Physics-Informed Neural Networks (PINNs) have emerged as an important class of numerical methods for solving partial differential equations (PDEs). However, during the late-stage optimization process, further parameter updates often yield diminishing accuracy improvements while increasing computational costs. To address this issue, this paper proposes a Physics-Informed Error Field Learning (PIEFL) framework for PINNs. Unlike conventional approaches that continuously approximate the solution field using a single network, PIEFL introduces an auxiliary error network after the primary network achieves satisfactory accuracy and shifts the learning objective from the solution field to the error field. By deriving error control equations under physical constraints, the error network learns the discrepancy between the current approximation and the exact solution, and the learned error correction is combined with the primary prediction to improve solution accuracy. The proposed framework avoids continuous optimization of the entire solution space and focuses computational resources on correcting existing prediction errors. Moreover, PIEFL requires no modification to the primary network architecture, making it compatible with existing PINN models and applicable as a general post-training optimization strategy. Numerical experiments on representative PDEs demonstrate that PIEFL achieves higher solution accuracy under the same computational budget, validating its effectiveness in improving the performance of PINNs.
Long-context Transformer inference increasingly relies on KV-cache compression or quantization. Prior rotation and transform-coding results suggest that the channel basis of each key/value vector affects how faithfully a fixed backend preserves model behavior. We introduce Codec-Gauge, a post-training cache-coordinate layer that learns small orthogonal channel transforms around existing compression and quantization backends. Its frequency-distribution objective combines a token-channel DCT spectral-centroid loss with a smooth rate proxy to concentrate KV energy in low-frequency codec-facing layouts. We evaluate actual compression and decompression using measured bytes and rolling compressed-history scoring. Across six models at $3$, $4$, and $6$ bits/value, learned gauges reduce zfp KL divergence by $44.0\%$ on average relative to raw coordinates and outperform random, Hadamard, DCT, and PCA/KLT controls. The same gauges improve quality preservation for block-uniform and KIVI-style quantization. Experiments on a 27B model and long-context task prompts reproduce the quality trend, while serial storage and timing measurements validate the implemented compressed-cache paths. These results establish cache-coordinate geometry as a practical post-training variable for improving compression fidelity without changing model weights, attention semantics, or backend coding rules.