The paper introduces SKG (Symplectic‑Kernel Gradient), a reasoning engine that integrates symplectic geometry, GENERIC, kernel methods, RKHS, group‑equivariant networks, and constrained mechanics into a unified framework for law‑induced inference and induction. It defines an octuple S = (M, g, ω, H, S, G, Δ, Π) capturing conservation laws, constitutive relations, symmetry groups, and axiom systems, and presents several theoretical results on metric reconstruction, kernel bandwidth determination, error decomposition, and function‑space reconstruction. The authors validate SKG on a 1‑D damped oscillator and a 2‑D rotation‑invariant system, reporting significant statistical performance (p < 0.01) across six unit tests and ten experiments.
arXiv:2607. 25624v1 Announce Type: new Abstract: Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top.
arXiv:2609. 18901v1 Announce Type: cross Abstract: In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+\xi_j$ or by a known physical constraint $Du^*=v$.
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.
arXiv:2608. 10172v1 Announce Type: new Abstract: Mechanistic interpretability explains models by identifying circuits inside them, but has no way to tell whether a circuit is a property of the model or an artifact of the method that found it.
arXiv:2606. 06171v1 Announce Type: cross Abstract: Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions.
arXiv:2609. 03129v1 Announce Type: cross Abstract: Several classical machine-learning methods, such as KRRs and SVRs, are both computationally and analytically tractable since their estimators either admit closed-form expressions or are obtained by minimizing convex training objectives; neither feature is generally available for deep neural networks.
arXiv:2104. 11547v5 Announce Type: replace-cross Abstract: Statistical models contain variables that are not random: parameters, treatments, environments, design points.
arXiv:2609. 21693v1 Announce Type: new Abstract: Equivalent finite parameterizations can represent the same functions and intrinsic norm yet induce different optimization algorithms.
arXiv:2609.06546v1 Announce Type: cross Abstract: Although machine learning can be used to predict the evolution of physical systems from data, a formulation that learns only the system state at each...
arXiv:2607. 07468v1 Announce Type: cross Abstract: We study the recovery of sparse functions from finite, noisy, and indirect observations in the framework of statistical inverse learning.
arXiv:2606. 08343v1 Announce Type: new Abstract: We introduce GENERIC-FNO, the first neural operator to embed the full GENERIC (metriplectic) structure of nonequilibrium thermodynamics -- reversible, energy-conserving dynamics and irreversible, entropy-producing dynamics coupled through the degeneracy conditions -- directly in function space.
The paper investigates observability in neural state‑space models, particularly the Mamba architecture, using tools from ordinary differential equations and control theory. It introduces several strategies—based on eigenvalues, roots of unity, permutations, Fourier transforms, and Vandermonde matrices—to enforce observability in high‑dimensional, learnable hidden states while maintaining computational efficiency. The authors also present a shared‑parameter construction for Mamba and a training algorithm that satisfies a Robbins‑Monro condition, contrasting it with classical procedures that fail to meet contraction requirements.