A Physics-informed Multi-resolution Neural Operator

2025-10-27

AI Summary

Operator learning on multi-resolution irregular meshes faces challenges of scarce high-quality labeled data and inconsistent sample resolutions. Method: This paper proposes a physics-informed, data-free neural operator framework. It extends the RINO architecture to a fully unsupervised setting: pre-trained basis functions encode arbitrarily discretized inputs into a latent space; an MLP models the operator mapping; and PDE constraints—formulated via finite differences—are embedded directly into the loss function as hard physical priors. Contribution/Results: The framework requires no input-output paired data. It significantly improves generalization and robustness across mixed coarse–fine meshes. In multiple multi-resolution numerical experiments—including Poisson, Darcy flow, and Navier–Stokes problems—the model stably predicts physical fields with accuracy comparable to traditional PDE solvers, while drastically reducing dependence on high-fidelity labeled datasets.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

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

The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to obtain in some real-world engineering applications. These datasets may be unevenly discretized from one realization to another, with the grid resolution varying across samples. In this study, we introduce a physics-informed operator learning approach by extending the Resolution Independent Neural Operator (RINO) framework to a fully data-free setup, addressing both challenges simultaneously. Here, the arbitrarily (but sufficiently finely) discretized input functions are projected onto a latent embedding space (i.e., a vector space of finite dimensions), using pre-trained basis functions. The operator associated with the underlying partial differential equations (PDEs) is then approximated by a simple multi-layer perceptron (MLP), which takes as input a latent code along with spatiotemporal coordinates to produce the solution in the physical space. The PDEs are enforced via a finite difference solver in the physical space. The validation and performance of the proposed method are benchmarked on several numerical examples with multi-resolution data, where input functions are sampled at varying resolutions, including both coarse and fine discretizations.

Problem

• Addresses operator learning with limited high-fidelity data
• Handles uneven and multi-resolution discretized training datasets
• Develops physics-informed neural operator for PDE solutions

Innovation

• Physics-informed operator learning without training data
• Latent embedding projection for arbitrary discretizations
• Multi-layer perceptron approximates PDE operator