Linearized subspace refinement framework to expose hidden accuracy in trained neural networks

Neural networks trained by gradient-based methods often exhibit optimization-induced accuracy plateaus in scientific machine learning tasks. We present Linearized Subspace Refinement (LSR), an architecture-agnostic post-training framework that exploits the local linearized model at a fixed trained state. By solving a reduced direct least-squares problem in a Jacobian-defined low-dimensional space, LSR computes a subspace-optimal linearized correction and yields a refined predictor with markedly improved accuracy. Across function approximation, data-driven operator learning, physics-informed operator fine-tuning, and noisy inverse problems, LSR shows that standard nonlinear training can remain far above this subspace-attainable error level. Similar accuracy plateaus persist even for the convex quadratic problem from local linearization when solved with standard iterative optimizers, identifying numerical ill-conditioning as a primary bottleneck. LSR frequently delivers order-of-magnitude error reductions, while the subspace rank provides an explicit capacity-control mechanism that balances correction strength, numerical stability, and noise sensitivity. Together, LSR exposes conditioning-limited attainable accuracy in trained-state linearized models and provides direct access to it.