arXiv:2608. 15976v1 Announce Type: new Abstract: A learning system can occupy execution states that are indistinguishable under every declared present-behavior readout yet respond differently to future training.
A learning system can occupy execution states that are indistinguishable under every declared present-behavior readout yet respond differently to future training. We formalize this through fiber fingerprints: controlled future-learning response laws restricted to present-behavior equivalence classes.
The paper presents a comprehensive framework for comparing parameter‑efficient fine‑tuning (PEFT) methods by treating each as a smooth map from a small parameter space into weight space, with a pretrained checkpoint as the base point. It identifies three key invariants—image near the base point, fibre dynamics, and base change—that explain expressivity, training dynamics, and initialization behavior. The authors prove numerous results about common PEFT techniques (e.g., LoRA, Householder adaptation) and classify 382 methods from 138 papers using seven coordinate descriptors.
arXiv:2607. 20887v1 Announce Type: cross Abstract: Model merging combines independently trained or fine-tuned models, but pairwise alignability does not imply globally consistent alignment.
The paper investigates the folded transonic flutter boundary of Isogai’s case A aerofoil, whose elastic axis is positioned half a chord ahead of the leading edge. Using coupled unsteady Euler simulations, a Gaussian‑process surrogate trained via batch active learning, and split conformal prediction, the authors map how the boundary changes as the elastic axis is moved, revealing that the fold exists only when the axis is more than ~0.4 chord ahead and closes between 0.39 and 0.33 chord ahead. Energy budget analyses explain the lower and upper branches of the fold, showing how shock phase and damping of the supersonic surface govern the flutter behavior.
HySPE introduces a hyperbolic symplectic positional encoding that uses dual shears to create a conformally symplectic contraction with two spectral decay rates per channel pair. It eliminates exponential drift by diagonalizing the operator and applying blockwise coordinate rebasing, ensuring length‑independent numerical bounds while matching the forward latency of RoPE. Experiments on TinyShakespeare and a 51M‑parameter Transformer show HySPE maintains stable perplexity at long sequence lengths, outperforming RoPE in extrapolation scenarios.
arXiv:2607. 24741v2 Announce Type: replace-cross Abstract: Positive two-marginal entropic optimal transport is solved by a nonlinear, positive, order-preserving, homogeneous Sinkhorn map.
The paper introduces a stability theory for Joint-Embedding Predictive Architectures (JEPAs), showing that training dynamics involve a driving force and a decay effect that determine representation collapse. By linearising the gradient flow, the authors derive a per‑mode stability ratio that separates data‑side and predictor‑side contributions, predicting a phase boundary confirmed across 800 configurations. Using this insight, they propose ResidualPred, a transformer predictor that biases attention toward the identity at initialization, improving representation rank and downstream accuracy on tabular and image benchmarks.
The paper investigates post‑hoc symmetry enforcement on pre‑trained machine‑learning models used for physics simulations. For finite, compact groups, simple group averaging (the Reynolds operator) projects predictors onto invariant or equivariant functions with non‑increasing risk guarantees, and numerical experiments confirm its effectiveness. The authors extend the framework to non‑compact groups, proposing three alternatives—Weyl unitary trick, data‑dependent orthogonal projection, and frame averaging—each evaluated theoretically and empirically on the unit‑rescaling and Lorentz groups; results show that only the orthogonal projection consistently improves performance, while the other methods help mainly in out‑of‑distribution or low‑data scenarios, and that training directly on invariant features can match or surpass post‑hoc projections.
The paper introduces an in‑span adaptation technique for reduced‑order models, where the reduced subspace is continually updated using the model’s own predictions via an incremental singular‑value decomposition with a forgetting factor. This creates a trajectory‑informed spectral preconditioner that reweights and realigns the basis without changing the subspace, enabling the model to better absorb future out‑of‑span corrections. The authors demonstrate the method on a 3‑D spiral example and nonlinear PDEs such as viscous Burgers and Fisher–KPP, and relate the approach to in‑context learning in dynamical systems.
arXiv:2606. 30789v1 Announce Type: new Abstract: Group Relative Policy Optimization (GRPO) has become a standard tool for improving the reasoning ability of large language models, yet its training dynamics are still described empirically: reward trajectories are fit with low-parameter functional forms whose constants carry no mechanistic meaning, and hyperparameter choices remain a matter of trial and error.
The paper introduces LatentLieODE, a neural ODE framework that learns linear Lie symmetries and first integrals from time‑series data such as videos. It uses an autoencoder to obtain a low‑dimensional latent space, fits linear infinitesimal generators to a Lie‑bracket residual, and trains a latent ODE to reconstruct observations. The method successfully identifies symmetries and conserved quantities in six benchmark systems, achieving high correlation with ground‑truth invariants and recovering a non‑abelian σ(3) algebra on the sphere.