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Fixed-Architecture Neural Networks Achieve Super-Expressive Approximation

Feng-Lei Fan, Ze-Yu Li, Chen-Yu Wang, Jian-Jun Wang· July 9, 2026 View original

Summary

This work demonstrates super-expressive approximation for fixed-architecture neural networks with explicit parameter bounds and elementary activations, resolving the issue of lacking quantitative parameter-error trade-offs in prior research. It uses the Chinese Remainder Theorem to construct networks that achieve specific parameter magnitudes for Lipschitz and H\"older-smooth functions.

Previous research has shown that neural networks of a fixed size can achieve "super-expressive" approximation capabilities, meaning they can approximate complex functions very accurately. However, these studies often lacked a clear, quantitative understanding of how the magnitude of the network's parameters relates to the approximation error. This new work addresses that gap by providing explicit characterizations of parameter bounds for fixed-architecture neural networks using elementary activation functions. The key innovation in this research is the use of the Chinese Remainder Theorem as a constructive encoding mechanism. This mathematical tool allows for the precise construction of neural networks with specific properties. For Lipschitz continuous functions defined on a D-dimensional unit hypercube, the researchers constructed a network with a width of `max{D,4}` and a depth of 5, providing explicit trade-offs between parameter magnitude and approximation error. Furthermore, for H\"older-smooth functions, a fixed network with a width of `max{2D, D+5N+1}` and a depth of `r + 9` was shown to achieve a parameter magnitude `P` where `log2 P` scales as `O(epsilon^(-2D/(r+gamma)) * log(1/epsilon))`. This result is significant because it offers a dual perspective to existing paradigms that focus on parameter-bounded but architecture-unbounded networks, providing a concrete understanding of how fixed-size networks can achieve high expressivity with controlled parameter sizes.

Why it matters

For AI engineers and researchers, understanding the explicit parameter-error trade-offs in fixed-architecture neural networks is crucial for designing efficient and robust models. This work provides theoretical guarantees that can guide the development of smaller, yet highly expressive, neural networks, particularly important for resource-constrained applications.

How to implement this in your domain

  1. 1Explore the theoretical underpinnings of super-expressive approximation to inform architectural design choices for neural networks.
  2. 2Consider the implications of explicit parameter bounds when designing models for deployment in environments with strict memory or computational constraints.
  3. 3Investigate the use of the Chinese Remainder Theorem or similar constructive encoding mechanisms for building highly efficient, fixed-size networks.
  4. 4Apply these theoretical insights to optimize the trade-off between model complexity and approximation accuracy in your deep learning projects.
  5. 5Benchmark the performance of fixed-architecture networks designed with these principles against more traditional, larger models for specific tasks.

Who benefits

AI HardwareEdge ComputingEmbedded SystemsRoboticsScientific Computing

Key takeaways

  • Fixed-architecture neural networks can achieve super-expressive approximation with explicit parameter bounds.
  • The Chinese Remainder Theorem is a key mechanism for constructing such efficient networks.
  • The research provides quantitative parameter-error trade-offs for Lipschitz and H\"older-smooth functions.
  • These findings are crucial for designing compact yet powerful neural networks for resource-limited applications.

Original post by Feng-Lei Fan, Ze-Yu Li, Chen-Yu Wang, Jian-Jun Wang

"arXiv:2607.06781v1 Announce Type: new Abstract: In this work, we investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations. While prior work demonstrated super-expressive approximation using fixed-size networks, they…"

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Originally posted by Feng-Lei Fan, Ze-Yu Li, Chen-Yu Wang, Jian-Jun Wang on X · view source

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