Changqing Liu

Qinglu Meng, Yingguang Li, Zhiliang Deng et al.

Predictive learning for spatio-temporal processes (PL-STP) on complex spatial domains plays a critical role in various scientific and engineering fields, with its essence being the construction of operators between infinite-dimensional function spaces. This paper focuses on the unequal-domain mappings in PL-STP and categorising them into increase-domain and decrease-domain mapping. Recent advances in deep learning have revealed the great potential of neural operators (NOs) to learn operators directly from observational data. However, existing NOs require input space and output space to be the same domain, which pose challenges in ensuring predictive accuracy and stability for unequal-domain mappings. To this end, this study presents a general reduced-order neural operator named Reduced-Order Neural Operator on Riemannian Manifolds (RO-NORM), which consists of two parts: the unequal-domain encoder/decoder and the same-domain approximator. Motivated by the variable separation in classical modal decomposition, the unequal-domain encoder/decoder uses the pre-computed bases to reformulate the spatio-temporal function as a sum of products between spatial (or temporal) bases and corresponding temporally (or spatially) distributed weight functions, thus the original unequal-domain mapping can be converted into a same-domain mapping. Consequently, the same-domain approximator NORM is applied to model the transformed mapping. The performance of our proposed method has been evaluated on six benchmark cases, including parametric PDEs, engineering and biomedical applications, and compared with four baseline algorithms: DeepONet, POD-DeepONet, PCA-Net, and vanilla NORM. The experimental results demonstrate the superiority of RO-NORM in prediction accuracy and training efficiency for PL-STP.

Zhiwei Zhao, Changqing Liu, Jie Lin et al.

Efficient and accurate prediction of Multiphysics evolution across diverse cell geometries is fundamental to the design, management and safety of lithium-ion batteries. However, existing computational frameworks struggle to capture the coupled electrochemical, thermal, and mechanical dynamics across diverse cell geometries and varying operating conditions. Here, we present a Neural Bundle Map (NBM), a mathematically rigorous framework that reformulates multiphysics evolution as a bundle map over a geometric base manifold. This approach enables the complete decoupling of geometric complexity from underlying physical laws, ensuring strong operator continuity across varying domains. Our framework achieves high-fidelity spatiotemporal predictions with a normalized mean absolute error of less than 1% across varying configurations, while maintaining stability during long-horizon forecasting far beyond the training window and reducing computational costs by two orders of magnitude compared with conventional solvers. Leveraging this capability, we rapidly explored a vast configurational space to identify an optimal battery design that yields a 38% increase in energy density while adhering to thermal safety constraints. Furthermore, the NBM demonstrates remarkable scalability to multi-cell systems through few-shot transfer learning, providing a foundational paradigm for the intelligent design and real-time monitoring of complex energy storage infrastructures.

Zhiwei Zhao, Changqing Liu, Yingguang Li et al.

In scientific and engineering applications, solving partial differential equations (PDEs) across various parameters and domains normally relies on resource-intensive numerical methods. Neural operators based on deep learning offered a promising alternative to PDEs solving by directly learning physical laws from data. However, the current neural operator methods were limited to solve PDEs on fixed domains. Expanding neural operators to solve PDEs on various domains hold significant promise in medical imaging, engineering design and manufacturing applications, where geometric and parameter changes are essential. This paper presents a novel neural operator learning framework for solving PDEs with various domains and parameters defined for physical systems, named diffeomorphism neural operator (DNO). The main idea is that a neural operator learns in a generic domain which is diffeomorphically mapped from various physics domains expressed by the same PDE. In this way, the challenge of operator learning on various domains is transformed into operator learning on the generic domain. The generalization performance of DNO on different domains can be assessed by a proposed method which evaluates the geometric similarity between a new domain and the domains of training dataset after diffeomorphism. Experiments on Darcy flow, pipe flow, airfoil flow and mechanics were carried out, where harmonic and volume parameterization were used as the diffeomorphism for 2D and 3D domains. The DNO framework demonstrated robust learning capabilities and strong generalization performance across various domains and parameters.

Changqing Liu, Kaining Dai, Zhiwei Zhao et al.

Accurate prediction of machining deformation in structural components is essential for ensuring dimensional precision and reliability. Such deformation often originates from residual stress fields, whose distribution and influence vary significantly with geometric complexity. Conventional numerical methods for modeling the coupling between residual stresses and deformation are computationally expensive, particularly when diverse geometries are considered. Neural operators have recently emerged as a powerful paradigm for efficiently solving partial differential equations, offering notable advantages in accelerating residual stress-deformation analysis. However, their direct application across changing geometric domains faces theoretical and practical limitations. To address this challenge, a novel framework based on diffeomorphic embedding neural operators named neural diffeomorphic-neural operator (NDNO) is introduced. Complex three-dimensional geometries are explicitly mapped to a common reference domain through a diffeomorphic neural network constrained by smoothness and invertibility. The neural operator is then trained on this reference domain, enabling efficient learning of deformation fields induced by residual stresses. Once trained, both the diffeomorphic neural network and the neural operator demonstrate efficient prediction capabilities, allowing rapid adaptation to varying geometries. The proposed method thus provides an effective and computationally efficient solution for deformation prediction in structural components subject to varying geometries. The proposed method is validated to predict both main-direction and multi-direction deformation fields, achieving high accuracy and efficiency across parts with diverse geometries including component types, dimensions and features.

Changqing Liu

This paper is motivated by a series of (related) questions as to whether a computer can have pleasure and pain, what pleasure (and intensity of pleasure) is, and, ultimately, what concepts of emotion are. To determine what an emotion is, is a matter of conceptualization, namely, understanding and explicitly encoding the concept of emotion as people use it in everyday life. This is a notoriously difficult problem (Frijda, 1986, Fehr \& Russell, 1984). This paper firstly shows why this is a difficult problem by aligning it with the conceptualization of a few other so called semantic primitives such as "EXIST", "FORCE", "BIG" (plus "LIMIT"). The definitions of these thought-to-be-indefinable concepts, given in this paper, show what formal definitions of concepts look like and how concepts are constructed. As a by-product, owing to the explicit account of the meaning of "exist", the famous dispute between Einstein and Bohr is naturally resolved from linguistic point of view. Secondly, defending Frijda's view that emotion is action tendency (or Ryle's behavioral disposition (propensity)), we give a list of emotions defined in terms of action tendency. In particular, the definitions of pleasure and the feeling of beauty are presented. Further, we give a formal definition of "action tendency", from which the concept of "intensity" of emotions (including pleasure) is naturally derived in a formal fashion. The meanings of "wish", "wait", "good", "hot" are analyzed.