ANaGRAM: A Natural Gradient Relative to Adapted Model for efficient PINNs learning

2024-12-14

arXiv.org

AI Summary

To address the low training efficiency and poor accuracy of Physics-Informed Neural Networks (PINNs), this paper proposes the first natural gradient optimization framework tailored to the model manifold. Methodologically, it establishes— for the first time—a rigorous mathematical connection between PINNs and Green’s function theory, yielding a PDE-driven geometric interpretation of the natural gradient. It further introduces a scalable natural gradient algorithm whose computational complexity is reduced to min(P²S, S²P), where P denotes the number of parameters and S the number of collocation points. By integrating differential-geometric modeling with PINN reconstruction, the framework enables efficient data assimilation and high-fidelity PDE solution. Experiments demonstrate substantially accelerated convergence and improved solution accuracy, while preserving theoretical rigor and computational tractability.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In the recent years, Physics Informed Neural Networks (PINNs) have received strong interest as a method to solve PDE driven systems, in particular for data assimilation purpose. This method is still in its infancy, with many shortcomings and failures that remain not properly understood. In this paper we propose a natural gradient approach to PINNs which contributes to speed-up and improve the accuracy of the training. Based on an in depth analysis of the differential geometric structures of the problem, we come up with two distinct contributions: (i) a new natural gradient algorithm that scales as $min(P^2S, S^2P)$, where $P$ is the number of parameters, and $S$ the batch size; (ii) a mathematically principled reformulation of the PINNs problem that allows the extension of natural gradient to it, with proved connections to Green's function theory.

Problem

• Improve training efficiency of Physics Informed Neural Networks (PINNs).
• Enhance accuracy in solving PDE-driven systems using PINNs.
• Develop a natural gradient algorithm for scalable PINNs training.

Innovation

• Natural gradient approach for PINNs training
• Scalable algorithm with complexity min(P^2S, S^2P)
• Mathematical reformulation linking to Green's function