Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms (1902.07896v1)

Abstract: We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with ReLU activation functions that achieve certain approximation rates. Second, we establish lower bounds for the approximation by ReLU neural networks for classes of Sobolev-regular functions. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations.

• The paper establishes that deep ReLU networks can approximate functions in Sobolev spaces with error ε using O(log(1/ε)) layers and O(ε^(-d/(n-s)) log(1/ε)) neurons.
• The paper shows that any architecture achieving these bounds must use at least Ω(ε^(-d/(2(n-k)))) weights, setting a theoretical lower limit on complexity.
• The paper introduces averaging techniques similar to Taylor expansions to analyze error in W^(s,p) norms, thereby bridging AI methods and numerical PDE analysis.

Error Bounds for Approximations with Deep ReLU Neural Networks in $W^{s,p}$ Norms

The paper "Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms" by Gühring, Kutyniok, and Petersen investigates the approximation capabilities of deep ReLU neural networks for functions that possess Sobolev regularity. It provides a rigorous analysis of the rates at which deep neural networks with Rectified Linear Unit (ReLU) activation functions approximate these functions in terms of Sobolev norms, which are critical for solving partial differential equations (PDEs) via numerical methods.

The authors extend existing theoretical frameworks—traditionally applied to simpler $L^\infty$ norms—by considering a broader class of Sobolev norms, specifically $W^{s,p}$ norms. This extension is pivotal as it aligns neural network approximation capabilities with the nuances of solving PDEs where Sobolev norms are more applicable due to their ability to capture both function values and their derivatives.

Main Contributions

1. Upper Complexity Bounds: The authors demonstrate that for any function $f$ belonging to a suitable Sobolev space $W^{n,p}([0,1]^d)$, deep ReLU networks can approximate $f$ within an error $\epsilon$ in the $W^{s,p}$ norms, where $0 \leq s \leq 1$. They establish that it is feasible to construct neural networks with a number of layers scaling as $\mathcal{O}(\log_2(1/\epsilon))$ and sizes scaling as $\mathcal{O}(\epsilon^{-d/(n-s)}\log_2(1/\epsilon))$. These results underscore the efficiency of deep ReLU networks relative to predictable increases in dimensional complexity.
2. Lower Complexity Bounds: They also prove that any architecture capable of realizing these approximation rates must have complexity proportional to $\Omega(\epsilon^{-d/(2(n-k))})$ weights for $k = 0,1$, putting forth a theoretical floor on the resource requirements for such tasks.
3. Derivation of Sobolev Norms: The paper introduces an intricate mathematical framework based on averaging techniques akin to Taylor expansions, tailored for Sobolev spaces. These are applied to dissect the approximation properties of neural network realizations further.

Implications and Future Directions

This research contributes to the mathematical foundations necessary for utilizing deep learning—particularly deep ReLU networks—in numerical solutions of PDEs, thus bridging areas traditionally dominated by finite element methods with machine learning. By moving from $L^\infty$ to $W^{s,p}$ norms, the results can potentially impact how neural networks are used in simulations and analyses where the factor of smoothness and differentiability plays a critical role.

From a practical standpoint, these theoretical advancements offer guidance on the expected computational requirements for deploying neural networks in high-dimensional and high-regularity contexts, highlighting trade-offs between network depth, breadth, and accuracy of approximation.

While the work makes significant strides, future exploration could address the curse of dimensionality more explicitly and seek methods to circumvent or mitigate its implications. Another avenue for advancement lies in refining these results for real-world scenarios where neural network weights are quantized or constrained by computational limits.

Ultimately, this research offers valuable insights and tools, reinforcing deep ReLU networks’ utility in areas demanding rigorous approximation guarantees. It sets a foundational bedrock on which further innovations in AI-driven numerical analysis can be constructed, particularly in applications where Sobolev-type regularity is indispensable.