Coordinate Encoding on Linear Grids for Physics-Informed Neural Networks

2026-03-23

AI Summary

This work addresses the slow convergence of physics-informed neural networks (PINNs) when solving partial differential equations, a challenge often attributed to spectral bias. To mitigate this issue, the authors propose a coordinate encoding mechanism based on axis-aligned independent linear grids, which employs natural cubic spline interpolation to construct a smooth coordinate mapping over local solution domains. This approach preserves derivative continuity while effectively alleviating spectral bias. The resulting method significantly enhances both the training efficiency and stability of PINNs. Numerical experiments demonstrate that the proposed strategy outperforms existing mesh-free PDE solvers in terms of convergence speed and computational cost.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

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: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

In solving partial differential equations (PDEs), machine learning utilizing physical laws has received considerable attention owing to advantages such as mesh-free solutions, unsupervised learning, and feasibility for solving high-dimensional problems. An effective approach is based on physics-informed neural networks (PINNs), which are based on deep neural networks known for their excellent performance in various academic and industrial applications. However, PINNs struggled with model training owing to significantly slow convergence because of a spectral bias problem. In this study, we propose a PINN-based method equipped with a coordinate-encoding layer on linear grid cells. The proposed method improves the training convergence speed by separating the local domains using grid cells. Moreover, it reduces the overall computational cost by using axis-independent linear grid cells. The method also achieves efficient and stable model training by adequately interpolating the encoded coordinates between grid points using natural cubic splines, which guarantees continuous derivative functions of the model computed for the loss functions. The results of numerical experiments demonstrate the effective performance and efficient training convergence speed of the proposed method.

Problem

• Physics-Informed Neural Networks
• spectral bias
• slow convergence
• partial differential equations
• training convergence

Innovation

• Physics-Informed Neural Networks
• Coordinate Encoding
• Linear Grid Cells
• Natural Cubic Splines
• Spectral Bias