Comparison of universal approximators incorporating partial monotonicity by structure.

Area of Science:

• Machine Learning
• Control Theory
• Artificial Intelligence

Background:

• Neural networks in control loops require guarantees beyond function approximation, including smoothness and monotonicity for stability.
• Loss of monotonicity in neural network control can lead to instability in control laws.
• Existing methods often struggle to enforce these critical properties structurally.

Purpose of the Study:

• To compare the Monotonic Multi-Layer Perceptron (MONMLP) and Monotonic MIN-MAX (MONMM) networks for incorporating partial monotonicity.
• To analyze the universal approximation capabilities of both MONMLP and MONMM networks for partially monotone functions.
• To investigate the trade-offs between approximation performance, training, and convergence for these networks.

Main Methods:

• Structural incorporation of partial monotonicity in neural network architectures.
• Comparative analysis of MONMLP and MONMM networks on various datasets.
• Evaluation of approximation error, training dynamics, and convergence rates.

Main Results:

• Both MONMLP and MONMM networks exhibit universal approximation capabilities for partially monotone functions.
• Dataset-specific advantages and disadvantages were observed for each network type.
• Differences in approximation performance, training efficiency, and convergence were identified.

Conclusions:

• MONMLP and MONMM offer viable, structure-based approaches for enforcing monotonicity in neural networks for control.
• The choice between MONMLP and MONMM depends on specific application requirements regarding performance, training, and convergence.
• Further research can explore hybrid approaches or optimizations for these monotonic neural networks.