arXiv subjects

Xuelian Jiang

Publications and source records attributed to Xuelian Jiang.

3 recordsLinked to original sources

arXiv2026

Novel Product Manifold Modeling and Orthogonality-Constrained Neural Network Solver for Parameterized Generalized Inverse Eigenvalue Problems

A parameterized orthogonality-constrained neural network is proposed for the first time to solve the parameterized generalized inverse eigenvalue problem (PGIEP) on product manifolds, offering a new perspective to address PGIEP. The key contributions are twofold. First, we construct a novel model for the PGIEP, where the optimization variables are located on the product of a Stiefel manifold and a Euclidean manifold. This model enables the application of optimization algorithms on the Stiefel manifold, a capability that is not achievable with existing models. Additionally, the gradient Lipschitz continuity of the objective function is proved. Second, a parameterized Stiefel multilayer perceptron (P-SMLP) that incorporates orthogonality constraints is proposed. Through hard constraints, P-SMLP enables end-to-end training without the need of alternating training between the two manifolds, providing a robust computational framework for generic PGIEPs. Numerical experiments demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2025

Lie Symmetry Net: Preserving Conservation Laws in Modelling Financial Market Dynamics via Differential Equations

This paper employs a novel Lie symmetries-based framework to model the intrinsic symmetries within financial market. Specifically, we introduce Lie symmetry net (LSN), which characterises the Lie symmetries of the differential equations (DE) estimating financial market dynamics, such as the Black-Scholes equation. To simulate these differential equations in a symmetry-aware manner, LSN incorporates a Lie symmetry risk derived from the conservation laws associated with the Lie symmetry operators of the target differential equations. This risk measures how well the Lie symmetries are realised and guides the training of LSN under the structural risk minimisation framework. Extensive numerical experiments demonstrate that LSN effectively realises the Lie symmetries and achieves an error reduction of more than one order of magnitude compared to state-of-the-art methods. The code is available at https://github.com/Jxl163/LSN_code.

math.AP↗

arXiv2024

Orthogonal Constrained Neural Networks for Solving Structured Inverse Eigenvalue Problems

This paper introduces a novel neural network for efficiently solving Structured Inverse Eigenvalue Problems (SIEPs). The main contributions lie in two aspects: firstly, a unified framework is proposed that can handle various SIEPs instances. Particularly, an innovative method for handling nonnegativity constraints is devised using the ReLU function. Secondly, a novel neural network based on multilayer perceptrons, utilizing the Stiefel layer, is designed to efficiently solve SIEP. By incorporating the Stiefel layer through matrix orthogonal decomposition, the orthogonality of similarity transformations is ensured, leading to accurate solutions for SIEPs. Hence, we name this new network Stiefel Multilayer Perceptron (SMLP). Furthermore, SMLP is an unsupervised learning approach with a lightweight structure that is easy to train. Several numerical tests from literature and engineering domains demonstrate the efficiency of SMLP.

math.NA↗