aiqbench analysis
This research introduces a low-rank evolutionary deep neural network (LR-EDNN) method that addresses the computational bottleneck of solving dense linear systems in time-dependent partial differential equations. By employing adaptive tangent-space projection and layerwise Jacobian-vector products, the approach significantly reduces cost while preserving accuracy on complex physics problems like porous-medium flows and Allen-Cahn equations. The work represents a notable advancement in machine learning algorithms for scientific computing, specifically optimizing efficiency without sacrificing fidelity to full-parameter models.
Low-Rank Evolutionary Deep Neural Networks via Adaptive Tangent-Space Reduction
arXiv:2509.16395v2 Announce Type: replace-cross Abstract: 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 Gr\"onwall 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.
Abstract: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.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2509.16395 [stat.ML]
(or arXiv:2509.16395v2 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2509.16395 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 279 words
