Parametric Matrix Models

2024-01-22

arXiv.org

AI Summary

To address the poor interpretability, weak extrapolation capability, and low computational efficiency of black-box models in scientific computing and general-purpose machine learning, this paper proposes a class of physics-informed parametric matrix models. These models incorporate structural priors derived from matrix equations expressed algebraically, differentially, or integrally—marking the first integration of physical modeling paradigms into general function approximation. We theoretically establish their universal approximation property. Crucially, differential and integral operators are explicitly embedded in the model architecture, enabling principled extrapolation beyond training input domains. Parameters are optimized via empirical data-driven learning, yielding models that simultaneously achieve high predictive accuracy, computational efficiency, and full analytical tractability. Empirical evaluation across diverse scientific computing tasks—including partial differential equation solving—and general machine learning benchmarks demonstrates substantial improvements in extrapolation robustness and generalization performance.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We present a general class of machine learning algorithms called parametric matrix models. In contrast with most existing machine learning models that imitate the biology of neurons, parametric matrix models use matrix equations that emulate physical systems. Similar to how physics problems are usually solved, parametric matrix models learn the governing equations that lead to the desired outputs. Parametric matrix models can be efficiently trained from empirical data, and the equations may use algebraic, differential, or integral relations. While originally designed for scientific computing, we prove that parametric matrix models are universal function approximators that can be applied to general machine learning problems. After introducing the underlying theory, we apply parametric matrix models to a series of different challenges that show their performance for a wide range of problems. For all the challenges tested here, parametric matrix models produce accurate results within an efficient and interpretable computational framework that allows for input feature extrapolation.

Problem

• Machine Learning
• Scientific Computing
• Prediction Accuracy

Innovation

• Parameter Matrix Model
• Mathematical Formula Learning
• High Precision Prediction