A universal linearized subspace refinement framework for neural networks

TL;DR

The paper introduces Linearized Subspace Refinement (LSR), a universal framework that improves neural network accuracy by solving a linear residual problem at a fixed network state, surpassing standard training methods without altering architectures.

Contribution

It presents a novel, architecture-agnostic refinement method that leverages linear residual models to enhance neural network predictions beyond traditional gradient-based training.

Findings

LSR achieves order-of-magnitude error reductions in various tasks.

Gradient-based training often fails to reach the attainable accuracy within the model's capacity.

Iterative LSR accelerates convergence and improves accuracy in operator-constrained problems.

Abstract

Neural networks are predominantly trained using gradient-based methods, yet in many applications their final predictions remain far from the accuracy attainable within the model's expressive capacity. We introduce Linearized Subspace Refinement (LSR), a general and architecture-agnostic framework that exploits the Jacobian-induced linear residual model at a fixed trained network state. By solving a reduced direct least-squares problem within this subspace, LSR computes a subspace-optimal solution of the linearized residual model, yielding a refined linear predictor with substantially improved accuracy over standard gradient-trained solutions, without modifying network architectures, loss formulations, or training procedures. Across supervised function approximation, data-driven operator learning, and physics-informed operator fine-tuning, we show that gradient-based training often fails to access this attainable accuracy, even when local linearization yields a convex problem. This observation indicates that loss-induced numerical ill-conditioning, rather than nonconvexity or model expressivity, can constitute a dominant practical bottleneck. In contrast, one-shot LSR systematically exposes accuracy levels not fully exploited by gradient-based training, frequently achieving order-of-magnitude error reductions. For operator-constrained problems with composite loss structures, we further introduce Iterative LSR, which alternates one-shot LSR with supervised nonlinear alignment, transforming ill-conditioned residual minimization into numerically benign fitting steps and yielding accelerated convergence and improved accuracy. By bridging nonlinear neural representations with reduced-order linear solvers at fixed linearization points, LSR provides a numerically grounded and broadly applicable refinement framework for supervised learning, operator learning, and scientific computing.