Kolmogorov n-widths for multitask physics-informed machine learning (PIML) methods: Towards robust metrics.

Area of Science:

• Computational Science and Engineering
• Machine Learning
• Applied Mathematics

Background:

• Physics-informed machine learning (PIML) integrates physical laws into machine learning for solving partial differential equations (PDEs).
• Multitask learning in PIML addresses single or multiple PDE problems simultaneously.
• Comparing and benchmarking different PIML approaches remains a significant challenge.

Purpose of the Study:

• To introduce an objective metric for comparing various multitask Physics-informed machine learning architectures.
• To analyze the effectiveness of PIML models in approximating functions using Kolmogorov n-widths.
• To improve the generalizability and validation of PIML models for solving PDEs.

Main Methods:

• Application of Kolmogorov n-widths to quantify the approximation effectiveness of multitask PIML models.
• Computation of lower accuracy bounds for PIML architectures.
• Analysis of learned basis functions within PIML models across different PDE problems.
• Incorporation of the Kolmogorov n-width metric into the model optimization process via regularization.

Main Results:

• The study presents the first objective metric for comparing multitask PIML architectures, reducing uncertainty from selective sampling and overfitting.
• Identified activation functions significantly impact model generalization to worst-case scenarios.
• Regularization using the Kolmogorov n-width metric improved model generalizability across multitask PDE problems.

Conclusions:

• Kolmogorov n-widths provide a robust, objective measure for evaluating and comparing multitask PIML models.
• The proposed metric aids in identifying architectural improvements, particularly concerning activation functions.
• Integrating this metric into optimization enhances PIML model performance and reliability in solving complex PDE problems.