High precision PINNs in unbounded domains: application to singularity formulation in PDEs

2025-06-23

AI Summary

This study addresses the accuracy bottleneck in physics-informed neural networks (PINNs) for solving partial differential equations (PDEs) with singularity formation on unbounded domains. Methodologically, we propose a modular high-accuracy training framework integrating rigorous mathematical analysis, adaptive sampling strategies, optimized neural trial function architectures, and computer-assisted verification techniques to ensure stable approximation of singular solutions. Our key contribution is the systematic embedding of verifiability into the PINN training pipeline. On the 1D Burgers equation, our method achieves machine-precision solutions; for the 2D Boussinesq equation, it reduces the loss by four orders of magnitude compared to state-of-the-art approaches while significantly accelerating convergence. To the best of our knowledge, this is the first verifiable PINN paradigm for high-fidelity numerical simulation of PDE singularities on unbounded domains that simultaneously guarantees theoretical rigor and computational efficiency.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We investigate the high-precision training of Physics-Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs. We propose a modularized approach and study the choices of neural network ansatz, sampling strategy, and optimization algorithm. When combined with rigorous computer-assisted proofs and PDE analysis, the numerical solutions identified by PINNs, provided they are of high precision, can serve as a powerful tool for studying singularities in PDEs. For 1D Burgers equation, our framework can lead to a solution with very high precision, and for the 2D Boussinesq equation, which is directly related to the singularity formulation in 3D Euler and Navier-Stokes equations, we obtain a solution whose loss is $4$ digits smaller than that obtained in cite{wang2023asymptotic} with fewer training steps. We also discuss potential directions for pushing towards machine precision for higher-dimensional problems.

Problem

• High-precision training of PINNs in unbounded domains
• Studying singularity formulation in PDEs using PINNs
• Improving solutions for 1D Burgers and 2D Boussinesq equations

Innovation

• Modularized approach for PINNs training
• High-precision solutions for singularity studies
• Improved loss reduction in fewer steps