symmetry-based degeneracy detection

This paper addresses the unified modeling of symmetries in machine learning. It proposes a framework grounded in differential geometry and convex optimization to (1) enforce known symmetries, (2) automatically discover unknown symmetries in models or data, and (3) actively induce symmetry breaking via user-specified candidate groups. The core contribution is the first formulation of symmetry imposition and discovery as dual linear-algebraic tasks, leveraging the Lie derivative to characterize fiberwise linear Lie group actions on vector bundles, and employing nuclear-norm relaxation to construct convex regularization terms. The method is broadly applicable to neural networks, dynamical system discovery, basis-function regression, and neural operators. Empirically, it significantly improves generalization performance and parameter efficiency—particularly in low-data regimes—while preserving geometric structure and interpretability.

This work addresses the challenge of balancing physical consistency and search efficiency in data-driven symbolic discovery of differential equations. We propose a physics-constrained embedding method grounded in differential invariants derived from Lie group symmetries. Our key innovation is the first use of such differential invariants—as opposed to conventional monomial bases—as fundamental building blocks for symbolic regression, ensuring that discovered equations inherently satisfy conservation laws and physical symmetries. Integrating sparse regression, genetic programming, and Lie group theory, we construct an efficient and interpretable equation discovery framework. Evaluated on fluid dynamics and reaction–diffusion systems, our approach successfully recovers concise, accurate, and physically self-consistent governing equations. It significantly improves discovery efficiency, generalizability across unseen conditions, and consistency with known physical principles.

This work addresses the challenge of efficiently identifying dynamical system parameters from a single trajectory when the underlying symmetry group is unknown. To this end, the authors propose an adaptive symmetry learning framework that, for the first time, automatically discovers the unknown symmetry group directly from a single trajectory. By integrating equivariant modeling of group actions with group representation theory, and leveraging the expansion properties of Cayley graphs for theoretical analysis, the method significantly reduces the required trajectory length. It achieves optimal sample efficiency—matching that of methods assuming known symmetries—even in the more challenging setting where symmetries are a priori unknown, thereby demonstrating both theoretical superiority and practical effectiveness.

This work addresses the problem of automatically discovering symmetries and corresponding conservation laws (e.g., energy, linear momentum, angular momentum) of physical systems directly from noisy, discrete trajectory data—without requiring velocity/momentum measurements or prior knowledge of the governing model structure. Methodologically, it introduces the first end-to-end framework for jointly learning discrete Lagrangian functions and their symmetry groups, integrating discrete variational integration, Lie group theory, and neural network parameterization, alongside a symmetry-driven loss function. It further proposes variational backward error analysis to rigorously unify discrete modeling with continuous physical constraints. Experiments demonstrate that the method significantly improves long-term trajectory prediction accuracy under noise, strictly preserves conservation quantities, and achieves superior qualitative and quantitative performance compared to state-of-the-art baselines.

Data-driven dimensionality reduction models for complex flows—such as the chaotic bursting regime in two-dimensional Kolmogorov turbulence—often neglect physical symmetries, leading to non-robust manifold estimation and inaccurate dynamical modeling. Method: We propose “symmetry-aware manifold learning”: a novel paradigm integrating Fourier-based symmetry identification, an implicit rank-minimizing autoencoder (with weight decay) for automatic low-dimensional invariant manifold dimension estimation, and equivariant neural ordinary differential equations to learn symmetry-preserving dynamics. Contribution/Results: This end-to-end framework rigorously enforces continuous and discrete group equivariance inherited from the Navier–Stokes equations. It significantly reduces data requirements while enhancing robustness in manifold dimension estimation, improving short-term trajectory prediction accuracy, and faithfully reproducing long-term statistical properties of the flow.

This work investigates whether intrinsic symmetries in training data induce conserved quantities during gradient flow training of neural networks. By integrating tools from differential geometry and dynamical systems theory, the study establishes—for the first time—a systematic connection between data symmetries and conservation laws in training dynamics, employing tensorized networks (including linear, polynomial, and Lightning Attention architectures) as an analytical framework. The analysis demonstrates that, under general non-polynomial losses, data symmetries do not yield additional conserved quantities; however, when combined with data augmentation under mean squared error (MSE) loss, novel conserved quantities emerge. This finding uncovers a distinctive conservation mechanism specific to MSE loss and offers a new perspective for understanding the dynamics of neural network training.

This work addresses the challenge that Lie point symmetries of stochastic differential equations (SDEs) are typically unknown and lack automated discovery methods from data. The authors propose LieStoNet, a novel framework that, for the first time, enables end-to-end learning of SDE Lie point symmetries directly from spatiotemporal trajectories without requiring predefined symmetry groups or templates. Built upon the SDE symmetry theory of Gaeta and Quintero, LieStoNet models drift and diffusion terms via neural networks and jointly uncovers symmetries of the associated Fokker–Planck equation. To guarantee that the learned infinitesimal generators form a valid Lie algebra, the method incorporates deterministic equation constraints, Lie bracket closure, and algebraic regularization. Evaluated on several canonical SDEs with known analytical symmetries, LieStoNet accurately recovers the symmetry generators, demonstrating its capability for interpretable symmetry discovery in noisy dynamical systems.

This work addresses the challenge of enforcing physical symmetries—such as rotational equivariance—in machine learning models without imposing explicit architectural constraints. The authors propose a general, architecture-agnostic approach that introduces a novel metric to quantify the degree of symmetry learning, employs spectral analysis to diagnose failure modes, and leverages targeted data augmentation to guide unconstrained Transformers—including graph neural networks and PointNet-style architectures—to progressively approximate equivariance across layers during training. Experiments demonstrate that injecting only the minimal necessary inductive bias substantially enhances physical fidelity, numerical stability, and predictive accuracy, while preserving the model’s expressive capacity.

This study addresses the construction of functions in algebraic combinatorics subject to stringent distributional constraints and the discovery of previously unknown combinatorial symmetries. To this end, we propose the SLURP framework, which integrates MapSeek-Functional and MapSeek-Symbolic approaches through alternating pseudo-label supervised learning, symbolic regression, and formal verification in Lean 4. The framework yields the first combinatorial interpretation of $q,t$-Narayana polynomials based on non-crossing partitions and provides a combinatorial proof of symmetry in previously unresolved cases by leveraging newly discovered statistics. All code and formalized results are publicly released to ensure reproducibility and rigorous verification.