2025/09/19 by Jiahao Zhang, Shiheng Zhang, Zhang, Jiahao +3 · 2 citations
Computer Science · Engineering · #Neural Networks and Applications #Industrial Vision Systems and Defect Detection #Metaheuristic Optimization Algorithms Research
paper · pdf · doi:10.48550/arxiv.2509.16395
Evolutionary deep neural networks (EDNNs) solve time-dependent partial differential equations by evolving the neural-network parameters sequentially in time through a local least-squares problem. Their main computational bottleneck is that each time step requires the solution of a dense linear system whose dimension equals the total number of trainable parameters. We propose a low-rank evolutionary deep neural network (LR-EDNN) method that reduces this cost through adaptive tangent-space projection. This construction replaces direct bilinear low-rank factor evolution by a linear reduced problem while preserving the sequential-in-time structure of EDNN. We construct the reduced Jacobian directly through layerwise Jacobian-vector products, without forming the full Jacobian. We further establish a finite-time comparison estimate: the deviation of the LR-EDNN trajectory from full EDNN is bounded by a discrete Grönwall accumulation of the local tangent-space projection defects, with amplification governed by the assumed Lipschitz and directional-coercivity constants. Numerical experiments on a porous-medium equation with drift, one- and two-dimensional Allen-Cahn equations, and two-dimensional viscous Burgers' equations demonstrate that LR-EDNN substantially reduces computational cost while maintaining the accuracy and qualitative fidelity of the full EDNN solver when the rank is chosen adequately.