The paper introduces a finite‑width geometric framework that explains how learned feature geometries are organized, transported, and selectively aligned in deep neural networks. It quantifies incompatibilities among weight‑generated covariance, gates, and backward sensitivities using three families of commutators, and provides exact layerwise identities that decompose these commutators into sources such as downstream transport, adjacent‑layer imbalance, and nonlinear gate‑covariance interactions. The study demonstrates that spectral alignment is a layer‑ and scale‑dependent compatibility phenomenon governed by transport, interaction, cancellation, and possible damping, rather than a universal consequence of training.
We develop a finite-width geometric framework describing how learned feature geometries are organized, transported, and selectively aligned in deep neural networks. Incompatibility among weight-genera...
arXiv:2603. 13751v2 Announce Type: replace Abstract: Physics-informed neural networks (PINNs) have achieved notable success in modeling dynamical systems governed by partial differential equations (PDEs).
arXiv:2609.08381v1 Announce Type: cross Abstract: Equivariant networks are commonly trained with Adam, yet recent work reports that matrix-structured optimizers such as Muon can perform better on the...
arXiv:2605. 18106v3 Announce Type: replace-cross Abstract: A striking geometric disparity has long persisted in the practice of deep learning.
arXiv:2605. 28983v2 Announce Type: replace-cross Abstract: In this paper, training a neural network is identified, exactly, as a search through Hamilton--Jacobi initial-value problems: each gradient step selects the initial data of a viscous Hamilton--Jacobi equation whose Hopf--Cole propagator best fits the observations; at inference, the input is the spatial point at which that solution is evaluated and the initial condition is already encoded in the weights.
The paper investigates how exact automatic differentiation behaves under hard‑ReLU gradient descent. It shows that while gradient‑descent states converge to the piecewise‑smooth gradient flow, the derivative of the training map does not, due to missing event‑time sensitivities captured by saltation matrices. The study demonstrates that for convex objectives, activation events can create large sensitivity gaps, and provides empirical evidence that event‑aware corrections are necessary for accurate flow derivatives.
arXiv:2607. 14018v1 Announce Type: cross Abstract: We investigate how each component of the Transformer feedforward block architecture design determines how much rank survives across depth at initialization.
arXiv:2607. 17696v1 Announce Type: cross Abstract: We develop an adjoint-sensitivity framework for positional influence in causal residual Transformers and separate unconditional analytic results from conditional boundary-shape conclusions.
We investigate how each component of the Transformer feedforward block architecture design determines how much rank survives across depth at initialization. We reinterpret skip connections and normalization, long understood as controlling magnitude, as mechanisms for preserving gradient rank across depth, since the very matrix multiplications and nonlinear activations that make the network expressive also reduce the rank.
arXiv:2606. 15892v1 Announce Type: new Abstract: Accurate interatomic potentials enable molecular dynamics of materials, molecules, and interfaces beyond density-functional-theory length and time scales.
arXiv:2607. 04993v1 Announce Type: cross Abstract: Many phenomena of deep learning are dynamical: they concern not only which minima exist, but how gradient descent reaches, avoids, or selects among them.
arXiv:2608.19021v2 Announce Type: replace Abstract: Global Covariance Pooling (GCP) improves deep networks by capturing second-order feature statistics, and is especially effective for fine-grained r...