ReLU Deep Neural Networks and Linear Finite Elements (1807.03973v2)

Abstract: In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the simplicial linear finite element method (FEM). We first consider the special case of FEM. By exploring the DNN representation of its nodal basis functions, we present a ReLU DNN representation of CPWL in FEM. We theoretically establish that at least $2$ hidden layers are needed in a ReLU DNN to represent any linear finite element functions in $\Omega \subseteq \mathbb{R}^d$ when $d\ge2$. Consequently, for $d=2,3$ which are often encountered in scientific and engineering computing, the minimal number of two hidden layers are necessary and sufficient for any CPWL function to be represented by a ReLU DNN. Then we include a detailed account on how a general CPWL in $\mathbb R^d$ can be represented by a ReLU DNN with at most $\lceil\log_2(d+1)\rceil$ hidden layers and we also give an estimation of the number of neurons in DNN that are needed in such a representation. Furthermore, using the relationship between DNN and FEM, we theoretically argue that a special class of DNN models with low bit-width are still expected to have an adequate representation power in applications. Finally, as a proof of concept, we present some numerical results for using ReLU DNNs to solve a two point boundary problem to demonstrate the potential of applying DNN for numerical solution of partial differential equations.

• The paper demonstrates that a minimum of two hidden layers in ReLU DNNs are both necessary and sufficient to represent any linear finite element function in domains where d ≥ 2.
• It establishes that ⌈log₂(d+1)⌉ hidden layers suffice to capture any CPWL function in ℝ^d, outlining a precise trade-off between network depth and neuron density.
• Numerical experiments validate the theoretical framework by showing that ReLU DNN discretizations can solve PDE problems more efficiently than traditional adaptive finite element methods with comparable degrees of freedom.

ReLU Deep Neural Networks and Linear Finite Elements: A Perspective on Representation and Applications

The paper "ReLU Deep Neural Networks and Linear Finite Elements" by Juncai He, Lin Li, Jinchao Xu, and Chunyue Zheng embarks on a comprehensive exploration of the relationship between deep neural networks (DNNs) employing rectified linear unit (ReLU) activation and continuous piecewise linear (CPWL) functions derived from simplicial linear finite element methods (FEM). This investigation bridges two dominant computational paradigms: DNNs in artificial intelligence and FEM in numerical analysis for partial differential equations (PDEs). The critical insight of this paper is its demonstration that DNNs with adequate depth can encapsulate the expressive range required for accurately representing FEM-based CPWL functions.

Theoretical Insights and Representation Capabilities

A core finding in this paper is that a minimum of two hidden layers in a ReLU DNN are both necessary and sufficient to represent any linear finite element function for domains in $\mathbb{R}^d$ when $d \ge 2$. This conclusion underscores the expressive capacity inherent in sufficiently deep neural networks compared to their shallow counterparts. Furthermore, extending the results to general CPWL functions in $\mathbb{R}^d$, the paper elucidates that $\lceil\log_2(d+1)\rceil$ hidden layers suffice for representing any CPWL function. The authors also provide estimates of the neuron counts needed, which, interestingly, highlight the trade-off between the number of layers and neuron density.

By deconstructing linear finite element representations into ReLU networks, the work not only solidifies theoretical insights into DNN expressiveness but also offers a constructive approach to establishing minimal DNN architectures needed for specific function classes. This approach is pivotal for understanding the minimum complexity required in neural architectures to achieve specific approximation tasks.

Numerical Results and Practical Implications

To validate the theoretical constructs, the authors present numerical results demonstrating the capability of ReLU DNNs in solving a two-point boundary value problem, showcasing the potential of DNNs as an alternative framework for tackling PDEs. This proof-of-concept example illustrates that DNN discretizations can achieve superior approximation quality over traditional adaptive finite element methods (AFEMs) while having a comparable number of degrees of freedom.

Implications for Artificial Intelligence and Beyond

The paper's insights extend into practical applications, particularly in the field of AI where DNNs are foundational. Understanding the minimal architecture necessary for representing specific function classes directly informs neural network design, potentially leading to more efficient training processes and reduced computational costs.

Moreover, the elucidation of ReLU DNNs' approximation properties could guide future research in machine learning where model interpretability and efficiency are paramount. Given the demonstrated equivalency with finite element functions, these findings could spur AI applications in domains requiring physical modeling and simulations, traditionally dominated by FEM methodologies.

Future Avenues of Research

While the paper rigorously tackles fundamental representation problems, it opens questions regarding optimal design strategies for DNNs beyond the theoretical lens, particularly concerning computational efficiency in high-dimensional spaces. Exploring the scalability issues and convergence properties of such DNN frameworks when applied to more complex PDE problems or real-world scenarios remains an enticing avenue for future research.

In summary, this paper provides rigorous analytical underpinnings for why and how deep architectures in neural networks succeed, particularly in applications traditionally associated with numerical solvers. The marriage of DNN theoretical properties and practical application in numerical PDEs as expounded in this work sets the stage for exciting interdisciplinary innovations bridging AI and numerical analysis.