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. 01582v1 Announce Type: cross Abstract: Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling.
The paper presents a data-driven method for discovering intrinsic Lie point symmetries of differential equation systems directly from sampled jet-space data. By formulating symmetry discovery as an operator-theoretic problem, the approach identifies admissible generators, separates them from redundant or trivial directions, and reconstructs the full symmetry algebra, including its Lie-bracket structure, on benchmark systems. The method demonstrates that explicit continuous-symmetry generators can be recovered from data generated by both exact and approximate differential systems.
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.
The paper presents an energy‑based reduced‑order model for micromagnetic magnetization dynamics that couples a convolutional autoencoder with a latent neural ODE. The latent dynamics are driven by a learned scalar potential via an antisymmetric operator and a symmetric dissipative operator, ensuring monotonic energy decrease while allowing motion along level sets. The model is trained solely on short trajectory windows without explicit physical labels, and shows that antisymmetric‑dissipative and deep‑quadratic energy formulations yield superior long‑term trajectory predictions compared to purely dissipative models.
arXiv:2606. 05957v1 Announce Type: new Abstract: Singular learning theory and information geometry have studied the same parameter spaces in mostly separate vocabularies: the former computes Bayesian invariants in resolved coordinates, the latter works in original coordinates under a non-degeneracy assumption that overparameterised models routinely violate.
arXiv:2607. 28939v1 Announce Type: new Abstract: Structure-preserving neural networks are essential for the long-term prediction of Hamiltonian systems from data.
arXiv:2607. 14937v1 Announce Type: cross Abstract: Recent foundation models (FMs) for zero-shot reconstruction of dynamical systems (DS) achieve strong out-of-domain generalization but provide little insight into the mechanisms that underlie their forecasts.
arXiv:2606. 03003v1 Announce Type: cross Abstract: A latent world model built from an equivariant encoder $E$ and an equivariant predictor $f$ inherits a provable symmetry of its training loss: when the world's dynamics genuinely carries a group $G$ acting on latents by an orthogonal representation $\rho(g)$, the one-step prediction relMSE is exactly invariant across the whole group, so fitting the dynamics on a restricted slice of orientations mathematically determines it on the entire orbit (j\v{u} y\=i f\v{a}n s\=an).
arXiv:2609.24737v1 Announce Type: new Abstract: We introduce Ananke, a representation-learning framework that scaffolds latent representations onto a structured product-torus prior, and its flagship...
arXiv:2607. 23501v1 Announce Type: new Abstract: Locating periodic solutions of chaotic dynamical systems normally requires an initial guess close enough to the target orbit for numerical continuation or gradient-based search to converge.
arXiv:2606. 27029v1 Announce Type: new Abstract: Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency.
We introduce Ananke, a representation-learning framework that scaffolds latent representations onto a structured product-torus prior, and its flagship visual backbone realization, Contractive Torus At...