A Physics-informed Multi-resolution Neural Operator

TL;DR

This paper presents a physics-informed neural operator framework that can learn from multi-resolution, unevenly discretized data without requiring high-fidelity training datasets, by projecting inputs into a latent space and enforcing PDEs via finite difference methods.

Contribution

It extends the RINO framework to a data-free setting, enabling operator learning from multi-resolution data with PDE constraints enforced through a finite difference solver.

Findings

Effective on multi-resolution numerical examples

Handles uneven discretizations without high-fidelity data

Achieves accurate PDE solutions across resolutions

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.