Learnable-Parameters-Incorporated Neural Dynamics for Time-Dependent System of Linear Equations

The time-dependent nature of real-world problems and the ubiquity of linear equations underscore the fundamental importance of time-dependent systems of linear equations (TDSLE). However, existing methods often lack specificity for particular tasks. Our work shows that incorporating specificity is able to enhance solution performance. Concretely, we leverage the similarity with the numerical discretizations of dynamical systems, which refers to deep residual networks that can be interpreted as Euler discretizations of dynamic systems. This similarity provides a foundation for the learnability of neural dynamics solvers' parameters. On this basis, this paper proposes a learnable-parameters-incorporated neural dynamics (LPIND) method for solving the TDSLE. Firstly, a dynamic system for solving TDSLE using a neural dynamics (ND) method is constructed. Secondly, a recurrent neural network (RNN) is used to infer parameters of the ND method, with these parameters trainable. The proposed LPIND method used to solve TDSLE is formulated as a linear decoupled dynamic system, and thus, parameters trained on the low-dimensional dataset can be generalized directly to solve high-dimensional problems. Theoretical analyses are given to ensure that the validity of the proposed LPIND method is not affected by the size of the data dimension. Numerical experiments and applications are provided to verify the effectiveness and generalization performance of the proposed LPIND method.