Towards a mathematical framework to inform neural network modelling via polynomial regression.

Area of Science:

• Machine Learning
• Statistical Modeling
• Artificial Intelligence

Background:

• Neural networks are powerful but often function as "black boxes", hindering error evaluation and dimensioning.
• Integrating traditional statistical methods with neural networks offers a path to increased transparency and interpretability.
• Bridging the gap between neural networks and statistical models is crucial for understanding complex data relationships.

Purpose of the Study:

• To explore a mathematical framework linking neural networks and polynomial regression.
• To derive an explicit expression for polynomial regression coefficients from neural network weights.
• To enhance the interpretability of single hidden layer neural networks in regression tasks.

Main Methods:

• Utilized a Taylor expansion approach to establish a relationship between neural networks and polynomial regression.
• Developed an explicit formula to calculate polynomial regression coefficients directly from neural network weights.
• Focused on single hidden layer neural networks for regression problems.

Main Results:

• Demonstrated that polynomial regression coefficients can be explicitly determined from neural network weights.
• Empirically validated the method using synthetic polynomial data, achieving near-identical predictions under specific conditions.
• The proposed method successfully generates polynomials that accurately approximate data locally when trained on polynomial-generated data.

Conclusions:

• The developed mathematical framework provides a method to interpret neural networks as polynomial regressions.
• The validity and performance depend on factors such as synaptic potential distribution and activation function choice.
• This approach offers a way to demystify neural networks, particularly in regression applications with polynomial data.