Dynamics as Code: On Model Compression via Dynamic System

Neural network compression is commonly achieved by pruning parameters based on local importance scores, e.g., magnitude-based pruning. We propose a complementary approach that compresses models by aggregating neurons with similar functional behavior rather than removing weights independently. Our method encodes a trained network as a polynomial ODE system and applies a lumping method called Approximate Forward Differential Equivalence to identify neurons with approximately matching induced dynamics. A single tolerance parameter, $\varepsilon$, controls the compression level and induces a smooth trade-off between model size and predictive accuracy. We evaluate the method on synthetic datasets derived from nonlinear dynamical systems with known ground-truth behavior and on public regression benchmarks. Across both settings, the proposed approach achieves substantial parameter reduction while preserving accuracy, and consistently compares favorably with magnitude-based pruning and Wanda at similar compression levels. These results suggest that differential equivalence-based aggregation is a principled and effective alternative to conventional weight-centric pruning.

Lossless compression can reduce the storage and movement of model weights without changing their floating-point values, but repeated statistical analysis and code construction add computational overhead. We study whether the statistical structure of exponents can be prepared once and reused. For this, we introduce NeuralZip, which groups chunks with similar exponent distributions, shares Huffman codes, and selectively represents recurring exponent tuples using packed exponents, thereby achieving additional moderate compression ratios. A setup chooses these representations before subsequent encodings, while every encoding still processes the current tensor values. In floating-point model checkpoints, post-setup compression is 1.81-21.33$\times$ faster than the baselines and achieves exact bit-to-bit reconstruction. We show that this setup can be precomputed and transferred from another compatible architecture, preserving similar compression ratios and avoiding the need to amortize setup costs. Therefore, compression adaptation is transferable and reusable. Training checkpoints demonstrate continued reuse as the weights evolve. Finally, GPU experiments reduce active memory usage by up to 27.5$\%$ while reproducing the logits exactly.

Compression is fundamental to intelligence. A model that can represent its training data as a short code has discovered regularities that enable generalization. Large neural networks may learn functions far simpler than their parameter counts suggest, but it is challenging to construct codes that realize this simplicity. Parameter-based methods such as quantization produce code lengths that scale with model size, insensitive to how much information the parameters store. Prequential coding bypasses this issue by compressing the training trajectory, but codes the exact data sequence regardless of how much the model learns, yielding large codes when the data has high entropy. We introduce requential coding, where a teacher model selects training samples drawn from the student's own distribution. The student's code records only these selections, which cost bits only where teacher and student disagree. The resulting code length is independent of parameter count and data entropy, and often orders of magnitude shorter than the prequential counterpart, with an advantage that grows with scale. This compression sheds light on phenomena inaccessible to prior compressors. Holding loss fixed, larger models and ensembles compress to much smaller sizes despite more parameters. Plugged into a PAC-Bayes bound, the requential code yields state-of-the-art generalization guarantees for billion-parameter LLMs, outperforming bounds built on aggressive post-training quantization even granted zero error. The bound tightens with scale in the compute-optimal regime, as models become increasingly compressible relative to dataset size. The same code predicts that models gradually overfit when trained for multiple epochs. It also isolates the learnable information in a dataset from its unpredictable, random content, revealing that lower-entropy text holds far more learnable structure than higher-entropy image data.