Physics-informed low-rank neural operators with application to parametric elliptic PDEs

2025-09-09

AI Summary

This work addresses the efficient approximation of high-dimensional parametric elliptic PDE solution operators on point cloud data. We propose a physics-informed low-rank neural operator framework that integrates an encoder-decoder architecture, low-rank kernel approximation, and physics-informed neural network (PINN) regularization. The method enables mesh-free, continuous-domain prediction and strictly enforces PDE constraints and boundary conditions—both in supervised and unsupervised settings. By eliminating reliance on discretized grids, it achieves significantly improved computational efficiency and generalization across high-dimensional parameter spaces compared to conventional numerical methods. Extensive experiments on the Poisson equation, variable-coefficient screened Poisson equation, and parametric Darcy flow demonstrate high accuracy, strong robustness to parameter variation, and favorable scalability. Our approach establishes a novel paradigm for constructing surrogate models of complex, parametric PDEs directly from unstructured geometric data.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

We present the Physics-Informed Low-Rank Neural Operator (PILNO), a neural operator framework for efficiently approximating solution operators of partial differential equations (PDEs) on point cloud data. PILNO combines low-rank kernel approximations with an encoder--decoder architecture, enabling fast, continuous one-shot predictions while remaining independent of specific discretizations. The model is trained using a physics-informed penalty framework, ensuring that PDE constraints and boundary conditions are satisfied in both supervised and unsupervised settings. We demonstrate its effectiveness on diverse problems, including function fitting, the Poisson equation, the screened Poisson equation with variable coefficients, and parameterized Darcy flow. The low-rank structure provides computational efficiency in high-dimensional parameter spaces, establishing PILNO as a scalable and flexible surrogate modeling tool for PDEs.

Problem

• Efficiently approximating solution operators of PDEs on point clouds
• Combining low-rank kernel approximations with encoder-decoder architecture
• Ensuring PDE constraints satisfaction in supervised and unsupervised settings

Innovation

• Physics-informed low-rank neural operator framework
• Combines low-rank kernel with encoder-decoder architecture
• Physics-informed penalty training for PDE constraints