities of DNNs. It establishes that the number of parameters required to achieve a certain accuracy depends mainly on the intrinsic dimension of the solution manifold, described by its Kolmogorov $N$-widths.

These contributions underscore the capability of DNNs to perform tasks typically associated with traditional numerical methods, like RBMs, but with added benefits brought by the flexibility and efficiency of neural networks, particularly in high-dimensional settings.

Implications and Future Directions

The implications of this research are multifaceted. Practically, it provides a pathway for using DNNs to solve parametric PDEs more efficiently, potentially transforming areas such as real-time physics simulations and optimization under uncertainty. Theoretically, this work extends the understanding of neural networks' capabilities in approximating complex functions beyond their traditional use cases.

For future developments, the paper suggests several directions: exploring non-linear and non-coercive PDEs, incorporating the learning process into the theoretical framework to assess required data generation, and applying these methods to convolutional and residual networks, which are prevalent in practical applications.

Overall, this research effectively bridges the gap between deep learning and numerical analysis, offering insights that could lead to more efficient and scalable algorithms for solving large-scale scientific and engineering problems.