image: a, Operators defined on Riemannian manifolds, where the input function and output function can be defined on the same or different Riemannian manifolds. The example for this illustration is the operator learning problem of the composite curing case, where the input temperature function and the output deformation function are both defined on the same manifold, the composite part. b, The framework of NORM, consists of two feature mapping layers (P and Q) and multiple L-layers. c, The structure of L-layer, consists of the encoder-approximator-decoder block, the linear transformation, and the non- linear activation function. d, Laplace-Beltrami Operator (LBO) eigenfunctions for the geometric domain (the composite part).
Credit: ©Science China Press
While artificial intelligence (AI) has made remarkable achievements in domains like image recognition and natural language processing, it encounters fundamental challenges when trying to deal with more intricate scientific principles and complex engineering situations that hold immense practical values for society:
Whether for the accurate prediction of plasma evolution in a tokamak device required to achieve controllable fusion, or the prediction of blood flow dynamics in the aorta required for the diagnosis and treatment of cardiovascular diseases in the medical field, or the prediction of aircraft pressure field for fuselage structure optimization, a common theoretical challenge lies in establishing the intrinsic connection and relationship between two functions defined on complex computational domains, namely operators on Riemannian manifolds.
Classical deep learning models, such as convolutional neural networks and transformers, are designed for learning mappings between discrete data, which are fundamentally incapable of learning mappings between functions defined on continuous domains. Recently, researchers proposed neural operators that could learn operators defined on Euclidean space, but they had theoretical limitations in dealing with real-life applications in irregular computational domains, including complex surfaces and solids, which are non-Euclidean in nature.
In response to this generic theoretical challenge, a research team led by Prof. Yingguang Li from Nanjing University of Aeronautics and Astronautics reported a new concept, Neural Operator on Riemannian Manifolds’ (NORM), which generalised existing neural operator theories from being limited to regular Euclidean spaces, to being applicable to complex irregular Riemannian manifolds. This is a significant step forward in breaking the theoretical limitations of existing neural operators and has the potential to solving complex problems in many fields of applications of the same nature and principle.
The proposed NORM was evaluated against the latest neural operator methods on learning solutions of three partial differential equations (PDEs), i.e., Darcy flow, pipe turbulence and heat transfer, and two engineering cases, namely composite workpiece deformation prediction and blood flow dynamics prediction. The experimental results demonstrated the superior performance of NORM in measures of relative prediction error and maximum prediction error. Notably, NORM exhibited even more exceptional performance in learning operators on 3D complex geometries.
“We also proved that NORM holds universal approximation property in learning operators on Riemannian manifolds. This breakthrough not only bridges the gap between existing neural operator research and practical applications, but also could potentially revolutionise more real-world applications that involves oprator learning on complex geometries, like aerodynamics, hemodynamics and sustainable nuclear fusion.” explains Prof. Yingguang Li, reseacher in the College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautic, who has extensive research experience in intelligent manufacturing for large-scale complex aerospace structural parts, underscores the transformative potential of this research.
This research was recently published in National Science Open. Other contributors include Prof. Xu Liu from Nanjing Tech University and Dr. Gengxiang Chen, Qinglu Meng, Lu Chen, Prof. Changqing Liu from Nanjing University of Aeronautics and Astronautics.
Journal
National Science Open